KAM for quasi-linear KdV - P Baldi; Massimiliano Berti; Riccardo Montalto (C. R. Math. Acad. Sci. Paris, 2014)
We prove the existence and stability of Cantor families of quasi-periodic, small-amplitude solutions of quasi-linear autonomous Hamiltonian perturbations of KdV.

KAM for Reversible Derivative Wave Equations - Massimiliano Berti; Luca Biasco; Michela Procesi (Arch. Ration. Mech. Anal., 2014)
We prove the existence of Cantor families of small amplitude, analytic, linearly stable quasi-periodic solutions of reversible derivative wave equations.

Laplace equation in a domain with a rectilinear crack: higher order derivatives of the energy with respect to the crack length - Gianni Dal Maso; Gianluca Orlando; Rodica Toader (SISSA, 2014)
We consider the weak solution of the Laplace equation in a planar domain with a straight crack, prescribing a homogeneous Neumann condition on the crack and a nonhomogeneous Dirichlet condition on the rest of the boundary. For every k we express the k-th derivative of the energy with respect to the crack length in terms of a finite number of coefficients of the asymptotic expansion of the solution near the crack tip and of a finite number of other parameters, which only depend on the shape of the domain.
Keywords: cracked domains, energy release rate, higher order derivatives, asymptotic expansion of solutions

Lecture notes on gradient flows and optimal transport - Sara Daneri; Giuseppe Savarè (Cambridge University Press, 2014)
We present a short overview on the strongest variational formulation for gradient flows of geodesically λ-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009.

Legendre duality on hypersurfaces in Kähler manifolds - Vittorio Martino (Walter de Gruyter and Co., 2014)
We give a sufficient condition on real strictly Levi-convex hypersurfaces M, embedded in four-dimensional Kähler manifolds V , such that Legendre duality can be performed. We consider the contact form onM whose kernel is the restriction of the holomorphic tangent space of V and show that if there exists a Legendrian Killing vector field v, then the dual form β(̇) := d(v, ̇) is a contact form on M with the same orientation than theta.

Linearized plastic plate models as Γ-limits of 3D finite elastoplasticity - Elisa Davoli (ESAIM: Control, Optimisation and Calculus of Variations, 2014)
The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of $Γ$-convergence, in the framework of finite plasticity. Denoting by $ε$ the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order $ε^{2 α -2}$, with $α ≥ 3$. According to the value of $α$, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized plate theory.

Lipschitz continuous viscosity solutions for a class of fully nonlinear equations on lie groups - Vittorio Martino; Annamaria Montanari (Springer, 2014)
In this paper, we prove existence and uniqueness of Lipschitz continuous viscosity solutions for Dirichlet problems involving a class a fully non-linear operators on Lie groups. In particular, we consider the elementary symmetric functions of the eigenvalues of the Hessian built with left-invariant vector fields.

Local and global minimality results for a nonlocal isoperimetric problem on R^N - Marco Bonacini; Riccardo Cristoferi (SIAM Journal on Mathematical Analysis, 2014)
We consider a nonlocal isoperimetric problem defined in the whole space R^N, whose nonlocal part is given by a Riesz potential with exponent $α\in(0, N-1)$. We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L^1-norm. This criterion provides the existence of a (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and allows to address several global minimality issues.
Keywords: Nonlocal isoperimetric problem

Local behavior of fractional p-minimizers - Agnese Di Castro; Tuomo Kuusi; Giampiero Palatucci (SISSA, 2014)
We extend the De Giorgi-Nash Moser theory to nonlocal, possibly degerate integro-differential operators
Keywords: fractional Sobolev spaces

On the Lp-differentiability of certain classes of functions - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (European Mathematical Society, 2014)
We prove the Lp-differentiability at almost every point for convolution products on ℝd of the form K*μ, where μ is bounded measure and K is a homogeneous kernel of degree 1-d. From this result we derive the Lp-differentiability for vector fields on R d whose curl and divergence are measures, and also for vector fields with bounded deformation.

Maximal generalized solution of eikonal equation - Sandro Zagatti (Elsevier, 2014)
We study the Dirichlet problem for the eikonal equation: 1/2 |∇u(x)|^2-a(x)=0 in Ω u(x)=(x) on Ω, without continuity assumptions on the map a(.). We find a class of maps a(.) contained in the space L∞(Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.

A model for crack growth with branching and kinking - Simone Racca (Asymptotic Analysis, 2014)
We study an evolution model for fractured elastic materials in the 2-dimensional case, for which the crack path is not assumed to be known a priori. We introduce some general assumptions on the structure of the fracture sets suitable to remove the restrictions on the regularity of the crack sets and to allow for kinking and branching to develop. In addition we define the front of the fracture and its velocity. By means of a time-discretization approach, we prove the existence of a continuous-time evolution that satisfies an energy inequality and a stability criterion. The energy balance also takes into account the energy dissipated at the front of the fracture. The stability criterion is stated in the framework of Griffith's theory, in terms of the energy release rate, when the crack grows at least at one point of its front.
Keywords: quasistatic crack evolution, branching, kinking, Griffith\\\'s criterion

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (Springer, 2014)
This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

A Moser-Trudinger inequality for the singular Toda system - Luca Battaglia; Andrea Malchiodi (Bull. Inst. Math. Acad. Sin., 2014)

New results on Gamma-limits of integral functionals - Nadia Ansini; Gianni Dal Maso; Caterina Ida Zeppieri (Elsevier, 2014)
Keywords: Gamma-convergence

Nonsingular Isogeometric Boundary Element Method for Stokes Flows in 3D - Luca Heltai; Marino Arroyo; Antonio DeSimone (Elsevier, 2014)
Isogeometric analysis (IGA) is emerging as a technology bridging Computer Aided Geometric Design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, and engineering analysis. In finite element and boundary element isogeometric methods (FE-IGA and IGA-BEM), the NURBS basis functions that de- scribe the geometry define also the approximation spaces. In the FE-IGA approach, the surfaces generated by the CAGD tools need to be extended to volumetric descriptions, a major open problem in 3D. This additional passage can be avoided in principle when the partial differential equations to be solved admit a formulation in terms of bound- ary integral equations, leading to Boundary Element Isogeometric Analysis (IGA-BEM). The main advantages of such an approach are given by the dimensionality reduction of the problem (from volumetric-based to surface-based), by the fact that the interface with CAGD tools is direct, and by the possibility to treat exterior problems, where the computational domain is infinite. By contrast, these methods produce system matrices which are full, and require the integration of singular kernels. In this paper we address the second point and propose a nonsingular formulation of IGA-BEM for 3D Stokes flows, whose convergence is carefully tested numerically. Standard Gaussian quadrature rules suffice to integrate the boundary integral equations, and carefully chosen known exact solutions of the interior Stokes problem are used to correct the resulting matrices, extending the work by Klaseboer et al. [27] to IGA-BEM.
Keywords: Isogeometric Analysis

On a quadratic functional for scalar conservation laws - Stefano Bianchini; Stefano Modena (Journal of Hyperbolic Differential Equations, 2014)
We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme.

Quadratic interaction functional for systems of conservation laws: a case study - Stefano Bianchini; Stefano Modena (Bulletin of the Institute of Mathematics of Academia Sinica (New Series), 2014)

Quasi-static crack growth in hydraulic fracture - Stefano Almi; Gianni Dal Maso; Rodica Toader (Nonlinear Analysis, 2014)
We present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption.

Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes - Gianni Dal Maso; Riccardo Scala (Journal of Dynamics and Differential Equations, 2014)
We introduce a model of dynamic visco-elasto-plastic evolution in the linearly elastic regime and prove an existence and uniqueness result. Then we study the limit of (a rescaled version of) the solutions when the data vary slowly. We prove that they converge, up to a subsequence, to a quasistatic evolution in perfect plasticity.

Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity - Elisa Davoli (Mathematical Models and Methods in Applied Sciences, 2014)
In this paper we deduce by $Γ$-convergence some partially and fully linearized quasistatic evolution models for thin plates, in the framework of finite plasticity. Denoting by $ε$ the thickness of the plate, we study the case where the scaling factor of the elasto-plastic energy is of order $ε^{2 α -2}$, with $α≥ 3$. These scalings of the energy lead, in the absence of plastic dissipation, to the Von Kármán and linearized Von Kármán functionals for thin plates. We show that solutions to the three-dimensional quasistatic evolution problems converge, as the thickness of the plate tends to zero, to a quasistatic evolution associated to a suitable reduced model depending on $α$.

Rate-independent damage in thermo-viscoelastic materials with inertia - Giuliano Lazzaroni; Riccarda Rossi; Marita Thomas; Rodica Toader (SISSA, 2014)
We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is Independent of temperature.

Reduced basis method for the Stokes equations in decomposable domains using greedy optimization - Laura Iapichino; Alfio Quarteroni; Gianluigi Rozza; Volkwein, Stefan (ECMI 2014 proceedings, 2014)

Reduced Order Methods for Modeling and Computational Reduction - Alfio Quarteroni; Gianluigi Rozza (MS&A, 2014)
This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics. Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects. This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.
Keywords: reduced order methods, MOR, ROM, POD, RB, greedy, CFD, Numerical Analysis

Reduction on characteristics for continuous of a scalar balance law - Giovanni Alberti; Stefano Bianchini; Laura Caravenna (AIMS Series on Applied Mathematics, vol. 8 (2014): 399 - 406, 2014)
Keywords: Method of characteristics

A robotic crawler exploiting directional frictional interactions: experiments, numerics, and derivation of a reduced model - Giovanni Noselli; Antonio DeSimone (Proceedings of the Royal Society A 470, 20140333 (2014), 2014)
We present experimental and numerical results for a model crawler which is able to extract net positional changes from reciprocal shape changes, i.e. ‘breathing-like’ deformations, thanks to directional, frictional interactions with a textured solid substrate, mediated by flexible inclined feet. We also present a simple reduced model that captures the essential features of the kinematics and energetics of the gait, and compare its predictions with the results from experiments and from numerical simulations.

SBV Regularity of Systems of Conservation Laws and Hamilton–Jacobi Equations - Stefano Bianchini (Springer, 2014)
We review the SBV regularity for solutions to hyperbolic systems of conservation laws and Hamilton-Jacobi equations. We give an overview of the techniques involved in the proof, and a collection of related problems concludes the paper.

Second Order Asymptotic Development for the Anisotropic Cahn-Hilliard Functional - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (SISSA, 2014)
The asymptotic behavior of an anisotropic Cahn-Hilliard functional with prescribed mass and Dirichlet boundary condition is studied when the parameter $ε$ that determines the width of the transition layers tends to zero. The double-well potential is assumed to be even and equal to $|s-1|^β$ near $s=1$, with $1<β<2$. The first order term in the asymptotic development by $Γ$-convergence is well-known, and is related to a suitable anisotropic perimeter of the interface. Here it is shown that, under these assumptions, the second order term is zero, which gives an estimate on the rate of convergence of the minimum values.
Keywords: Gamma-convergence, Cahn-Hilliard functional, phase transitions

Shape control of active surfaces inspired by the movement of euglenids - Marino Arroyo; Antonio DeSimone (Elsevier, 2014)
We examine a novel mechanism for active surface morphing inspired by the cell body deformations of euglenids. Actuation is accomplished through in-plane simple shear along prescribed slip lines decorating the surface. Under general non-uniform actuation, such local deformation produces Gaussian curvature, and therefore leads to shape changes. Geometrically, a deformation that realizes the prescribed local shear is an isometric embedding. We explore the possibilities and limitations of this bio-inspired shape morphing mechanism, by first characterizing isometric embeddings under axisymmetry, understanding the limits of embeddability, and studying in detail the accessibility of surfaces of zero and constant curvature. Modeling mechanically the active surface as a non-Euclidean plate (NEP), we further examine the mechanism beyond the geometric singularities arising from embeddability, where mechanics and buckling play a decisive role. We also propose a non-axisymmetric actuation strategy to accomplish large amplitude bending and twisting motions of elongated cylindrical surfaces. Besides helping understand how euglenids delicately control their shape, our results may provide the background to engineer soft machines.

Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows - F. Ballarin; Andrea Manzoni; Gianluigi Rozza; Sandro Salsa (Springer, 2014)
Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.

Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms - Sara Daneri; Aldo Pratelli (Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2014)
We show that a planar bi-Lipschitz orientation-preserving homeomorphism can be approximated in the W1,p norm, together with its inverse, with an orientation-preserving homeomorphism which is piecewise affine or smooth.

Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics - Giuliano Lazzaroni; Riccarda Rossi; Marita Thomas; Rodica Toader (SISSA, 2014)
This note deals with the analysis of a model for partial damage, where the rateindependent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubicek [1] with the methods from Lazzaroni/Rossi/Thomas/Toader [2] and extend the analysis to the setting of inhomogeneous time-dependent Dirichlet data.

Some remarks on the seismic behaviour of embedded cantilevered retaining walls - Riccardo Conti; F. Burali D'Arezzo; Giulia M.B. Viggiani (Thomas Telford, 2014)
This paper is a numerical investigation of the physical phenomena that control the dynamic behaviour of embedded cantilevered retaining walls. Recent experimental observations obtained from centrifuge tests have shown that embedded cantilevered retaining walls experience permanent displacements even before the acceleration reaches its critical value, corresponding to full mobilisation of the soil strength. The motivation for this work stems from the need to incorporate these observations in simplified design procedures. A parametric study was carried out on a pair of embedded cantilevered walls in dry sand, subjected to real earthquakes scaled at different values of the maximum acceleration. The results of these analyses indicate that, for the geotechnical design of the wall, the equivalent acceleration to be used in pseudo-static calculations can be related to the maximum displacement that the structure can sustain, and can be larger than the maximum acceleration expected at the site. For the structural design of the wall, it is suggested that the maximum bending moments of the wall can be computed using a realistic distribution of contact stress and a conservative value of the pseudo-static acceleration, taking into account two-dimensional amplification effects near the walls.

Spontaneous division and motility in active nematic droplets - Luca Giomi; Antonio DeSimone (American Physical Society, 2014)
We investigate the mechanics of an active droplet endowed with internal nematic order and surrounded by an isotropic Newtonian fluid. Using numerical simulations we demonstrate that, due to the interplay between the active stresses and the defective geometry of the nematic director, this system exhibits two of the fundamental functions of living cells: spontaneous division and motility, by means of self-generated hydrodynamic flows. These behaviors can be selectively activated by controlling a single physical parameter, namely, an active variant of the capillary number.

Stability of equilibrium configurations for elastic films in two and three dimensions - Marco Bonacini (Advances in Calculus of Variations, 2014)
We establish a local minimality sufficiency criterion, based on the strict positivity of the second variation, in the context of a variational model for the epitaxial growth of elastic films. Our result holds also in the three-dimensional case and for a general class of nonlinear elastic energies. Applications to the study of the local minimality of flat morphologies are also shown.
Keywords: Epitaxially strained elastic films

Stabilized reduced basis method for parametrized advection-diffusion PDEs - Pacciarini, P.; Gianluigi Rozza (Computer Methods in Applied Mechanics and Engineering, 2014)
In this work, we propose viable and efficient strategies for the stabilization of the reduced basis approximation of an advection dominated problem. In particular, we investigate the combination of a classic stabilization method (SUPG) with the Offline-Online structure of the RB method. We explain why the stabilization is needed in both stages and we identify, analytically and numerically, which are the drawbacks of a stabilization performed only during the construction of the reduced basis (i.e. only in the Offline stage). We carry out numerical tests to assess the performances of the ``double'' stabilization both in steady and unsteady problems, also related to heat transfer phenomena.

Stabilized reduced basis method for parametrized scalar advection-diffusion problems at higher Péclet number: Roles of the boundary layers and inner fronts - Pacciarini, P.; Gianluigi Rozza (11th World Congress on Computational Mechanics, WCCM 2014, 5th European Conference on Computational Mechanics, ECCM 2014 and 6th European Conference on Computational Fluid Dynamics, ECFD 2014, 2014)
Advection-dominated problems, which arise in many engineering situations, often require a fast and reliable approximation of the solution given some parameters as inputs. In this work we want to investigate the coupling of the reduced basis method - which guarantees rapidity and reliability - with some classical stabilization techiques to deal with the advection-dominated condition. We provide a numerical extension of the results presented in [1], focusing in particular on problems with curved boundary layers and inner fronts whose direction depends on the parameter.

Steady nearly incompressible vector elds in 2D: chain rule and renormalization - Stefano Bianchini; N.A. Gusev (SISSA, 2014)

Structure of entropy solutions to general scalar conservation laws in one space dimension - Stefano Bianchini; Lei Yu (Journal of Mathematical Analysis and Applications, 2014)

Swelling dynamics of a thin elastomeric sheet under uniaxial pre-stretch - Alessandro Lucantonio; Paola Nardinocchi; Howard A. Stone (American Institute of Physics, 2014)
It has been demonstrated experimentally that pre-stretch affects the swelling of an elastomeric membrane when it is exposed to a solvent. We study theoretically the one-dimensional swelling of a pre-stretched thin elastomeric sheet, bonded to an impermeable rigid substrate, to quantify the influence of pre-stretch. We show that the solvent uptake increases when pre-stretch increases, both at equilibrium and during the swelling transient, where it exhibits two different scaling regimes. The coupling between the solvent uptake and pre-stretch may be practically exploited to design soft actuators where the swelling-induced deformations can be controlled by varying the pre-stretch.

Swelling-induced and controlled curving in layered gel beams - Alessandro Lucantonio; Paola Nardinocchi; Matteo Pezzulla (Royal Society of London, 2014)
We describe swelling-driven curving in originally straight and non-homogeneous beams. We present and verify a structural model of swollen beams, based on a new point of view adopted to describe swelling-induced deformation processes in bilayered gel beams, that is based on the split of the swelling-induced deformation of the beam at equilibrium into two components, both depending on the elastic properties of the gel. The method allows us to: (i) determine beam stretching and curving, once assigned the characteristics of the solvent bath and of the non-homogeneous beam, and (ii) estimate the characteristics of non-homogeneous flat gel beams in such a way as to obtain, under free-swelling conditions, three-dimensional shapes. The study was pursued by means of analytical, semi-analytical and numerical tools; excellent agreement of the outcomes of the different techniques was found, thus confirming the strength of the method.

The topology of a subspace of the Legendrian curves on a closed contact 3-manifold - Ali Maalaoui; Vittorio Martino (Advanced Nonlinear Studies, 2014)
In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S 1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.

A uniqueness result for the continuity equation in two dimensions: dedicated to constantine dafermos on the occasion of his 70th birthday - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (European Mathematical Society; Springer Verlag, 2014)
We characterize the autonomous, divergence-free vector fields b on the plane such that the Cauchy problem for the continuity equation ∂tu +div(bu) = 0 admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential f associated to b. As a corollary we obtain uniqueness under the assumption that the curl of b is a measure. This result can be extended to certain nonautonomous vector fields b with bounded divergence.

A variational approach to statics and dynamics of elasto-plastic systems - Riccardo Scala (SISSA, 2014)
We prove some existence results for dynamic evolutions in elasto-plasticity and delamination. We study the limit as the data vary very slowly and prove convergence results to quasistatic evolutions. We model dislocations by mean of currents, we introduce the space of deformations in the presence of dislocations and study the graphs of these maps. We prove existence results for minimum problems. We study the properties of minimizers.
Keywords: delamination

A variational model for the quasi-static growth of fractional dimensional brittle fractures - Simone Racca; Rodica Toader (European Mathematical Society, 2014)
We propose a variational model for the irreversible quasi-static evolution of brittle fractures having fractional Hausdorff dimension in the setting of two-dimensional antiplane and plane elasticity. The evolution along such irregular crack paths can be obtained as $Γ$-limit of evolutions along one-dimensional cracks when the fracture toughness tends to zero.
Keywords: Variational models

A weighted empirical interpolation method: A priori convergence analysis and applications - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (EDP Sciences, 2014)
We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.

Where best to place a Dirichlet condition in an anisotropic membrane? - Paolo Tilli; Davide Zucco (SISSA, 2014)
We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with non constant coefficients, over a fixed domain $Ω$. Dirichlet conditions are imposed along $\partialΩ$ and, in addition, along a set $Σ$ of prescribed length ($1$-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $Σ$ in order to maximize the first eigenvalue. We characterize the limit distribution of the optimal sets, as their prescribed length tends to infinity, via $Γ$-convergence.

Ambrosio-Tortorelli approximation of cohesive fracture models in linearized elasticity - Matteo Focardi; Flaviana Iurlano (SISSA, 2013)
We provide an approximation result in the sense of $\Gamma$-convergence for cohesive fracture energies of the form \[ \int_\Omega \mathscr{Q}_1(e(u))\,dx+a\,\mathcal{H}^{n-1}(J_u)+b\,\int_{J_u}\mathscr{Q}_0^{1/2}([u]\odot\nu_u)\,d\mathcal{H}^{n-1}, \] where $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary, $\mathscr{Q}_0$ and $\mathscr{Q}_1$ are coercive quadratic forms on ${\mathbb M}^{n\times n}_{sym}$, $a,\,b$ are positive constants, and $u$ runs in the space of fields $SBD^2(\Omega)$ , i.e., it's a special field with bounded deformation such that its symmetric gradient $e(u)$ is square integrable, and its jump set $J_u$ has finite $(n-1)$-Hausdorff measure in ${\mathbb R}^n$. The approximation is performed by means of Ambrosio-Tortorelli type elliptic regularizations, the prototype example being \[ \int_\Omega\Big(v|e(u)|^2+\frac{(1-v)^2}{\varepsilon}+{\gamma\,\varepsilon}|\nabla v|^2\Big)\,dx, \] where $(u,v)\in H^1(\Omega,{\mathbb R}^n){\times} H^1(\Omega)$, $\varepsilon\leq v\leq 1$ and $\gamma>0$.
Keywords: Functions of bounded deformation

Analytical validation of a continuum model for epitaxial growth with elasticity on vicinal surfaces - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (Springer, 2013)
In this paper it is shown existence of weak solutions of a variational inequality derived from the continuum model introduced by Xiang [7, formula (3.62)] (see also the work of Xiang and E [8] and Xu and Xiang [9]) to describe the self-organization of terraces and steps driven by misfit elasticity between a film and a substrate in heteroepitaxial growth. This model is obtained as a continuum limit of discrete theories of Duport, Politi, and Villain [3] and Tersoff, Phang, Zhang, and Lagally[6].
Keywords: singular nonlinear parabolic equations, Hilbert transform, thin films

An Approximation Result for Generalised Functions of Bounded Deformation and Applications to Damage Problems - Flaviana Iurlano (SISSA, 2013)
Keywords: Functions of bounded deformation

Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length - Paolo Tilli; Davide Zucco (Society for Industrial and Applied Mathematics, 2013)
We consider the problem of maximizing the first eigenvalue of the $p$-Laplacian (possibly with nonconstant coefficients) over a fixed domain $Ω$, with Dirichlet conditions along $\partialΩ$ and along a supplementary set $Σ$, which is the unknown of the optimization problem. The set $Σ$, which plays the role of a supplementary stiffening rib for a membrane $Ω$, is a compact connected set (e.g., a curve or a connected system of curves) that can be placed anywhere in $\overline{Ω}$ and is subject to the constraint of an upper bound $L$ to its total length (one-dimensional Hausdorff measure). This upper bound prevents $Σ$ from spreading throughout $Ω$ and makes the problem well-posed. We investigate the behavior of optimal sets $Σ_L$ as $L→∞$ via $Γ$-convergence, and we explicitly construct certain asymptotically optimal configurations. We also study the behavior as $p→∞$ with $L$ fixed, finding connections with maximum-distance problems related to the principal frequency of the $∞$-Laplacian.

Attainment results for nematic elastomers - Virginia Agostiniani; Gianni Dal Maso; Antonio DeSimone (SISSA, 2013)
We consider a class of non-quasiconvex frame indifferent energy densities which includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition. Other attainment results, both for the nonlinear and the linearized model, are obtained by using the theory of convex integration introduced by Mueller and Sverak in the context of crystalline solids.

A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices - Denis Devaud; Andrea Manzoni; Gianluigi Rozza (Comptes Rendus Mathematique. Volume 351, Issue 15-16, August 2013, Pages 593-598, 2013)
We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB-ANOVA-RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB-ANOVA-RB procedure computationally more feasible.
Keywords: Partial differential equations

Common dynamical features of sensory adaptation in photoreceptors and olfactory sensory neurons. - Giovanna De Palo; Giuseppe Facchetti; Monica Mazzolini; Anna Menini; Vincent Torre; Claudio Altafini (Nature. Scientific Reports 3, Article number: 1251, Published : 13 February 2013, 2013)
Sensory systems adapt, i.e., they adjust their sensitivity to external stimuli according to the ambient level. In this paper we show that single cell electrophysiological responses of vertebrate olfactory receptors and of photoreceptors to different input protocols exhibit several common features related to adaptation, and that these features can be used to investigate the dynamical structure of the feedback regulation responsible for the adaptation. In particular, we point out that two different forms of adaptation can be observed, in response to steps and to pairs of pulses. These two forms of adaptation appear to be in a dynamical trade-off: the more adaptation to a step is close to perfect, the slower is the recovery in adaptation to pulse pairs and viceversa. Neither of the two forms is explained by the dynamical models currently used to describe adaptation, such as the integral feedback model.

Concentration of solutions for a singularly perturbed mixed problem in non-smooth domains - Serena Dipierro (Journal of Differential Equations, 2013)
We consider a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions in a bounded domain $Ω \subset \mathbb{R}^n$ whose boundary has an $(n−2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n−2}$, we prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.
Keywords: Finite-dimensional reductions; Local inversion; Singularly perturbed elliptic problems

The Conformal Willmore Functional: A Perturbative Approach - Andrea Mondino (Journal of Geometric Analysis, 2013)
The conformal Willmore functional (which is conformal invariant in general Riemannian manifolds $(M,g)$ is studied with a perturbative method: the Lyapunov–Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3,g_ε)$ – where $g_ε$ is a metric close and asymptotic to the Euclidean one. With the same technique a non-existence result is proved in general Riemannian manifolds $(M,g)$ of dimension three.

Connected Sum Construction for σk-Yamabe Metrics - Giovanni Catino; Lorenzo Mazzieri (Journal of Geometric Analysis 23, nr.2 (2013), pages 812-854, 2013)
In this paper we produce families of Riemannian metrics with positive constant $σ_k$-curvature equal to $2^{-k} {n \choose k}$ by performing the connected sum of two given compact {\em non degenerate} $n$--dimensional solutions $(M_1,g_1)$ and $(M_2,g_2)$ of the (positive) $σ_k$-Yamabe problem, provided $2 ≤ 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.

Crawlers in viscous environments: linear vs nonlinear rheology - Antonio DeSimone; Federica Guarnieri; Giovanni Noselli; Amabile Tatone (International Journal of Non-Linear Mechanics 56, 142-147 (2013), 2013)
We study model self-propelled crawlers which derive their propulsive capabilities from the tangential resistance to motion offered by the environment. Two types of relationships between tangential forces and slip velocities are considered: a linear, Newtonian one and a nonlinear one of Bingham-type. Different behaviors result from the two different rheologies. These differences and their implications in terms of motility performance are discussed. Our aim is to develop new tools and insight for future studies of cell motility by crawling.

The curvature: a variational approach - Andrei A. Agrachev; Davide Barilari; Luca Rizzi (SISSA, 2013)
The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.
Keywords: Crurvature, subriemannian metric, optimal control problem

The deal.II Library, Version 8.1 - Wolfgang Bangerth; Timo Heister; Luca Heltai; G. Kanschat; Martin Kronbichler; Matthias Maier; Bruno Turcksin; T. D. Young (SISSA, 2013)
This paper provides an overview of the new features of the finite element library deal.II version 8.0.

Defect annihilation and proliferation in active nematics - Luca Giomi; Mark J. Bowick; Xu Ma; M. Cristina Marchetti (SISSA, 2013)
Liquid crystals inevitably possess topological defect excitations generated\r\nthrough boundary conditions, applied fields or in quenches to the ordered\r\nphase. In equilibrium pairs of defects coarsen and annihilate as the uniform\r\nground state is approached. Here we show that defects in active liquid crystals\r\nexhibit profoundly different behavior, depending on the degree of activity and\r\nits contractile or extensile character. While contractile systems enhance the\r\nannihilation dynamics of passive systems, extensile systems act to drive\r\ndefects apart so that they swarm around in the manner of topologically\r\nwell-characterized self-propelled particles. We develop a simple analytical\r\nmodel for the defect dynamics which reproduces the key features of both the\r\nnumerical solutions and recent experiments on microtuble-kinesin assemblies.

Dislocation dynamics in crystals: a macroscopic theory in a fractional Laplace setting - Serena Dipierro; Giampiero Palatucci; Enrico Valdinoci (SISSA, 2013)
We consider an evolution equation arising in the Peierls-Nabarro model for crystal dislocation. We study the evolution of such dislocation function and show that, at a macroscopic scale, the dislocations have the tendency to concentrate at single points of the crystal, where the size of the slip coincides with the natural periodicity of the medium. These dislocation points evolve according to the external stress and an interior repulsive potential.
Keywords: nonlocal Allen-Cahn equation

Early phase of plasticity-related gene regulation and SRF dependent transcription in the hippocampus - Giovanni Iacono; Claudio Altafini; Vincent Torre (PloS one. Volume 8, Issue 7, July 2013 : e68078, 2013)
Hippocampal organotypic cultures are a highly reliable in vitro model for studying neuroplasticity: in this paper, we analyze the early phase of the transcriptional response induced by a 20 µM gabazine treatment (GabT), a GABA-Ar antagonist, by using Affymetrix oligonucleotide microarray, RT-PCR based time-course and chromatin-immuno-precipitation. The transcriptome profiling revealed that the pool of genes up-regulated by GabT, besides being strongly related to the regulation of growth and synaptic transmission, is also endowed with neuro-protective and pro-survival properties. By using RT-PCR, we quantified a time-course of the transient expression for 33 of the highest up-regulated genes, with an average sampling rate of 10 minutes and covering the time interval [10:90] minutes. The cluster analysis of the time-course disclosed the existence of three different dynamical patterns, one of which proved, in a statistical analysis based on results from previous works, to be significantly related with SRF-dependent regulation (p-value<0.05). The chromatin immunoprecipitation (chip) assay confirmed the rich presence of working CArG boxes in the genes belonging to the latter dynamical pattern and therefore validated the statistical analysis. Furthermore, an in silico analysis of the promoters revealed the presence of additional conserved CArG boxes upstream of the genes Nr4a1 and Rgs2. The chip assay confirmed a significant SRF signal in the Nr4a1 CArG box but not in the Rgs2 CArG box.

Epitaxially strained elastic films: the case of anisotropic surface energies - Marco Bonacini (ESAIM Control. Optim. Calc. Var. 19 (2013) 167-189, 2013)
In the context of a variational model for the epitaxial growth of strained elastic films, we study the effects of the presence of anisotropic surface energies in the determination of equilibrium configurations. We show that the threshold effect that describes the stability of flat morphologies in the isotropic case remains valid for weak anisotropies, but is no longer present in the case of highly anisotropic surface energies, where we show that the flat configuration is always a local minimizer of the total energy. The main tool used to obtain these results is a minimality criterion based on the positivity of the second variation.

Existence and symmetry results for a Schrodinger type problem involving the fractional Laplacian - Serena Dipierro; Giampiero Palatucci; Enrico Valdinoci (Le Matematiche (Catania), Vol. 68 (2013), no. 1: 201-216, 2013)
This paper deals with the following class of nonlocal Schr\"odinger equations $$ \displaystyle (-Δ)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space $H^s(\mathbb{R}^N)$. Our results are in clear accordance with those for the classical local counterpart, that is when $s=1$.

An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime - Aleks Jevnikar (Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2013)

Fields of bounded deformation for mesoscopic dislocations - Nicolas Van Goethem (SISSA, 2013)
In this paper we discuss the consequences of the distributional approach to dislocations in terms of the mathematical properties\\r\\nof the auxiliary model fields such as displacement and displacement gradient which are obtained directly from \\r\\nthe main model field here considered as the linear strain. We show that these fields cannot be introduced rigourously without \\r\\nthe introduction of gauge fields, or equivalently, without cuts in the Riemann foliation associated to the dislocated crystal.\\r\\nIn a second step we show that the space of bounded deformations follows from the distributional approach in a natural way and \\r\\ndiscuss the reasons why it is adequate to model dislocations. The case of dislocation clusters is also addressed, as it represents an important issue in industrial crystal growth while from a mathematical point of view, peculiar phenomena might appear at the set of accumulation points. \\r\\nThe elastic-plastic decomposition of the strain within this approach is also given a precise meaning.

Fracture models as Gamma-limits of damage models - Gianni Dal Maso; Flaviana Iurlano (Communications on Pure and Applied Analysis 12 (2013) 1657-1686, 2013)
We analyze the asymptotic behavior of a variational model for damaged elastic materials. This model depends on two small parameters, which govern the width of the damaged regions and the minimum elasticity constant attained in the damaged regions. When these parameters tend to zero, we find that the corresponding functionals Gamma-converge to a functional related to fracture mechanics. The corresponding problem is brittle or cohesive, depending on the asymptotic ratio of the two parameters.

Generalized Sturm-Liouville boundary conditions for first order differential systems in the plane - Alessandro Fonda; Maurizio Garrione (Topol. Methods Nonlinear Anal., 2013)
We study asymptotically positively homogeneous first order systems in the plane, with boundary conditions which are positively homogeneous, as well. Defining a generalized concept of Fučík spectrum which extends the usual one for the scalar second order equation, we prove existence and multiplicity of solutions. In this way, on one hand we extend to the plane some known results for scalar second order equations (with Dirichlet, Neumann or Sturm-Liouville boundary conditions), while, on the other hand, we investigate some other kinds of boundary value problems, where the boundary points are chosen on a polygonal line, or in a cone. Our proofs rely on the shooting method.

An improved geometric inequality via vanishing moments, with applications to singular Liouville equations - Mauro Bardelloni; Andrea Malchiodi (Communications in Mathematical Physics 322, nr.2 (2013): 415-452, 2013)

KAM theory for the Hamiltonian derivative wave equation - Massimiliano Berti; Luca Biasco; Michela Procesi (Annales Scientifiques de l'Ecole Normale Superieure, 2013)
We prove an infinite dimensional KAM theorem which implies the existence of Can- tor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. © 2013 Société Mathématique de France.

Lipschitz Classification of Almost-Riemannian Distances on Compact Oriented Surfaces - Ugo Boscain; Grégoire Charlot; Roberta Ghezzi; Mario Sigalotti (Journal of Geometric Analysis, 2013)
Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We consider the Carnot–Carathéodory distance canonically associated with an almost-Riemannian structure and study the problem of Lipschitz equivalence between two such distances on the same compact oriented surface. We analyze the generic case, allowing in particular for the presence of tangency points, i.e., points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a characterization of the Lipschitz equivalence class of an almost-Riemannian distance in terms of a labeled graph associated with it.

Macroscopic contact angle and liquid drops on rough solid surfaces via homogenization and numerical simulations - Cacace, S.; Antonin Chambolle; Antonio DeSimone; Livio Fedeli (ESAIM: Mathematical Modelling and Numerical Analysis, 2013)

Minimal partitions and image classification using a gradient-free perimeter approximation - Samuel Amstutz; Nicolas Van Goethem; Antonio André Novotny (SISSA, 2013)
In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $Γ$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.
Keywords: Image classification, deblurring, optimal partitions, perimeter approximation

Minimality and stability results for a class of free-discontinuity and nonlocal isoperimetric problems - Marco Bonacini (SISSA, 2013)
Keywords: free-discontinuity problems

The Monge Problem for Distance Cost in Geodesic Spaces - Stefano Bianchini; Fabio Cavalletti (Communications in Mathematical Physics, 2013)
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

Multiplicity result for a nonhomogeneous Yamabe type equation involving the Kohn Laplacian - Ali Maalaoui; Vittorio Martino (Journal of Mathematical Analysis and Applications. Volume 399, Issue 1, 1 March 2013, Pages 333-339, 2013)
Keywords: CR-Yamabe

A New Quadratic Potential for Scalar Conservation Laws - Stefano Bianchini; Stefano Modena (Oberwolfach Reports, 2013)

The nonlinear multidomain model: a new formal asymptotic analysis. - Stefano Amato; Giovanni Bellettini; Maurizio Paolini (Geometry Partial Differential Equations – proceedings, CRM Series (15), 2013., 2013)
We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m- equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies.
Keywords: bidomain model, anisotropic mean curvature, star-shaped combination

A note on KAM theory for quasi-linear and fully nonlinear forced KdV - P Baldi; Massimiliano Berti; Riccardo Montalto (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 3: 437–450, 2013)
We present the recent results in [3] concerning quasi-periodic solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities the solutions are linearly stable. The proofs are based on a combination of di erent ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a di erential operator with constant coe cients plus a bounded remainder. These transformations are obtained by changes of variables induced by di eomorphisms of the torus and pseudo-di erential operators. (iii) A reducibility KAM scheme, which completes the reduction to constant coe cients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues.
Keywords: KAM for PDEs

A note on non-homogeneous hyperbolic operators with low-regularity coefficients - Ferruccio Colombini; Francesco Fanelli (2013)
In this paper we obtain an energy estimate for a complete strictly hyperbolic operator with second order coefficients satisfying a log-Zygmund-continuity condition with respect to $t$, uniformly with respect to $x$, and a log-Lipschitz-continuity condition with respect to $x$, uniformly with respect to $t$.

One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls - Gianni Dal Maso; Antonio DeSimone; Marco Morandotti (SISSA, 2013)
In this paper we study a mathematical model of one-dimensional swimmers performing a planar motion while fully immersed in a viscous fluid. The swimmers are assumed to be of small size, and all inertial effects are neglected. Hydrodynamic interactions are treated in a simplified way, using the local drag approximation of resistive force theory. We prove existence and uniqueness of the solution of the equations of motion driven by shape changes of the swimmer. Moreover, we prove a controllability result showing that given any pair of initial and final states, there exists a history of shape changes such that the resulting motion takes the swimmer from the initial to the final state. We give a constructive proof, based on the composition of elementary maneuvers (straightening and its inverse, rotation, translation), each of which represents the solution of an interesting motion planning problem. Finally, we prove the existence of solutions for the optimal control problem of finding, among the histories of shape changes taking the swimmer from an initial to a final state, the one of minimal energetic cost.

Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions - Alberto Boscaggin; Fabio Zanolin (Advanced Nonlinear Studies, 2013)

Periodic bouncing solutions for nonlinear impact oscillators - Alessandro Fonda; Andrea Sfecci (Advanced Nonlinear Studies, 2013)

Planar Hamiltonian systems at resonance: the Ahmad–Lazer–Paul condition - Alberto Boscaggin; Maurizio Garrione (Nonlinear Differential Equations and Applications NoDEA, 2013)
We consider the planar Hamiltonian system\$\$Ju^{\backslashprime} = \backslashnabla F(u) + \backslashnabla_u R(t,u), \backslashquad t \backslashin [0,T], \backslash,u \backslashin \backslashmathbb{R}^2,\$\$with F(u) positive and positively 2-homogeneous and \$\${\backslashnabla_{u}R(t, u)}\$\$sublinear in u. By means of an Ahmad-Lazer-Paul type condition, we prove the existence of a T-periodic solution when the system is at resonance. The proof exploits a symplectic change of coordinates which transforms the problem into a perturbation of a linear one. The relationship with the Landesman–Lazer condition is analyzed, as well.

Quadratic cohomology - Andrei A. Agrachev (SISSA, 2013)
We study homological invariants of smooth families of real quadratic forms as\r\na step towards a \"Lagrange multipliers rule in the large\" that intends to\r\ndescribe topology of smooth vector functions in terms of scalar Lagrange\r\nfunctions.

A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence - Elisa Davoli; Maria Giovanna Mora (Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2013)
The subject of this paper is the rigorous derivation of a quasistatic evolution model for a linearly elastic–perfectly plastic thin plate. As the thickness of the plate tends to zero, we prove via Γ-convergence techniques that solutions to the three-dimensional quasistatic evolution problem of Prandtl–Reuss elastoplasticity converge to a quasistatic evolution of a suitable reduced model. In this limiting model the admissible displacements are of Kirchhoff–Love type and the stretching and bending components of the stress are coupled through a plastic flow rule. Some equivalent formulations of the limiting problem in rate form are derived, together with some two-dimensional characterizations for suitable choices of the data.
Keywords: -convergence; Perfect plasticity; Prandtl–Reuss plasticity; Quasistatic evolution; Rate-independent processes; Thin plates

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants - Gianluigi Rozza; Phuong Huynh; Andrea Manzoni (Numerische Mathematik, 2013, 2013)
In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in a ffinely parametrized geometries, focusing on the role played by the Brezzi\\\'s and Babu ska\\\'s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an a ne parametric dependence enabling to perform competitive Off ine-Online splitting in the computational\\r\\nprocedure and a rigorous a posteriori error estimation on eld variables.\\r\\nThe combination of these three factors yields substantial computational savings which are at the basis of an e fficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identi cation). In particular, in this work we focus on i) the stability of the reduced basis approximation based on the Brezzi\\\'s saddle point theory and the introduction of a supremizer operator on the pressure terms, ii) a rigorous a posteriori error estimation procedure for velocity and pressure elds based on the Babu ska\\\'s inf-sup constant (including residuals calculations), iii) the computation of a lower bound of the stability constant, and iv) di erent options for the reduced basis spaces construction. We present some illustrative results for both\\r\\ninterior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette \\r\\nflows, a channel contraction and a simple flow control problem around a curved obstacle.
Keywords: parametrized Stokes equations

Reduced Basis Approximation for the Structural-Acoustic Design based on Energy Finite Element Analysis (RB-EFEA) - Denis Devaud; Gianluigi Rozza (CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches, 2013)

A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (SISSA, 2013)

A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (SISSA, 2013)

Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics - Toni Lassila; Andrea Manzoni; Gianluigi Rozza (SISSA, 2013)

Remarks on the Moser–Trudinger inequality - Gabriele Mancini; Luca Battaglia (Advances in Nonlinear Analysis, 2013)
We extend the Moser-Trudinger inequality to any Euclidean domain satisfying Poincaré's inequality. We find out that the same equivalence does not hold in general for conformal metrics on the unit ball, showing counterexamples. We also study the existence of extremals for the Moser-Trudinger inequalities for unbounded domains, proving it for the infinite planar strip.

Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces - Ugo Boscain; Dario Prandi (2013)

Softly Constrained Films - Luca Giomi (SISSA, 2013)
The shape of materials is often subject to a number of geometric constraints\r\nthat limit the size of the system or fix the structure of its boundary. In soft\r\nand biological materials, however, these constraints are not always hard, but\r\nare due to other physical mechanisms that affect the overall force balance. A\r\ncapillary film spanning a flexible piece of wire or a cell anchored to a\r\ncompliant substrate by mean of adhesive contacts are examples of these softly\r\nconstrained systems in the macroscopic and microscopic world. In this article I\r\nreview some of the important mathematical and physical developments that\r\ncontributed to our understanding of shape formation in softly constrained films\r\nand their recent application to the mechanics of adherent cells.

Some models of crack growth in brittle materials - Simone Racca (SISSA, 2013)
Keywords: Quasi-static crack evolution

Some open problems - Andrei A. Agrachev (SISSA, 2013)
We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.
Keywords: Geometry

Some remarks on the viscous approximation of crack growth - Giuliano Lazzaroni; Rodica Toader (Discrete Contin. Dyn. Syst. Ser. S, 2013)
We describe an existence result for quasistatic evolutions of cracks in antiplane elasticity obtained in [16] by a vanishing viscosity approach, with free (but regular enough) crack path. We underline in particular the motivations for the choice of the class of admissible cracks and of the dissipation potential. Moreover, we extend the result to a model with applied forces depending on time.
Keywords: Variational models

The splitting theorem in non-smooth context - Nicola Gigli (2013)
We prove that an infinitesimally Hilbertian $CD(0,N)$ space containing a line splits as the product of $R$ and an infinitesimally Hilbertian $CD(0,N −1)$ space. By ‘infinitesimally Hilbertian’ we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.

Stabilization of Stochastic Quantum Dynamics via Open and Closed Loop Control - Claudio Altafini; Francesco Ticozzi; K. Nishio (IEEE Transactions on Automatic Control. Volume 58, Issue 1, 2013, Article number6228517, Pages 74-85, 2013)
In this paper, we investigate parametrization-free solutions of the problem of quantum pure state preparation and subspace stabilization by means of Hamiltonian control, continuous measurement, and quantum feedback, in the presence of a Markovian environment. In particular, we show that whenever suitable dissipative effects are induced either by the unmonitored environment, or by non-Hermitian measurements, there is no need for feedback, as open-loop time-invariant control is sufficient to achieve stabilization of the target set in probability. Constructive necessary and sufficient conditions on the form of the control Hamiltonian can be provided in this case. When time-invariant control is not sufficient, state stabilization can be attained by the addition of filtering-based feedback control

A stable and adaptive semi-Lagrangian potential model for unsteady and nonlinear ship-wave interactions - Andrea Mola; Luca Heltai; Antonio DeSimone (Engineering Analysis with Boundary Elements, 37(1):128 – 143, 2013., 2013)
We present an innovative numerical discretization of the equations of inviscid potential flow for the simulation of three dimensional unsteady and nonlinear water waves generated by a ship hull advancing in water. The equations of motion are written in a semi-Lagrangian framework, and the resulting integro-diff erential equations are discretized in space via an adaptive iso-parametric collocation Boundary Element Method, and in time via adaptive implicit Backward Di erentiation Formulas (BDF) with variable step and variable order. When the velocity of the advancing ship hull is non-negligible, the semi-Lagrangian formulation (also known as Arbitrary Lagrangian Eulerian formulation, or ALE) of the free surface equations contains dominant transport terms which are stabilized with a Streamwise Upwind Petrov-Galerkin (SUPG) method. The SUPG stabilization allows automatic and robust adaptation of the spatial discretization with unstructured quadrilateral grids. Preliminary results are presented where we compare our numerical model with experimental results on the case of a Wigley hull advancing in calm water with fi xed sink and trim.
Keywords: Unsteady ship-wave interaction

Stable determination of a body immersed in a fluid: the nonlinear stationary case - Andrea Ballerini (Applicable Analysis, 2013)
We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the stationary Navier–Stokes equations, from a single measurement of force and velocity on a portion of the boundary. We obtain an estimate of stability of log–log type.

The structure and regularity of admissible BV solutions to hyperbolic conservation laws in one space dimension - Lei Yu (SISSA, 2013)
This thesis is devoted to the study of the qualitative properties of admissible BV solutions to the strictly hyperbolic conservation laws in one space dimension by using wave-front tracking approximation. This thesis consists of two parts: • SBV-like regularity of vanishing viscosity BV solutions to strict hyperbolic systems of conservation laws. • Global structure of admissible BV solutions to strict hyperbolic conservation laws.

Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions - Alberto Boscaggin; Fabio Zanolin (Discrete & Continuous Dynamical Systems - A, 2013)
We study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.
Keywords: lower and upper solutions; parameter dependent equations; Periodic solutions; Poincaré-Birkhoff twist theorem; subharmonic solutions

On Sudakov's type decomposition of transference plans with norm costs - Stefano Bianchini; Sara Daneri (SISSA, 2013)

A variational Analysis of the Toda System on Compact Surfaces - Andrea Malchiodi; David Ruiz (Communications on Pure and Applied Mathematics, Volume 66, Issue 3, March 2013, Pages 332-371, 2013)
In this paper we consider the Toda system of equations on a compact surface. We will give existence results by using variational methods in a non coercive case. A key tool in our analysis is a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u_1, u_2.

On 2-step, corank 2 nilpotent sub-Riemannian metrics - Davide Barilari; Ugo Boscain; Jean-Paul Gauthier (SIAM J. Control Optim., 50 (2012) 559–582, 2012)
In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics\\r\\nthat are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2 sub-Riemannian metric.

Asymptotics of the s-perimeter as s →0 - Serena Dipierro; Alessio Figalli; Giampiero Palatucci; Enrico Valdinoci (Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790, 2012)
We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $Ω$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $Ω$. Moreover, we construct examples of sets for which the limit does not exist.

On the behaviour of flexible retaining walls under seismic actions - Riccardo Conti; G.S.P. Madabhushi; Giulia M.B. Viggiani (Geotechnique, Volume 62, Issue 12, December 2012, Pages 1081-1094, 2012)
This paper describes an experimental investigation of the behaviour of embedded retaining walls under seismic actions. Nine centrifuge tests were carried out on reduced-scale models of pairs of retaining walls in dry sand, either cantilevered or with one level of props near the top. The experimental data indicate that, for maximum accelerations that are smaller than the critical limit equilibrium value, the retaining walls experience significant permanent displacements under increasing structural loads, whereas for larger accelerations the walls rotate under constant internal forces. The critical acceleration at which the walls start to rotate increases with increasing maximum acceleration. No significant displacements are measured if the current earthquake is less severe than earthquakes previously experienced by the wall. The increase of critical acceleration is explained in terms of redistribution of earth pressures and progressive mobilisation of the passive strength in front of the wall. The experimental data for cantilevered retaining walls indicate that the permanent displacements of the wall can be reasonably predicted adopting a Newmark-type calculation with a critical acceleration that is a fraction of the limit equilibrium value.
Keywords: Centrifuge modelling

Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (Mathematical Modelling and Numerical Analysis, in press, 2012-13, 2012)
We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework.
Keywords: shape optimization

On a class of vector fields with discontinuity of divide-by-zero type and its applications - Roberta Ghezzi; Alexey O. Remizov (Journal of dynamical and control systems, 2012)
We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.

A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group. - Andrea Malchiodi; Paul Yang; Jih-Hsin Cheng; JennFang Hwang (Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198, 2012)
In this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 1

Computing optimal strokes for low reynolds number swimmers - Antonio DeSimone; Luca Heltai; François Alouges; Lefebvre-Lepot Aline (Natural locomotion in fluids and on surfaces : swimming, flying, and sliding / editors Stephen Childress, Anette Hosoi, William W. Schultz, and Z. Jane Wang, editors,, 2012)
We discuss connections between low-Reynolds-number swimming and geometric control theory, and present a general algorithm for the numerical computation of energetically optimal strokes. As an illustration of our approach, we show computed motility maps and optimal strokes for two model swimmers.
Keywords: Numerical analysis.

Concentration on circles for nonlinear Schrödinger–Poisson systems with unbounded potentials vanishing at infinity - Bonheure, Denis; Di Cosmo, Jonathan; Mercuri, Carlo (Communications in Contemporary Mathematics, 2012)
The present paper is devoted to weighted nonlinear Schrödinger–Poisson systems with potentials possibly unbounded and vanishing at infinity. Using a purely variational approach, we prove the existence of solutions concentrating on a circle.

Conservation of Geometric Structures for Non-Homogeneous Inviscid Incompressible Fluids - Francesco Fanelli (Communications in Partial Differential Equations, 2012)
In this article we get a result on propagation of geometric properties for solutions of the non-homogeneous incompressible Euler system in any dimension N ≥ 2. In particular, we investigate conservation of striated and conormal regularity, which generalize the 2-D structure of vortex patches. The results we get are only local in time, even for N = 2; however, we provide an explicit lower bound for the lifespan of the solution. In the case of physical dimension N = 2 or 3, we investigate also propagation of Hölder regularity in the interior of a bounded domain.

Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density - Maria Giovanna Mora; Lucia Scardia (Elsevier, 2012)
The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness $h$ of the plate goes to zero. More precisely, it is shown that critical points of the nonlinear elastic functional $\mathcal E^h$, whose energies (per unit thickness) are bounded by $Ch^4$, converge to critical points of the $Γ$-limit of $h^{-4}\mathcal E^h$. This is proved under the physical assumption that the energy density $W(F)$ blows up as $\det F→0$.

Convex pencils of real quadratic forms - Antonio Lerario (Discrete and Computational Geometry, Volume 48, Issue 4, December 2012, Pages 1025-1047, 2012)
We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RP^n (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere S^n; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)≤ 2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular X. In the nondegenerate case we also prove the bound on each specific Betti number b_k(X)≤ 2(k+2).

Crawling motility through the analysis of model locomotors: two case studies - Antonio DeSimone; Amabile Tatone (The European Physical Journal E, Volume 35, Issue 9, September 2012, Article number85, 2012)
We study model locomotors on a substrate, which derive their propulsive capabilities from the tangential (viscous or frictional) resistance offered by the substrate. Our aim is to develop new tools and insight for future studies of cellular motility by crawling and of collective bacterial motion. The purely viscous case (worm) is relevant for cellular motility by crawling of individual cells. We re-examine some recent results on snail locomotion in order to assess the role of finely regulated adhesion mechanisms in crawling motility. Our main conclusion is that such regulation, although well documented in several biological systems, is not indispensable to accomplish locomotion driven by internal deformations, provided that the crawler may execute sufficiently large body deformations. Thus, there is no snail theorem. Namely, the crawling analog of the scallop theorem of low Reynolds number hydrodynamics does not hold for snail-like crawlers. The frictional case is obtained by assuming that the viscous coefficient governing tangential resistance forces, which act parallel and in the direction opposite to the velocity of the point to which they are applied, depends on the normal force acting at that point. We combine these surface interactions with inertial effects in order to investigate the mechanisms governing the motility of a bristle-robot. This model locomotor is easily manufactured and has been proposed as an effective tool to replicate and study collective bacterial motility.

Detection of transcriptional triggers in the dynamics of microbial growth: application to the respiratory-versatile bacterium Shewanella oneidensis - Q Beg; Mattia Zampieri; N Klitgord; S Collins; M Serres; Daniel Segrè; Claudio Altafini (Nucleic Acids Research, Volume 40, Issue 15, August 2012, Pages 7132-7149, 2012)
The capacity of microorganisms to respond to variable external conditions requires a coordination of environment-sensing mechanisms and decisionmaking regulatory circuits. Here, we seek to understand the interplay between these two processes by combining high-throughput measurement of time-dependent mRNA profiles with a novel computational approach that searches for key genetic triggers of transcriptional changes. Our approach helped us understand the regulatory strategies of a respiratorily versatile bacterium with promising bioenergy and bioremediation applications, Shewanella oneidensis, in minimal and rich media. By comparing expression profiles across these two conditions, we unveiled components of the transcriptional program that depend mainly on the growth phase. Conversely, by integrating our time-dependent data with a previously available large compendium of static perturbation responses, we identified transcriptional changes that cannot be explained solely by internal network dynamics, but are rather triggered by specific genes acting as key mediators of an environment-dependent response. These transcriptional triggers include known and novel regulators that respond to carbon, nitrogen and oxygen limitation. Our analysis suggests a sequence of physiological responses, including a coupling between nitrogen depletion and glycogen storage, partially recapitulated through dynamic flux balance analysis, and experimentally confirmed by metabolite measurements. Our approach is broadly applicable to other systems

A dynamical feedback model for adaptation in the olfactory transduction pathway - Giovanna De Palo; Anna Boccaccio; Andrew Miri; Anna Menini; Claudio Altafini (Biophysical Journal. Volume 102, Issue 12, 20 June 2012, Pages 2677-2686, 2012)
Olfactory transduction exhibits two distinct types of adaptation, which we denote multipulse and step adaptation. In terms of measured transduction current, multipulse adaptation appears as a decrease in the amplitude of the second of two consecutive responses when the olfactory neuron is stimulated with two brief pulses. Step adaptation occurs in response to a sustained steplike stimulation and is characterized by a return to a steady-state current amplitude close to the prestimulus value, after a transient peak. In this article, we formulate a dynamical model of the olfactory transduction pathway, which includes the kinetics of the CNG channels, the concentration of Ca ions flowing through them, and the Ca-complexes responsible for the regulation. Based on this model, a common dynamical explanation for the two types of adaptation is suggested. We show that both forms of adaptation can be well described using different time constants for the kinetics of Ca ions (faster) and the kinetics of the feedback mechanisms (slower). The model is validated on experimental data collected in voltage-clamp conditions using different techniques and animal species.

Exploring the low-energy landscape of large-scale signed social networks - Giuseppe Facchetti; Giovanni Iacono; Claudio Altafini (Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. Volume 86, Issue 3, 26 September 2012, Article number036116, 2012)
Analogously to a spin glass, a large-scale signed social network is characterized by the presence of disorder, expressed in this context (and in the social network literature) by the concept of structural balance. If, as we have recently shown, the signed social networks currently available have a limited amount of true disorder (or frustration), it is also interesting to investigate how this frustration is organized, by exploring the landscape of near-optimal structural balance. What we obtain in this paper is that while one of the networks analyzed shows a unique valley of minima, and a funneled landscape that gradually and smoothly worsens as we move away from the optimum, another network shows instead several distinct valleys of optimal or near-optimal structural balance, separated by energy barriers determined by internally balanced subcommunities of users, a phenomenon similar to the replica-symmetry breaking of spin glasses. Multiple, essentially isoenergetic, arrangements of these communities are possible. Passing from one valley to another requires one to destroy the internal arrangement of these balanced subcommunities and then to reform it again. It is essentially this process of breaking the internal balance of the subcommunities which gives rise to the energy barriers.

A formula for Popp\'s volume in sub-Riemannian geometry - Luca Rizzi; Davide Barilari (Analysis and Geometry in Metric Spaces, vol. 1 (2012), pages : 42-57, 2012)
For an equiregular sub-Riemannian manifold M, Popp\'s volume is a smooth\r\nvolume which is canonically associated with the sub-Riemannian structure, and\r\nit is a natural generalization of the Riemannian one. In this paper we prove a\r\ngeneral formula for Popp\'s volume, written in terms of a frame adapted to the\r\nsub-Riemannian distribution. As a first application of this result, we prove an\r\nexplicit formula for the canonical sub-Laplacian, namely the one associated\r\nwith Popp\'s volume. Finally, we discuss sub-Riemannian isometries, and we prove\r\nthat they preserve Popp\'s volume. We also show that, under some hypotheses on\r\nthe action of the isometry group of M, Popp\'s volume is essentially the unique\r\nvolume with such a property.
Keywords: subriemannian, volume, Popp, control

A Fully Coupled Immersed Finite Element Method for Fluid Structure Interaction via the Deal.II Library - Luca Heltai; Saswati Roy; Francesco Costanzo (SISSA, 2012)
We present the implementation of a solution scheme for fluid-structure\\r\\ninteraction problems via the finite element software library deal.II. The\\r\\nsolution scheme is an immersed finite element method in which two independent discretizations are used for the fluid and immersed deformable body. In this type of formulation the support of the equations of motion of the fluid is extended to cover the union of the solid and fluid domains. The equations of motion over the extended solution domain govern the flow of a fluid under the action of a body force field. This body force field informs the fluid of the presence of the immersed solid. The velocity field of the immersed solid is the restriction over the immersed domain of the velocity field in the extended equations of motion. The focus of this paper is to show how the determination of the motion of the immersed domain is carried out in practice. We show that our implementation is general, that is, it is not dependent on a specific choice of the finite element spaces over the immersed solid and the extended fluid domains. We present some preliminary results concerning the accuracy of the proposed method.
Keywords: Finite Element Method; Immersed Boundary Method; Immersed Finite Element Method

Gamma-convergence and H-convergence of linear elliptic operators - Nadia Ansini; Gianni Dal Maso; Caterina Ida Zeppieri (Journal de Mathématiques Pures et Appliquées, Available online 12 September 2012, 2012)
Keywords: Linear elliptic operators

A general method for the existence of periodic solutions of differential systems in the plane - Alessandro Fonda; Andrea Sfecci (Journal of Differential Equations, 2012)
We propose a general method to prove the existence of periodic solutions for planar systems of ordinary differential equations, which can be used in many different circumstances. Applications are given to some nonresonant cases, even for systems with superlinear growth in some direction, or with a singularity. Systems “at resonance” are also considered, provided a Landesman–Lazer type of condition is assumed.
Keywords: Nonlinear dynamics; Periodic solutions

Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs - Toni Lassila; Andrea Manzoni; Alfio Quarteroni; Gianluigi Rozza (Springer, Indam Series, Vol. 4, 2012, 2012)
The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates.
Keywords: solution manifold

On the Hausdorff volume in sub-Riemannian geometry - Andrei A. Agrachev; Davide Barilari; Ugo Boscain (Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388, 2012)
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative\r\nof the spherical Hausdorff measure with respect to a smooth volume. We prove\r\nthat this is the volume of the unit ball in the nilpotent approximation and it\r\nis always a continuous function. We then prove that up to dimension 4 it is\r\nsmooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4\r\non every smooth curve) but in general not C^5. These results answer to a\r\nquestion addressed by Montgomery about the relation between two intrinsic\r\nvolumes that can be defined in a sub-Riemannian manifold, namely the Popp and\r\nthe Hausdorff volume. If the nilpotent approximation depends on the point (that\r\nmay happen starting from dimension 5), then they are not proportional, in\r\ngeneral.

Hybridization in nanostructured DNA monolayers probed by AFM: theory versus experiment - Alessandro Bosco; Fouzia Bano; Pietro Parisse; Loredana Casalis; Antonio DeSimone; Cristian Micheletti (Nanoscale. 2012 Mar; 4(5):1734-41, 2012)
Nanografted monolayers (NAMs) of DNA show novel physico-chemical properties that make them ideally suited for advanced biosensing applications. In comparison with alternative solid-phase techniques for diagnostic DNA detection, NAMs have the advantage of combining a small size with a high homogeneity of the DNA surface coverage. These two properties favour the extreme miniaturization and ultrasensitivity in high-throughput biosensing devices. The systematic use of NAMs for quantitative DNA (and protein) detection has so far suffered from the lack of a control on key fabrication parameters, such as the ss- or ds-DNA surface coverage. Here we report on a combined experimental-computational study that allows us to estimate the surface density of the grafted DNA by analyzing the sample mechanical response, that is the DNA patch height vs. applied tip load curves. It is shown that the same analysis scheme can be used to detect the occurrence of hybridization with complementary strands in solution and estimate its efficiency. Thanks to these quantitative relationships it is possible to use a single AFM-based setup to: (i) fabricate a DNA NAM, (ii) control the DNA surface coverage, and (iii) characterize its level of hybridization helping the design of NAMs with pre-determined fabrication parameters.

Introduction to Riemannian and sub-Riemannian geometry - Andrei A. Agrachev; Davide Barilari; Ugo Boscain (SISSA, 2012)

Linear elasticity obtained from finite elasticity by Gamma-convergence under weak coerciveness conditions - Virginia Agostiniani; Gianni Dal Maso; Antonio DeSimone (Ann. Inst. H. Poincare Anal. Non Lineaire, 2012)
The energy functional of linear elasticity is obtained as G-limit of suitable rescalings of the energies of finite elasticity...
Keywords: Nonlinear elasticity

Mathematical and numerical modeling of liquid crystal elastomer phase transition and deformation - Mariarita De Luca; Antonio DeSimone (Materials Research Society Symposium Proceedings. Volume 1403, 2012, Pages 125-130, 2012)
Liquid crystal (in particular, nematic) elastomers consist of cross-linked flexible polymer chains with embedded stiff rod molecules that allow them to behave as a rubber and a liquid crystal. Nematic elastomers are characterized by a phase transition from isotropic to nematic past a temperature threshold. They behave as rubber at high temperature and show nematic behavior below the temperature threshold. Such transition is reversible. While in the nematic phase, the rod molecules are aligned along the direction of the "nematic director". This molecular rearrangement induces a stretch in the polymer chains and hence macroscopic spontaneous deformations. The coupling between nematic order parameter and deformation gives rise to interesting phenomena with a potential for new interesting applications. In the biological field, the ability to considerably change their length makes them very promising as artificial muscles actuators. Their tunable optical properties make them suitable, for example, as lenses for new imaging systems. We present a mathematical model able to describe the behavior of nematic elastomers and numerical simulations reproducing such peculiar behavior. We use a geometrically linear version of the Warner and Terentjev model [1] and consider cooling experiments and stretching experiments in the direction perpendicular to the one of the director at cross-linking.
Keywords: Artificial muscle

Modeling and control of quantum systems: An introduction - Claudio Altafini; Francesco Ticozzi (IEEE Transactions on Automatic Control. Volume 57, Issue 8, 2012, Article number6189035, Pages 1898-1917, 2012)
The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

The Monge problem in Wiener space - Fabio Cavalletti (Calculus of Variations and Partial Differential Equations, 2012)
We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.

Nonlinear thin-walled beams with a rectangular cross-section-Part I - Lorenzo Freddi; Maria Giovanna Mora; Roberto Paroni (Math. Models Methods Appl. Sci. 22, 1150016 (2012), 2012)
Our aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section, starting from three-dimensional nonlinear elasticity. The different limit models are distinguished by the different scaling of the elastic energy and of the ratio between the sides of the cross-section. In this paper we report the first part of our results.

A nonresonance condition for radial solutions of a nonlinear Neumann elliptic problem - Andrea Sfecci (Nonlinear Analysis: Theory, Methods & Applications, 2012)
We prove an existence result for radial solutions of a Neumann elliptic problem whose nonlinearity asymptotically lies between the first two eigenvalues. To this aim, we introduce an alternative nonresonance condition with respect to the second eigenvalue which, in the scalar case, generalizes the classical one, in the spirit of Fonda et al. (1991) [2]. Our approach also applies for nonlinearities which do not necessarily satisfy a subcritical growth assumption.
Keywords: Neumann problem; Nonresonance; Radial solutions; Time-map

Non-uniqueness results for critical metrics of regularized determinants in four dimensions - Matthew Gursky; Andrea Malchiodi (Communications in Mathematical Physics, Volume 315, Issue 1, September 2012, Pages 1-37, 2012)
The regularized determinant of the Paneitz operator arises in quantum gravity (see Connes 1994, IV.4.$γ$). An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in Branson (1996). A similar formula holds for Cheeger's half-torsion, which plays a role in self-dual field theory (see Juhl, 2009), and is defined in terms of regularized determinants of the Hodge laplacian on $p$-forms ($p < n/2$). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in Chang and Yang (1995), Branson, Chang, and Yang (1992), and Gursky (1997). We also study entire solutions of the Euler-Lagrange equation of $\log \det P$ and the half-torsion $τ_h$ on $\mathbb{R}^4 \setminus {0}$, and show the existence of two families of periodic solutions. One of these families includes Delaunay-type solutions.

Numerical modelling of installation effects for diaphragm walls in sand - Riccardo Conti; Luca de Sanctis; Giulia M.B. Viggiani (Acta Geotechnica, Volume 7, Issue 3, September 2012, Pages 219-237, 2012)
The scopes of this work are to study the mechanisms of load transfer and the deformations of the ground during slurry trenching and concreting in dry sand and to evaluate their effects on service structural loads, wall deflections and ground displacements behind the wall caused by subsequent excavation. A series of three-dimensional finite element analyses was carried out modelling the installation of diaphragm walls consisting of panels of different length. The soil was modelled as either linearly elastic-perfectly plastic or incrementally non-linear (hypoplastic) with elastic strain range. Plane strain analyses of diaphragm walls of identical cross section were also carried out in which wall installation was either modelled or the wall was wished in place (WIP). The analyses predict ground movements consistent with the experimental observations both in magnitude and trend. The results also show that the maximum horizontal wall deflections and structural loads reduce with increasing panel aspect ratio towards a minimum which is about twice the value computed for WIP analyses. Panel aspect ratios should be larger than about three to take advantage of the three-dimensional effects. The pattern and magnitude of surface vertical displacements obtained from linearly elastic-perfectly plastic analyses, no matter whether three- or two-dimensional, are unrealistic.
Keywords: Constitutive relations

Ogden-type energies for nematic elastomers - Virginia Agostiniani; Antonio DeSimone (International Journal of Non-Linear mechanics, 2012)
Ogden-type extensions of the free-energy densities currently used to model the mechanical behavior of nematic elastomers are proposed and analyzed. Based on a multiplicative decomposition of the deformation gradient into an elastic and a spontaneous or remanent part, they provide a suitable framework to study the stiffening response at high imposed stretches. Geometrically linear versions of the models (Taylor expansions at order two) are provided and discussed. These small strain theories provide a clear illustration of the geometric structure of the underlying energy landscape (the energy grows quadratically with the distance from a non-convex set of spontaneous strains or energy wells). The comparison between small strain and finite deformation theories may also be useful in the opposite direction, inspiring finite deformation generalizations of small strain theories currently used in the mechanics of active and phase-transforming materials. The energy well structure makes the free-energy densities non-convex. Explicit quasi-convex envelopes are provided, and applied to compute the stiffening response of a specimen tested in plane strain extension experiments (pure shear).
Keywords: Nonlinear elasticity

One-signed harmonic solutions and sign-changing subharmonic solutions to scalar second order differential equations - Alberto Boscaggin (Advanced Nonlinear Studies, 2012)

Optimal Transport with Branching Distance Costs and the Obstacle Problem - Fabio Cavalletti (SIAM Journal on Mathematical Analysis, 2012)

Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight - Alberto Boscaggin; Fabio Zanolin (Journal of Differential Equations, 2012)
We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODEu″+λa(t)g(u)=0,λ>0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and a(t) is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities g(x) when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.
Keywords: Critical points; Necessary conditions; Pairs of positive solutions; Periodic solutions

Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces - Alessandro Fonda; Andrea Sfecci (Differential Integral Equations, 2012)

Periodic solutions to superlinear planar Hamiltonian systems - Alberto Boscaggin (Portugaliae Mathematica, 2012)
We prove the existence of infinitely many periodic (harmonic and subharmonic) solutions to planar Hamiltonian systems satisfying a suitable superlinearity condition at infinity. The proof relies on the Poincare-Birkhoff fixed point theorem.

Positive periodic solutions of second order nonlinear equations with indefinite weight: Multiplicity results and complex dynamics - Alberto Boscaggin; Fabio Zanolin (Journal of Differential Equations, 2012)
We prove the existence of a pair of positive T-periodic solutions as well as the existence of positive subharmonic solutions of any order and the presence of chaotic-like dynamics for the scalar second order ODEu″+aλ,μ(t)g(u)=0, where g(x) is a positive function on R+, superlinear at zero and sublinear at infinity, and aλ,μ(t) is a T-periodic and sign indefinite weight of the form λa+(t)−μa−(t), with λ,μ>0 and large.
Keywords: Complex dynamics; Poincaré map; Positive periodic solutions; Subharmonics

Predicting and characterizing selective multiple drug treatments for metabolic diseases and cancer. - Giuseppe Facchetti; Claudio Altafini; Mattia Zampieri (BMC Systems Biology. 29 August 2012, Page 115, 2012)
Background: In the field of drug discovery, assessing the potential of multidrug therapies is a difficult task because of the combinatorial complexity (both theoretical and experimental) and because of the requirements on the selectivity of the therapy. To cope with this problem, we have developed a novel method for the systematic in silico investigation of synergistic effects of currently available drugs on genome-scale metabolic networks. The algorithm finds the optimal combination of drugs which guarantees the inhibition of an objective function, while minimizing the side effect on the overall network. Results: Two different applications are considered: finding drug synergisms for human metabolic diseases (like diabetes, obesity and hypertension) and finding antitumoral drug combinations with minimal side effect on the normal human metabolism.The results we obtain are consistent with some of the available therapeutic indications and predict some new multiple drug treatments.A cluster analysis on all possible interactions among the currently available drugs indicates a limited variety on the metabolic targets for the approved drugs. Conclusion: The in silico prediction of drug synergism can represent an important tool for the repurposing of drug in a realistic perspective which considers also the selectivty of the therapy. Moreover, for a more profitable exploitation of drug-drug interactions, also drugs which show a too low efficacy but which have a non-common mechanism of action, can be reconsider as potential ingredients of new multicompound therapeutic indications.Needless to say the clues provided by a computational study like ours need in any case to be thoroughly evaluated experimentally.

Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution - Gianni Dal Maso; Antonio DeSimone; Francesco Solombrino (Calculus of variations and partial differential equations 44 (2012) 495-541, 2012)
Cam-Clay plasticity is a well established model for the description of the mechanics of fine grained soils. As solutions can develop discontinuities in time, a weak notion of solution, in terms of a rescaled time s , has been proposed in [8] to give a meaning to this discontinuous evolution. In this paper we first prove that this rescaled evolution satisfies the flow-rule for the rate of plastic strain, in a suitable measure-theoretical sense. In the second part of the paper we consider the behavior of the evolution in terms of the original time variable t . We prove that the unrescaled solution satisfies an energy-dissipation balance and an evolution law for the internal variable, which can be expressed in terms of integrals depending only on the original time. Both these integral identities contain terms concentrated on the jump times, whose size can only be determined by looking at the rescaled formulation.

Quasistatic evolution in non-associative plasticity - the cap models - Jean-Francois Babadjian; Gilles A. Francfort; Maria Giovanna Mora (SIAM Journal on Mathematical Analysis 44, nr. 1 (2012) 245-292, 2012)
Non-associative elasto-plasticity is the working model of plasticity for soil and rocks mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollification of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled.
Keywords: Elasto-plasticity

Reduction strategies for PDE-constrained oprimization problems in Haemodynamics - Gianluigi Rozza; Andrea Manzoni; Federico Negri (European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.), Vienna, Austria, 10-14 sept. 2012, 2012)
Solving optimal control problems for many different scenarios obtained by varying a set of parameters in the state system is a computationally extensive task. In this paper we present a new reduced framework for the formulation, the analysis and the numerical solution of parametrized PDE-constrained optimization problems. This framework is based on a suitable saddle-point formulation of the optimal control problem and exploits the reduced basis method for the rapid and reliable solution of parametrized PDEs, leading to a relevant computational reduction with respect to traditional discretization techniques such as the finite element method. This allows a very efficient evaluation of state solutions and cost functionals, leading to an effective solution of repeated optimal control problems, even on domains of variable shape, for which a further (geometrical) reduction is pursued, relying on flexible shape parametrization techniques. This setting is applied to the solution of two problems arising from haemodynamics, dealing with both data reconstruction and data assimilation over domains of variable shape,\\r\\nwhich can be recast in a common PDE-constrained optimization formulation.
Keywords: inverse problems

Resonance at the first eigenvalue for first-order systems in the plane: vanishing Hamiltonians and the Landesman-Lazer condition - Maurizio Garrione (Differential Integral Equations, 2012)

Reverse engineering the euglenoid movement - Marino Arroyo; Luca Heltai; Daniel Millán; Antonio DeSimone (Proceedings of the National Academy of Sciences of the United States of America. Volume 109, Issue 44, 30 October 2012, Pages 17874-17879, 2012)
Euglenids exhibit an unconventional motility strategy amongst unicellular eukaryotes, consisting of large-amplitude highly concerted deformations of the entire body (euglenoid movement or metaboly). A plastic cell envelope called pellicle mediates these deformations. Unlike ciliary or flagellar motility, the biophysics of this mode is not well understood, including its efficiency and molecular machinery. We quantitatively examine video recordings of four euglenids executing such motions with statistical learning methods. This analysis reveals strokes of high uniformity in shape and pace. We then interpret the observations in the light of a theory for the pellicle kinematics, providing a precise understanding of the link between local actuation by pellicle shear and shape control. We systematically understand common observations, such as the helical conformations of the pellicle, and identify previously unnoticed features of metaboly. While two of our euglenids execute their stroke at constant body volume, the other two exhibit deviations of about 20% from their average volume, challenging current models of low Reynolds number locomotion. We find that the active pellicle shear deformations causing shape changes can reach 340%, and estimate the velocity of the molecular motors. Moreover, we find that metaboly accomplishes locomotion at hydrodynamic efficiencies comparable to those of ciliates and flagellates. Our results suggest new quantitative experiments, provide insight into the evolutionary history of euglenids, and suggest that the pellicle may serve as a model for engineered active surfaces with applications in microfluidics.
Keywords: microswimmers

On robust Lie-algebraic stability conditions for switched linear systems - Andrei A. Agrachev; Yurij Baryshnikov; Daniel Liberzon (Systems and Control Letters. Volume 61, Issue 2, February 2012, Pages 347-353, 2012)
This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novelty feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters.

SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension - Stefano Bianchini; Laura Caravenna (Communications in Mathematical Physics 313 (2012) 1-33, 2012)

SBV regularity for Hamilton-Jacobi equations with Hamiltonian depending on (t,x) - Stefano Bianchini; Daniela Tonon (Siam Journal on Mathematical Analysis, 2012)

SBV regularity of genuinely nonlinear hyperbolic systems of conservation laws in one space dimension - Stefano Bianchini (Acta Mathematica Scientia, Volume 32, Issue 1, January 2012, Pages 380-388, 2012)
The problem of the presence of Cantor part in the derivative of a solution to a hyperbolic system of conservation laws is considered. An overview of the techniques involved in the proof is given, and a collection of related problems concludes the paper. Key words hyperbolic systems; conservation laws; SBV; regularity
Keywords: Hyperbolic systems

SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension - Stefano Bianchini; Lei Yu (Rend. Istit. Mat. Univ. Trieste, 2012)

SBV-like regularity for Hamilton-Jacobi equations with a convex Hamiltonian - Stefano Bianchini; Daniela Tonon (Journal of Mathematical Analysis and Applications, 2012)

Second order approximations of quasistatic evolution problems in finite dimension - Virginia Agostiniani (Discrete & Continuous Dynamical Systems - A, 2012)
In this paper, we study the limit, as ε goes to zero, of a particular solution of the equation $ε^2A\ddot u^ε(t)+εB\dot u^ε(t)+\nabla_xf(t,u^ε(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^ε(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.
Keywords: discrete approximations; perturbation methods; saddle-node bifurcation; Singular perturbations; vanishing viscosity

Self-propelled micro-swimmers in a Brinkman fluid - Marco Morandotti (Journal of Biological Dynamics, 2012)
We prove an existence, uniqueness, and regularity result for the motion of a self-propelled micro-swimmer in a particulate viscous medium, modelled as a Brinkman fluid. A suitable functional setting is introduced to solve the Brinkman system for the velocity field and the pressure of the fluid by variational techniques. The equations of motion are written by imposing a self-propulsion constraint, thus allowing the viscous forces and torques to be the only ones acting on the swimmer. From an infinite-dimensional control on the shape of the swimmer, a system of six ordinary differential equations for the spatial position and the orientation of the swimmer is obtained. This is dealt with standard techniques for ordinary differential equations, once the coefficients are proved to be measurable and bounded. The main result turns out to extend an analogous result previously obtained for the Stokes system.

Simulation-based uncertainty quantification of human arterial network hemodynamics - Peng Chen; Alfio Quarteroni; Gianluigi Rozza (International Journal Numerical Methods Biomedical Engineering, 2012)
This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time.
Keywords: uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction

Some applications of the SBV Regularity Theorem for entropy solutions of 1D scalar conservation laws to ConvectionTheory and sticky particles - Daniela Tonon (Riv. Mat. Univ. Parma, 2012)
We show how it is possible to apply the SBV Regularity Theorem for entropy solutions of one-dimensional scalar conservation laws, proved by Ambrosio and De Lellis, to Convection Theory and sticky particles. In the multi-dimensional case we present a counterexample which prevent us from using the same approach.

A stable semi-lagrangian potential method for the simulation of ship interaction with unsteady and nonlinear waves - Andrea Mola; Luca Heltai; Antonio DeSimone (17th Int. Conf. Ships Shipp. Res., 2012)

Sub-Riemannian structures on 3D Lie groups - Andrei A. Agrachev; Davide Barilari (Journal of Dynamical and Control Systems. Volume 18, Issue 1, January 2012, Pages 21-44, 2012)
We give a complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. As a corollary we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^+(\mathbb{R})× S^1$, where $A^+(\mathbb{R})$ denotes the group of orientation preserving affine maps on the real line.

Systems of Quadratic Inequalities - Andrei A. Agrachev; Antonio Lerario (Proceedings of the London Mathematical Society, Volume 105, Issue 3, September 2012, Pages 622-660, 2012)
We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2 of the spectral sequence and give a simple explicit formula for the differential d_2.

Topological sensitivity analysis for high order elliptic operators - Samuel Amstutz; Antonio André Novotny; Nicolas Van Goethem (SISSA, 2012)
The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m>=1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.
Keywords: Topological derivative, Elliptic operators, Polarization tensor

Variational implementation of immersed finite element methods - Luca Heltai; Francesco Costanzo (Computer Methods in Applied Mechanics and Engineering. Volume 229-232, 1 July 2012, Pages 110-127, 2012)
Dirac-delta distributions are often crucial components of the solid-fluid coupling operators in immersed solution methods for fluid-structure interaction (FSI) problems. This is certainly so for methods like the Immersed Boundary Method (IBM) or the Immersed Finite Element Method (IFEM), where Dirac-delta distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-delta distributions, either formally or practically. This has been shown in the Finite Element Immersed Boundary Method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-delta distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be and either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non- matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization.
Keywords: Turbulent flow

A Viscosity-driven crack evolution - Simone Racca (Advances in Calculus of Variations 5 (2012) 433-483, 2012)
We present a model of crack growth in brittle materials which couples dissipative effects on the crack tip and viscous effects. We consider the 2 -dimensional antiplane case with pre-assigned crack path, and firstly prove an existence result for a rate-dependent evolution problem by means of time-discretization. The next goal is to describe the rate-independent evolution as limit of the rate-dependent ones when the dissipative and viscous effects vanish. The rate-independent evolution satisfies a Griffith’s criterion for the crack growth, but, in general, it does not fulfil a global minimality condition; its fracture set may exhibit jump discontinuities with respect to time. Under suitable regularity assumptions, the quasi-static crack growth is described by solving a finite-dimensional problem.

Weighted barycentric sets and singular Liouville equations on compact surfaces - Alessandro Carlotto; Andrea Malchiodi (Journal of Functional Analysis 262 (2012) 409-450, 2012)
Given a closed two dimensional manifold, we prove a general existence result\\r\\nfor a class of elliptic PDEs with exponential nonlinearities and negative Dirac\\r\\ndeltas on the right-hand side, extending a theory recently obtained for the\\r\\nregular case. This is done by global methods: since the associated Euler\\r\\nfunctional is in general unbounded from below, we need to define a new model\\r\\nspace, generalizing the so-called space of formal barycenters and\\r\\ncharacterizing (up to homotopy equivalence) its very low sublevels. As a\\r\\nresult, the analytic problem is reduced to a topological one concerning the\\r\\ncontractibility of this model space. To this aim, we prove a new functional\\r\\ninequality in the spirit of [16] and then we employ a min-max scheme based on a cone-style construction, jointly with the blow-up analysis given in [5] (after\\r\\n[6] and [8]). This study is motivated by abelian Chern- Simons theory in\\r\\nself-dual regime, or from the problem of prescribing the Gaussian curvature in\\r\\npresence of conical singularities (hence generalizing a problem raised by\\r\\nKazdan and Warner in [26]).

Axial symmetry of some steady state solutions to nonlinear Schrödinger equations - Changfeng Gui; Andrea Malchiodi; Haoyuan Xu; Paul Yang (Proc. Amer. Math. Soc. 139 (2011), 1023-1032, 2011)
In this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space.
Keywords: Nonlinear Schrödinger equation

Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry - Andrei A. Agrachev; Paul Lee (SISSA, 2011)
We prove a Bishop volume comparison theorem and a Laplacian comparison\r\ntheorem for three dimensional contact subriemannian manifolds with symmetry.

A class of existence results for the singular Liouville equation - Alessandro Carlotto; Andrea Malchiodi (Comptes Rendus Mathematique 349 (2011) 161-166, 2011)
We consider a class of elliptic PDEs on closed surfaces with exponential nonlinearities and Dirac deltas on the right-hand side. The study arises from abelian Chern–Simons theory in self-dual regime, or from the problem of prescribing the Gaussian curvature in presence of conical singularities. A general existence result is proved using global variational methods: the analytic problem is reduced to a topological problem concerning the contractibility of a model space, the so-called space of formal barycenters, characterizing the very low sublevels of a suitable functional.

Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential - David Ruiz; Giusi Vaira (Rev. Mat. Iberoamericana, 2011)

Compactness by maximality - Sandro Zagatti (2011)
We derive a compactness property in the Sobolev space $W^{1,1}(Ø)$ in order to study the Dirichlet problem for the eikonal equation \begin{displaymath} \begin{cases} \ha |\n u(x)|^2 - a(x) = 0 & \ in \ Ø\cr u(x)=φ(x) & \ on \ \partial Ø, \end{cases} \end{displaymath} without continuity assumptions on the map $a$.

Concentration of solutions for a singularly perturbed Neumann problem in non-smooth domains - Serena Dipierro (Annales de l'I.H.P. Analyse non linéaire, 2011)

Crack growth with non-interpenetration : a simplified proof for the pure Neumann problem - Gianni Dal Maso; Giuliano Lazzaroni (Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1219-1231, 2011)
We present a recent existence result concerning the quasi-static evolution of cracks in hyperelastic brittle materials, in the frame-work of finite elasticity with non-interpenetration. In particular, here we consider the problem where no Dirichlet conditions are imposed, the boundary is traction-free, and the body is subject only to time-dependent volume forces. This allows us to present the main ideas of the proof in a simpler way, avoiding some of the technicalities needed in the general case, studied in.

Critical points of the Moser-Trudinger functional - Francesca De Marchis; Andrea Malchiodi; Luca Martinazzi (SISSA, 2011)
Keywords: Moser-Trudinger inequality

Cytoskeletal actin networks in motile cells are critically self-organized systems synchronized by mechanical interactions - Luca Cardamone; Alessandro Laio; Rajesh Shahapure; Antonio DeSimone (PNAS 108 (2011) 13978, 2011)

A Decomposition Theorem for BV functions - Stefano Bianchini; Daniela Tonon (Communications on Pure and Applied Analysis, 2011)

Dimensional Reduction and Approximation of Measures and Weakly Differentiable Homeomorphisms - Sara Daneri (SISSA, 2011)
This thesis is devoted to the study of two different problems: the properties of the disintegration of the Lebesgue measure on the faces of a convex function and the existence of smooth approximations of bi-Lipschitz orientation-preserving homeomorphisms in the plane.

Double resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations - Alessandro Fonda; Maurizio Garrione (Journal of Differential Equations, 2011)
We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman–Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer–Fučik spectrum.
Keywords: Double resonance; Landesman–Lazer conditions; Nonlinear planar systems

Embedding theorems and existence results for nonlinear Schrödinger–Poisson systems with unbounded and vanishing potentials - Bonheure, Denis; Mercuri, Carlo (Journal of Differential Equations, 2011)
<span style="color: #2e2e2e; font-family: NexusSerif, Georgia, 'Times New Roman', Times, STIXGeneral, 'Cambria Math', 'Lucida Sans Unicode', 'Microsoft Sans Serif', 'Segoe UI Symbol', 'Arial Unicode MS', serif; font-size: 18px;">Motivated by existence results for positive solutions of non-autonomous nonlinear Schrödinger–Poisson systems with potentials possibly unbounded or vanishing at infinity, we prove embedding theorems for weighted Sobolev spaces. We both consider a general framework and spaces of radially symmetric functions when assuming radial symmetry of the potentials.

Energy release rate and stress intensity factor in antiplane elasticity - Giuliano Lazzaroni; Rodica Toader (Journal de Mathematiques Pures et Appliquees 95 (2011) 565-584, 2011)
In the setting of antiplane linearized elasticity, we show the existence of the stress intensity factor and its relation with the energy release rate when the crack path is a C1,1 curve. Finally, we show that the energy release rate is continuous with respect to the Hausdorff convergence in a class of admissible cracks.

Ennio De Giorgi and Γ-convergence - Gianni Dal Maso (Discrete and Continuous Dynamical Systems - Series A 31 (2011) 1017-1021, 2011)
Γ-convergence was introduced by Ennio De Giorgi in a series of papers published between 1975 and 1983. In the same years he developed many applications of this tool to a great variety of asymptotic problems in the calculus of variations and in the theory of partial differential equations.

An Estimate on the Flow Generated by Monotone Operators - Stefano Bianchini; Matteo Gloyer (Communications in Partial Differential Equations 36 (2011) 777-796, 2011)

An Existence and Uniqueness Result for the Motion of Self-Propelled Microswimmers - Gianni Dal Maso; Antonio DeSimone; Marco Morandotti (SIAM J. Math. Anal., 2011)
We present an analytical framework to study the motion of micro-swimmers in a viscous fluid. Our main result is that, under very mild regularity assumptions, the change of shape determines uniquely the motion of the swimmer. We assume that the Reynolds number is very small, so that the velocity field of the surrounding, infinite fluid is governed by the Stokes system and all inertial effects can be neglected. Moreover, we enforce the self propulsion constraint (no external forces and torques). Therefore, Newton\\\'s equations of motion reduce to the vanishing of the viscous drag force and torque acting on the body. By exploiting an integral representation of viscous force and torque, the equations of motion can be reduced to a system of six ordinary differential equations. Variational techniques are used to prove the boundedness and measurability of its coefficients, so that classical results on ordinary differential equations can be invoked to prove existence and uniqueness of the solution.

Existence for wave equations on domains with arbitrary growing cracks - Gianni Dal Maso; Cristopher J. Larsen (Rend. Lincei Mat. Appl. 22 (2011) 387-408, 2011)
In this paper we formulate and study scalar wave equations on domains with arbitrary growing cracks. This includes a zero Neumann condition on the crack sets, and the only assumptions on these sets are that they have bounded surface measure and are growing in the sense of set inclusion. In particular, they may be dense, so the weak formulations must fall outside of the usual weak formulations using Sobolev spaces. We study both damped and undamped equations, showing existence and, for the damped equation, uniqueness and energy conservation.
Keywords: Wave equation

Fracture and plastic models as Gamma-limits of damage models under different regimes - Flaviana Iurlano (Advances in Calculus of Variations., to appear., 2011)
We consider a variational model for damaged elastic materials. This model depends on three small parameters, which are related to the cost of the damage, to the width of the damaged regions, and to the minimum elasticity constant attained in the damaged regions. As these parameters tend to zero, our models Gamma-converge to a model for brittle fracture, for fracture with a cohesive zone, or for perfect plasticity, depending on the asymptotic ratios of the three parameters.

Gamma-convergence of energies for nematic elastomers in the small strain limit - Virginia Agostiniani; Antonio DeSimone (Continuum. Mech. Therm., 2011)
We study two variational models recently proposed in the literature to describe the mechanical behaviour of nematic elastomers either in the fully nonlinear regime or in the framework of a geometrically linear theory. We show that, in the small strain limit, the energy functional of the first one I\\\"-converges to the relaxation of the second one, a functional for which an explicit representation formula is available.
Keywords: Liquid crystals

Generalised functions of bounded deformation - Gianni Dal Maso (J. Eur. Math. Soc. (JEMS), to appear, 2011)
We introduce the space GBD of generalized functions of bounded deformation and study the structure properties of these functions: the rectifiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for GBD, which leads to a compactness result for the space GSBD of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational problems arising in fracture mechanics in the framework of linearized elasticity.
Keywords: free discontinuity problems, special functions of bounded deformation, jump set, rec- tifiability, slicing, approximate differentiability

Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds - Andrei A. Agrachev; Paul Lee (SISSA, 2011)

The geometry of Maximum Principle - Andrei A. Agrachev; Revaz Gamkrelidze (Proceedings of the Steklov Institute of mathematics. vol. 273 (2011), page: 5-27 ; ISSN: 0081-5438, 2011)
An invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.

Homology invariants of quadratic maps - Antonio Lerario (SISSA, 2011)
Given a real projective algebraic set X we could hope that the equations describing it can give some information on its topology, e.g. on the number of its connected components. Unfortunately in the general case this hope is too vague and there is no direct way to extract such information from the algebraic description of X: Even the problem to decide whether X is empty or not is far from an easy visualization and requires some complicated algebraic machinery. A fi rst step observation is that as long as we are interested only in the topology of X, we can replace, using some Veronese embedding, the original ambient space with a much bigger RPn and assume that X is cut by quadratic equations. The price for this is the increase of the number of equations de ning our set; the advantage is that quadratic polynomials are easier to handle and our hope becomes more concrete...

Infinitely many positive solutions for a Schrödinger–Poisson system - Pietro d’Avenia; Alessio Pomponio; Giusi Vaira (Nonlinear Analysis: Theory, Methods & Applications, 2011)
We are interested in the existence of infinitely many positive solutions of the Schrödinger–Poisson system −Δu+u+V(|x|)ϕu=|u|p−1u,x∈R3,−Δϕ=V(|x|)u2,x∈R3, where V(|x|) is a positive bounded function, 1<p<5 and V(r
Keywords: Non-autonomous Schrödinger–Poisson system; Perturbation method

An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations - Sandro Zagatti (Journal of Convex Analysis 18 (2011) 1141-1170, 2011)
We devote the \\\\textit{integro-extremization} method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations \\\\begin{displaymath} \\\\begin{cases} F(Du)=0 & \\\\quad \\\\textrm{in} \\\\quad\\\\O\\\\cr u(x)=\\\\varphi(x) & \\\\quad \\\\textrm{for} \\\\quad x\\\\in \\\\partial \\\\O, \\\\end{cases} \\\\end{displaymath} with a particular interest for non coercive hamiltonians $F$, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations \\\\begin{displaymath} \\\\begin{cases} \\\\frac{\\\\partial u}{\\\\partial t}+ F(\\\\nabla u)=0 & \\\\quad \\\\textrm{in} \\\\quad ]0,T[\\\\times \\\\O\\\\cr u(0,x)=\\\\eta(x) & \\\\quad \\\\textrm{for} \\\\quad x\\\\in\\\\O \\\\cr u(t,x)=\\\\psi(x) & \\\\quad \\\\textrm{for} \\\\quad (t,x)\\\\in[0,T]\\\\times \\\\partial \\\\O. \\\\end{cases} \\\\end{displaymath} We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian $F$, on the domain $\\\\O$ and on the boundary datum, without any growth assumptions on $F$.

Invariant manifolds for a singular ordinary differential equation - Stefano Bianchini; Laura Spinolo (Journal of Differential Equations 250 (2011) 1788-1827, 2011)

Invariants, volumes and heat kernels in sub-Riemannian geometry - Davide Barilari (SISSA, 2011)
Sub-Riemannian geometry can be seen as a generalization of Riemannian geometry under non-holonomic constraints. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators (see [32, 57, 70, 92] and references therein) and many problems of geometric measure theory (see for instance [18, 79]). In applications it appears in the study of many mechanical problems (robotics, cars with trailers, etc.) and recently in modern elds of research such as mathematical models of human behaviour, quantum control or motion of self-propulsed micro-organism (see for instance [15, 29, 34])\\r\\nVery recently, it appeared in the eld of cognitive neuroscience to model the\\r\\nfunctional architecture of the area V1 of the primary visual cortex, as proposed by Petitot in [87, 86], and then by Citti and Sarti in [51]. In this context, the sub-Riemannian heat equation has been used as basis to new applications in image reconstruction (see [35]).
Keywords: Sub-Riemannian geometry

Large Time Existence for Thin Vibrating Plates - Helmut Abels; Maria Giovanna Mora; Stefan Müller (Communication in Partial Differential Equations 36 (2011) 2062-2102, 2011)
We construct strong solutions for a nonlinear wave equation for a thin vibrating plate described by nonlinear elastodynamics. For sufficiently small thickness we obtain existence of strong solutions for large\\r\\ntimes under appropriate scaling of the initial values such that the limit system as h --> 0 is either the nonlinear von Karman plate equation or the linear fourth order Germain-Lagrange equation. In the case of the\\r\\nlinear Germain-Lagrange equation we even obtain a convergence rate of the three-dimensional solution to the solution of the two-dimensional linear plate equation.

The matching property of infinitesimal isometries on elliptic surfaces and elasticity on thin shells - Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad (Archive for Rational Mechanics and Analysis 200 (2011) 1023-1050, 2011)
Using the notion of Γ-convergence, we discuss the limiting behavior of the three-dimensional nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h β with 2 < β < 4. We establish that, for the given scaling regime, the limiting theory reduces to linear pure bending. Two major ingredients of the proofs are the density of smooth infinitesimal isometries in the space of W 2,2 first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.

Metastable equilibria of capillary drops on solid surfaces: a phase field approach - Livio Fedeli; Turco, Alessandro; Antonio DeSimone (Continuum Mechanics and Thermodynamics, 2011)
We discuss a phase field model for the numerical simulation of metastable equilibria of capillary drops resting on rough solid surfaces and for the description of contact angle hysteresis phenomena in wetting. The model is able to reproduce observed transitions of drops on micropillars from Cassie–Baxter to Wenzel states. When supplemented with a dissipation potential which describes energy losses due to frictional forces resisting the motion of the contact line, the model can describe metastable states such as drops in equilibrium on vertical glass plates. The reliability of the model is assessed by a detailed comparison of its predictions with experimental data on the maximal size of water drops that can stick on vertical glass plates which have undergone different surface treatments.

A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION - Giuliano Lazzaroni; Rodica Toader ({MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES}, 2011)
{In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.}
Keywords: Brittle fracture; Crack propagation; energy derivative; energy release rate; free-discontinuity problems; Griffith's criterion; local minimizers; stress intensity factor}; vanishing viscosity; {Variational models

The Monge Problem in Geodesic Spaces - Stefano Bianchini; Fabio Cavalletti (Nonlinear Conservation Laws and Applications, 2011)
We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.