Multiplicity of solutions for a mean field equation on compact surfaces - Francesca De Marchis (Boll. Unione Mat. Ital.(9), 2011)
We consider a scalar field equation on compact surfaces which has variational structure. When the surface is a torus and a physical parameter ρ belongs to $(8π, 4π^2 )$ we show under some extra assumptions that, as conjectured in [9], the functional admits at least three saddle points other than a local minimum.

New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces - Andrea Malchiodi; David Ruiz (Geometric and Functional Analysis 21 (2011) 1196-1217, 2011)
We consider a singular Liouville equation on a compact surface, arising from the study of Chern-Simons vortices in a self dual regime. Using new improved versions of the Moser-Trudinger inequalities (whose main feature is to be scaling invariant) and a variational scheme, we prove new existence results.

Nonlinear resonance: a comparison between Landesman-Lazer and Ahmad-Lazer-Paul conditions - Alessandro Fonda; Maurizio Garrione (Advanced Nonlinear Studies, 2011)
We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

Nonlinear thin-walled beams with a rectangular cross-section - Part II - Lorenzo Freddi; Maria Giovanna Mora; Roberto Paroni (SISSA, 2011)
In this paper we report the second part of our results concerning the rigorous derivation of a hierarchy of one-dimensional models for thin-walled beams with rectangular cross-section..
Keywords: Thin-walled cross-section beams

Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces - Massimiliano Berti; Michela Procesi (Duke Mathematical Journal, 2011)
We develop linear and nonlinear harmonic analysis on compact Lie groups and homogeneous spaces relevant for the theory of evolutionary Hamiltonian PDEs. A basic tool is the theory of the highest weight for irreducible representations of compact Lie groups. This theory provides an accurate description of the eigenvalues of the Laplace-Beltrami operator as well as the multiplication rules of its eigenfunctions. As an application, we prove the existence of Cantor families of small amplitude time-periodic solutions for wave and Schr¨odinger equations with differentiable nonlinearities. We apply an abstract Nash-Moser implicit function theorem to overcome the small divisors problem produced by the degenerate eigenvalues of the Laplace operator. We provide a new algebraic framework to prove the key tame estimates for the inverse linearized operators on Banach scales of Sobolev functions.

A note on a superlinear indefinite Neumann problem with multiple positive solutions - Alberto Boscaggin (Journal of Mathematical Analysis and Applications, 2011)
We prove the existence of three positive solutions for the Neumann problem associated to u″+a(t)uγ+1=0, assuming that a(t) has two positive humps and ∫0Ta−(t)dt is large enough. Actually, the result holds true for a more general class of superlinear nonlinearities.
Keywords: Indefinite weight; Nonlinear boundary value problems; positive solutions; Shooting method

Numerical Strategies for Stroke Optimization of Axisymmetric Microswimmers - François Alouges; Antonio DeSimone; Luca Heltai (Mathematical Models and Methods in Applied Sciences 21 (2011) 361-387, 2011)
We propose a computational method to solve optimal swimming problems, based on the boundary integral formulation of the hydrodynamic interaction between swimmer and surrounding fluid and direct constrained minimization of the energy consumed by the swimmer. We apply our method to axisymmetric model examples. We consider a classical model swimmer (the three-sphere swimmer of Golestanian et al.) as well as a novel axisymmetric swimmer inspired by the observation of biological micro-organisms.
Keywords: Optimal swimming

Osservazioni sui teoremi di inversione globale - Antonio Ambrosetti (Rendiconti Lincei - Matematica e Applicazioni 22 (2011) 3-15, 2011)
Some global inversion theorems with applications to semilinear elliptic equation are discussed.

A planar bi-Lipschitz extension Theorem - Sara Daneri; Aldo Pratelli (2011)

Planar loops with prescribed curvature: existence, multiplicity and uniqueness results - Roberta Musina (Proceedings of the American Mathematical Society 139 (2011) 4445-4459, 2011)
Keywords: Plane curves

A proof of Sudakov theorem with strictly convex norms - Laura Caravenna (Mathematische Zeitschrift 268 (2011) 371-407, 2011)
We establish a solution to the Monge problem in Rn, with an asymmetric, strictly convex norm cost function, when the initial measure is absolutely continuous. We focus on the strategy, based on disintegration of measures, initially proposed by Sudakov. As known, there is a gap to fill. The missing step is completed when the unit ball is strictly convex, but not necessarily differentiable nor uniformly convex. The key disintegration is achieved following a similar proof for a variational problem.

Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications - Pierluigi Cesana; Antonio DeSimone (Journal of the Mechanics and Physics of Solids 59 (2011) 787-803, 2011)
We provide some explicit formulas for the quasiconvex envelope of energy densities for nematic elastomers in the small strain regime and plane strain conditions. We then demonstrate their use as a powerful tool for the interpretation of mechanical experiments.

Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach - Filippo Cagnetti; Rodica Toader (ESAIM: COCV 17 (2011) 1-27, 2011)
A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [6] is recovered. In this case, the convergence of the discrete time approximations is improved.

Quasistatic crack growth in finite elasticity with Lipschitz data - Giuliano Lazzaroni (ANNALI DI MATEMATICA PURA ED APPLICATA, 2011)
{We extend the recent existence result of Dal Maso and Lazzaroni (Ann Inst H Poincare Anal Non Lineaire 27:257-290, 2010) for quasistatic evolutions of cracks in finite elasticity, allowing for boundary conditions and external forces with discontinuous first derivatives.}
Keywords: Brittle fracture; Crack propagation; Energy minimization; Finite elasticity; free-discontinuity problems; Griffith's criterion; Non-interpenetration}; Polyconvexity; Quasistatic evolution; Rate-independent processes; {Variational models

Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling - Gianni Dal Maso; Antonio DeSimone; Francesco Solombrino (Calculus of Variations and Partial Differential Equations 40 (2011) 125-181, 2011)
Cam-Clay nonassociative plasticity exhibits both hardening and softening behaviour, depending on the loading. For many initial data the classical formulation of the quasistatic evolution problem has no smooth solution. We propose here a notion of generalized solution, based on a viscoplastic approximation. To study the limit of the viscoplastic evolutions we rescale time, in such a way that the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized through an energy-dissipation balance, that can be written in a natural way using the rescaled time. As shown in [4] and [6], the proposed solution may be discontinuous with respect to the original time. Our formulation allows to compute the amount of viscous dissipation occurring instantaneously at each discontinuity time.
Keywords: Cam-Clay plasticity

Quasistatic evolution of sessile drops and contact angle hysteresis - Giovanni Alberti; Antonio DeSimone (Arch. Rational Mech. Anal. 202 (2011) 295-348, 2011)
We consider the classical model of capillarity coupled with a rate-independent dissipation mechanism due to frictional forces acting on the contact line, and prove the existence of quasistatic evolutions with prescribed initial configuration. We also discuss in detail some explicit solutions to show that the model does account for contact angle hysteresis, and to compare its predictions with experimental observations.

Resonance and Landesman-Lazer conditions for first order systems in R^2 - Maurizio Garrione (Le Matematiche, 2011)
The first part of the paper surveys the concept of resonance for $T$-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in $\mathbbR^2$ where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (for the case with friction). The results have been obtained with A. Fonda, and have been published in [8].

Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem - Alberto Boscaggin; Maurizio Garrione (Nonlinear Analysis: Theory, Methods & Applications, 2011)
In the general setting of a planar first order system (0.1)u′=G(t,u),u∈R2, with G:[0,T]×R2→R2, we study the relationships between some classical nonresonance conditions (including the Landesman–Lazer one) — at infinity and, in the unforced case, i.e. G(t,0)≡0, at zero — and the rotation numbers of “large” and “small” solutions of (0.1), respectively. Such estimates are then used to establish, via the Poincaré–Birkhoff fixed point theorem, new multiplicity results for T-periodic solutions of unforced planar Hamiltonian systems Ju′=∇uH(t,u) and unforced undamped scalar second order equations x″+g(t,x)=0. In particular, by means of the Landesman–Lazer condition, we obtain sharp conclusions when the system is resonant at infinity.
Keywords: Multiple periodic solutions; Poincaré–Birkhoff theorem; Resonance; Rotation number

SBV regularity for Hamilton-Jacobi equations in R^n - Stefano Bianchini; Camillo De Lellis; Roger Robyr (Arch. Rational Mech. Anal. 200 (2011) 1003-1021, 2011)
In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad in\quad Ω\subset \mathbb{R}× \mathbb{R}^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb{R}^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(Ω)$.

Singular perturbation models in phase transitions for second order materials - Milena Chermisi; Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (Indiana Univ. Math. J. 60 (2011) 367-409, 2011)
A variational model proposed in the physics literature to describe the onset of pattern formation in two-component bilayer membranes and amphiphilic monolayers leads to the analysis of a Ginzburg-Landau type energy with a negative term depending on the first derivative of the phase function. Scaling arguments motivate the study of the family of second order singular perturbed energies Fe having a negative term depending on the first derivative of the phase function. Here, the asymptotic behavior of Fe is studied using G-convergence techniques. In particular, compactness results and an integral representation of the limit energy are obtained.

On the Space of Symmetric Operators with Multiple Ground States - Andrei A. Agrachev (Functional Analysis and its Applications, Volume 45, Issue 4, December 2011, Pages 241-251, 2011)
We study homological structure of the filtrations of the spaces of self-adjoint operators by the multiplicity of the ground state. We consider only operators acting in a finite dimensional complex or real Hilbert space but infinite dimensional generalizations are easily guessed.
Keywords: Multiple eigenvalue

The sphere and the cut locus at a tangency point in two-dimensional almost-Riemannian geometry - Bernard Bonnard; Grégoire Charlot; Roberta Ghezzi; Gabriel Janin (Journal of Dynamical and Control Systems, 2011)
We study the tangential case in 2-dimensional almost-Riemannian geometry. We\\r\\nanalyse the connection with the Martinet case in sub-Riemannian geometry. We\\r\\ncompute estimations of the exponential map which allow us to describe the\\r\\nconjugate locus and the cut locus at a tangency point. We prove that this last\\r\\none generically accumulates at the tangency point as an asymmetric cusp whose branches are separated by the singular set.

Structure of level sets and Sard-type properties of Lipschitz maps - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (SISSA, 2011)

Subharmonic solutions of planar Hamiltonian systems: a rotation number approach - Alberto Boscaggin (Advanced Nonlinear Studies, 2011)

Subharmonic solutions of planar Hamiltonian systems via the Poincaré́-Birkhoff theorem - Alberto Boscaggin (Le Matematiche, 2011)
We revisit some recent results obtained in [1] about the existence of subharmonic solutions for a class of (nonautonomous) planar Hamiltonian systems, and we compare them with the existing literature. New applications to undamped second order equations are discussed, as well.

Supercritical conformal metrics on surfaces with conical singularities - Mauro Bardelloni; Francesca De Marchis; Andrea Malchiodi (Int Math Res Notices (2011) 2011 (24): 5625-5643, 2011)
We study the problem of prescribing the Gaussian curvature on surfaces with conical singularities in supercritical regimes. Using a Morse-theoretical approach we prove a general existence theorem on surfaces with positive genus, with a generic multiplicity result.

Thin-walled beams with a cross-section of arbitrary geometry: derivation of linear theories starting from 3D nonlinear elasticity - Elisa Davoli (2011)
The subject of this paper is the rigorous derivation of lower dimensional models for a nonlinearly elastic thin-walled beam whose cross-section is given by a thin tubular neighbourhood of a smooth curve. Denoting by h and δ_h, respectively, the diameter and the thickness of the cross-section, we analyse the case where the scaling factor of the elastic energy is of order ε_h^2, with ε_h/δ_h^2 \rightarrow l \in [0, +\infty). Different linearized models are deduced according to the relative order of magnitude of δ_h with respect to h.

The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity - Helmut Abels; Maria Giovanna Mora; Stefan Müller (Calculus of Variations and Partial Differential Equations 41 (2011) 241-259, 2011)
The asymptotic behaviour of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness $h$ of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of $h$, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von K\\\\\\\'arm\\\\\\\'an plate equation.

Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter - Alessandro Selvitella (Nonlinear Analysis: Theory, Methods & Applications, 2011)
We study least energy solutions of a quasilinear Schrödinger equation with a small parameter. We prove that the ground state is nondegenerate and unique up to translations and phase shifts using bifurcation theory.
Keywords: Bifurcation theory; Nonlinear Schrödinger equations; Stationary solutions

A uniqueness result for the continuity equation in two dimensions - Giovanni Alberti; Stefano Bianchini; Gianluca Crippa (SISSA, 2011)

The well-posedness issue for the density-dependent Euler equations in endpoint Besov spaces - Raphaël Danchin; Francesco Fanelli (Journal de Mathématiques Pures et Appliquées, 2011)
This work is the continuation of the recent paper (Danchin, 2010) [9] devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type B∞,rs embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Hölder spaces C1,α and of the endpoint Besov space B∞,11. For such data and under the non-vacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda (1984) [2]. In the last part of the paper, we give lower bounds for the lifespan of a solution. In dimension two, we point out that the lifespan tends to infinity when the initial density tends to be a constant. This is, to our knowledge, the first result of this kind for the density-dependent incompressible Euler equations. Résumé Ce travail complète lʼarticle récent (Danchin, 2010) [9] consacré au système dʼEuler incompressible à densité variable. Lorsque lʼétat initial ne comporte pas de vide, on montre ici que le système est bien posé dans tous les espaces de Besov B∞,rs inclus dans lʼensemble des fonctions lipschitziennes. Ce cadre fonctionnel contient en particulier les espaces de Hölder C1,α et lʼespace de Besov limite B∞,11. On établit également un critère de prolongement dans lʼesprit de celui de Beale, Kato et Majda (1984) [2] pour le cas homogène. Dans la dernière partie de lʼarticle, on donne des minorations pour le temps de vie des solutions du système. En dimension deux, on montre que ce temps de vie tend vers lʼinfini lorsque la densité tend à être homogène. À notre connaissance, il sʼagit du premier résultat de ce type pour le système dʼEuler incompressible à densité variable.
Keywords: Blow-up criterion; Critical regularity; Incompressible Euler equations; Lifespan; Nonhomogeneous inviscid fluids

Almost-Riemannian Geometry from a Control Theoretical Viewpoint - Roberta Ghezzi (SISSA, 2010)

Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions - Jesus Garcia Azorero; Andrea Malchiodi; Luigi Montoro; Ireneo Peral (Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 37-56, 2010)
In this paper we carry on the study of asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions, started in the first paper. Here we are mainly interested in the analysis of the location and shape of least energy solutions when the singular perturbation parameter tends to zero. We show that in many cases they coincide with the new solutions produced in.

Concentration of solutions for some singularly perturbed mixed problems. Part I: existence results - Jesus Garcia Azorero; Andrea Malchiodi; Luigi Montoro; Ireneo Peral (Arch. Ration. Mech. Anal. 196 (2010) 907-950, 2010)
In this paper we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. 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.

Continuity of optimal control costs and its application to weak KAM theory - Andrei A. Agrachev; Paul Lee (Calculus of Variations and Partial Differential Equations. Volume 39, Issue 1, 2010, Pages 213-232, 2010)
We prove continuity of certain cost functions arising from optimal control of\\r\\naffine control systems. We give sharp sufficient conditions for this\\r\\ncontinuity. As an application, we prove a version of weak KAM theorem and\\r\\nconsider the Aubry-Mather problems corresponding to these systems.

Convergence of equilibria of thin elastic rods under physical growth conditions for the energy density - Elisa Davoli; Maria Giovanna Mora (2010)
The subject of this paper is the study of the asymptotic behaviour of the equilibrium configurations of a nonlinearly elastic thin rod, as the diameter of the cross-section tends to zero. Convergence results are established assuming physical growth conditions for the elastic energy density and suitable scalings of the applied loads, that correspond at the limit to different rod models: the constrained linear theory, the analogous of von Kármán plate theory for rods, and the linear theory.

The disintegration of the Lebesgue measure on the faces of a convex function - Laura Caravenna; Sara Daneri (J. Funct. Anal. 258 (2010) 3604-3661, 2010)
We consider the disintegration of the Lebesgue measure on the graph of a convex function f:\\\\Rn-> \\\\R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure of the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we also prove that a Green-Gauss formula for these directions holds on special sets.

Dynamics control by a time-varying feedback - Andrei A. Agrachev; Marco Caponigro (Journal of Dynamical and Control Systems. Volume 16, Issue 2, April 2010, Pages :149-162, 2010)
We consider a smooth bracket generating control-affine system in R^d and show that any orientation preserving diffeomorphism of R^d can be approximated, in the very strong sense, by a diffeomorphism included in the flow generated by a time-varying feedback control which is polynomial with respect to the state variables and trigonometric-polynomial with respect to the time variable.
Keywords: Discrete-time dynamics

Estimates on path functionals over Wasserstein Spaces - Stefano Bianchini; Alessio Brancolini (SIAM J. Math. Anal. 42 (2010) 1179-1217, 2010)
In this paper we consider the class a functionals (introduced in [Brancolini, Buttazzo, and Santambrogio, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 415-434] $\\\\mathcal{G}_{r,p}$ defined on Lipschitz curves $\\\\gamma$ valued in the $p$-Wasserstein space. The problem considered is the following: given a measure $\\\\mu$, give conditions in order to assure the existence of a curve $\\\\gamma$ such that $\\\\gamma(0)=\\\\mu$, $\\\\gamma(1)=\\\\delta_{x_0}$, and $\\\\mathcal{G}_{r,p}(\\\\gamma)<+\\\\infty$. To this end, new estimates on $\\\\mathcal{G}_{r,p}(\\\\mu)$ are given, and a notion of dimension of a measure (called path dimension) is introduced: the path dimension specifies the values of the parameters $(r,p)$ for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions.

On the Euler-Lagrange equation for a variational problem : the general case II - Stefano Bianchini; Matteo Gloyer (Math. Z. 265 (2010) 889-923, 2010)

Exact reconstruction of damaged color images using a total variation model - Irene Fonseca; Giovanni Leoni; Francesco Maggi; Massimiliano Morini (Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 1291-1331, 2010)
In this paper the reconstruction of damaged piecewice constant color images is studied using a RGB total variation based model for colorization/inpainting. In particular, it is shown that when color is known in a uniformly distributed region, then reconstruction is possible with maximal fidelity.

Existence of planar curves minimizing length and curvature - Ugo Boscain; Grégoire Charlot; Francesco Rossi (Proc. Steklov Inst. Math. 270 (2010) 43-56, 2010)
In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\\\\int \\\\sqrt{1+K_\\\\gamma^2} ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions.\\nFor this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles.\\nWe instead prove existence of minimizers for the \\\"time-reparameterized\\\" functional $$\\\\int \\\\| \\\\dot\\\\gamma(t) \\\\|\\\\sqrt{1+K_\\\\ga^2} dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles.

Gene expression analysis of the emergence of epileptiform activity after focal injection of kainic acid into mouse hippocampus. - Dario Motti; Caroline Le Duigou; Nicole Chemaly; Lucia Wittner; Dejan Lazarevic; Helena Krmac; Troels Torben Marstrand; Eivind Valen; Remo Sanges; Elia Stupka; Albin Sandelin; Enrico Cherubini; et al. (The European journal of neuroscience. 2010 Oct; 32(8):1364-79, 2010)
We report gene profiling data on genomic processes underlying the progression towards recurrent seizures after injection of kainic acid (KA) into the mouse hippocampus. Focal injection enabled us to separate the effects of proepileptic stimuli initiated by KA injection. Both the injected and contralateral hippocampus participated in the status epilepticus. However, neuronal death induced by KA treatment was restricted to the injected hippocampus, although there was some contralateral axonal degeneration. We profiled gene expression changes in dorsal and ventral regions of both the injected and contralateral hippocampus. Changes were detected in the expression of 1526 transcripts in samples from three time-points: (i) during the KA-induced status epilepticus, (ii) at 2 weeks, before recurrent seizures emerged, and (iii) at 6 months after seizures emerged. Grouping genes with similar spatio-temporal changes revealed an early transcriptional response, strong immune, cell death and growth responses at 2 weeks and an activation of immune and extracellular matrix genes persisting at 6 months. Immunostaining for proteins coded by genes identified from array studies provided evidence for gliogenesis and suggested that the proteoglycan biglycan is synthesized by astrocytes and contributes to a glial scar. Gene changes at 6 months after KA injection were largely restricted to tissue from the injection site. This suggests that either recurrent seizures might depend on maintained processes including immune responses and changes in extracellular matrix proteins near the injection site or alternatively might result from processes, such as growth, distant from the injection site and terminated while seizures are maintained.

Generic multiplicity for a scalar field equation on compact surfaces - Francesca De Marchis (Journal of Functional Analysis, 2010)
We prove generic multiplicity of solutions for a scalar field equation on compact surfaces via Morse inequalities. In particular our result improves significantly the multiplicity estimate which can be deduced from the degree-counting formula in Chen and Lin (2003) [12]. Related results are derived for the prescribed Q-curvature equation.
Keywords: Generic multiplicity; Geometric PDE's; Morse inequalities; Scalar field equations

A global compactness result for the p-Laplacian involving critical nonlinearities - Mercuri, Carlo; Willem, Michel (Discrete & Continuous Dynamical Systems-A, 2010)
We prove a representation theorem for Palais-Smale sequences involving the p-Laplacian and critical nonlinearities. Applications are given to a critical problem.

Homogeneous binary trees as ground states of quantum critical Hamiltonians - Pietro Silvi; Vittorio Giovannetti; Simone Montangero; Matteo Rizzi; J. Ignacio Cirac; Rosario Fazio (Phys. Rev. A 81 (2010) 062335, 2010)
Many-body states whose wave-function admits a representation in terms of a uniform binary-tree tensor decomposition are shown to obey to power-law two-body correlations functions. Any such state can be associated with the ground state of a translational invariant Hamiltonian which, depending on the dimension of the systems sites, involve at most couplings between third-neighboring sites. A detailed analysis of their spectra shows that they admit an exponentially large ground space.

Homogeneous multiscale entanglement renormalization ansatz tensor networks for quantum critical systems - Matteo Rizzi; Simone Montangero; Pietro Silvi; Vittorio Giovannetti; Rosario Fazio (New J. Phys. 12 (2010) 075018, 2010)
In this paper, we review the properties of homogeneous multiscale entanglement renormalization ansatz (MERA) to describe quantum critical systems.We discuss in more detail our results for one-dimensional (1D) systems (the Ising and Heisenberg models) and present new data for the 2D Ising model. Together with the results for the critical exponents, we provide a detailed description of the numerical algorithm and a discussion of new optimization\\nstrategies. The relation between the critical properties of the system and the tensor structure of the MERA is expressed using the formalism of quantum channels, which we review and extend.

Homogenization of fiber reinforced brittle material: the intermediate case - Gianni Dal Maso; Caterina Ida Zeppieri (Adv. Calc. Var. 3 (2010) 345-370, 2010)
We derive a cohesive fracture model by homogenizing a periodic composite material whose microstructure is characterized by the presence of brittle inclusions in a reticulated unbreakable elastic structure.

Invariant Lagrange submanifolds of dissipative systems - Andrei A. Agrachev (Russian Mathematical Surveys. Volume 65, Issue 5, 2010, Pages: 977-978, 2010)
We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) =\\r\\n0, q \\\\in M, on a compact manifold M .

New approximation results for free discontinuity problems - Flaviana Iurlano (Università degli Studi di Trieste and SISSA, 2010)

Nonlocal character of the reduced theory of thin films with higher order perturbations - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni (Adv. Calc. Var. 3 (2010) 287-319, 2010)

A normal form for generic 2-dimensional almost-Riemannian structures at a tangency point - Ugo Boscain; Grégoire Charlot; Roberta Ghezzi (arXiv preprint arXiv:1008.5036, 2010)

On the number of positive solutions of some semilinear elliptic problems - Antonio Ambrosetti (2010)

On optimality of c-cyclically monotone transference plans - Stefano Bianchini; Laura Caravenna (Comptes Rendus Mathematique 348 (2010) 613-618, 2010)
Abstract. This note deals with the equivalence between the optimality of a transport plan for the Monge-Kantorovich problem and the condition of c-cyclical monotonicity, as an outcome of the construction in [7]. We emphasize the measurability assumption on the hidden structure of linear preorder, applied also to extremality and uniqueness problems. Resume. Dans la presente note nous decrivons brievement la construction introduite dans [7] a propos de l\\\'equivalence entre l\\\'optimalite d\\\'un plan de transport pour le probleme de Monge-Kantorovich et la condition de monotonie c-cyclique ainsi que d\\\'autres sujets que cela nous amene a aborder. Nous souhaitons mettre en evidence l\\\'hypothese de mesurabilite sur la structure sous-jacente de pre-ordre lineaire.

Optimally swimming Stokesian Robots - François Alouges; Antonio DeSimone; Luca Heltai; Aline Lefebvre; Benoit Merlet (2010)
We study self propelled stokesian robots composed of assemblies of balls, in dimen-\\nsions 2 and 3, and prove that they are able to control their position and orientation. This is a result of controllability, and its proof relies on applying Chow\\\'s theorem in an analytic framework, similarly to what has been done in [3] for an axisymmetric system swimming along the axis of symmetry. However, we simplify drastically\\nthe analyticity result given in [3] and apply it to a situation where more complex swimmers move either in a plane or in three-dimensional space, hence experiencing also rotations. We then focus our attention on energetically optimal strokes, which we are able to compute numerically. Some examples of computed optimal strokes are discussed in detail.

A Phase Field Approach to Wetting and Contact Angle Hysteresis Phenomena - Antonio DeSimone; Livio Fedeli; Turco, Alessandro (IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, 2010)
We discuss a phase field model for the numerical simulation of contact angle hysteresis phenomena in wetting. The performance of the model is assessed by comparing its predictions with experimental data on the critical size of drops that can stick on a vertical glass plate.

Poles of Integrale Tritronquee and Anharmonic Oscillators. Asymptotic localization from WKB analysis - Davide Masoero (Nonlinearity. vol. 23, (2010), page 2501-2507, 2010)
Poles of integrale tritronquee are in bijection with cubic oscillators that admit the simultaneous solutions of two quantization conditions. We show that the poles lie near the solutions of a pair of Bohr-Sommerfeld quantization conditions (the Bohr-Sommerfeld-Boutroux system): the distance between a pole and the corresponding solution of the Bohr-Sommerfeld-Boutroux system vanishes asymptotically.

Positive solutions for some non-autonomous Schrödinger–Poisson systems - Giovanna Cerami; Giusi Vaira (Journal of Differential Equations, 2010)

Projective Reeds-Shepp car on $S^2$ with quadratic cost - Ugo Boscain; Francesco Rossi (ESAIM COCV 16 (2010) 275-297, 2010)
Fix two points $x,\\\\bar{x}\\\\in S^2$ and two directions (without orientation) $\\\\eta,\\\\bar\\\\eta$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $$ J[\\\\gamma]=\\\\int_0^T g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t))+\\nK^2_{\\\\gamma(t)}g_{\\\\gamma(t)}(\\\\dot\\\\gamma(t),\\\\dot\\\\gamma(t)) ~dt$$ along all smooth curves starting from $x$ with direction $\\\\eta$ and ending in $\\\\bar{x}$ with direction $\\\\bar\\\\eta$. Here $g$ is the standard Riemannian metric on $S^2$ and $K_\\\\gamma$ is the corresponding geodesic curvature.\\nThe interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1).\\nWe compute the global solution for this problem: an interesting feature is that some optimal geodesics present cusps. The cut locus is a stratification with non trivial topology.

Quasistatic crack growth in elasto-plastic materials: the two-dimensional case - Gianni Dal Maso; Rodica Toader (Arch. Ration. Mech. Anal. 196 (2010) 867-906, 2010)
We study a variational model for the quasistatic evolution of elasto-plastic materials with cracks in the case of planar small strain associative elasto-plasticity.

Quasistatic crack growth in finite elasticity with non-interpenetration - Gianni Dal Maso; Giuliano Lazzaroni (Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290, 2010)
We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking\\ninto account the non-interpenetration condition.

Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case - Gianni Dal Maso; Francesco Solombrino (Netw. Heterog. Media 5 (2010) 97-132, 2010)
We study the spatially uniform case of the problem of quasistatic evolution in small strain nonassociative elastoplasticity (Cam-Clay model). Through the introdution of a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE\\\'s in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity, the slow dynamics, when the strain evolves smoothly on the yield surface and plastic flow is produced, and the fast dynamics, which may happen only in the softening regime, where\\nviscous solutions exhibit a jump across a heteroclinic orbit of an auxiliary system.
Keywords: Cam-Clay plasticity

Quasistatic evolution for plasticity with softening: The spatially homogeneous case - Francesco Solombrino (Discrete & Continuous Dynamical Systems - A, 2010)
The spatially uniform case of the problem of quasistatic evolution in small strain associative elastoplasticity with softening is studied. Through the introdution of a viscous approximation, the problem reduces to determine the limit behaviour of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. We see that the limit dynamics presents, for a generic choice of the initial data, the alternation of three possible regimes (elastic regime, slow dynamics, fast dynamics), which is determined by the sign of two scalar indicators, whose explicit expression is given.
Keywords: plasticity with softening; rate independent processes

The role of membrane viscosity in the dynamics of fluid membranes - Marino Arroyo; Antonio DeSimone; Luca Heltai (2010)
Fluid membranes made out of lipid bilayers are the fundamental separation structure in eukaryotic cells. Many physiological processes rely on dramatic shape and topological changes (e.g. fusion, fission) of fluid membrane systems. Fluidity is key to the versatility and constant reorganization of lipid bilayers. Here, we study the role of the membrane intrinsic viscosity, arising from the friction of the lipid molecules as they rearrange to accommodate shape changes, in the dynamics of morphological changes of fluid vesicles. In particular, we analyze the competition between the membrane viscosity and the viscosity of the bulk fluid surrounding the vesicle as the dominant dissipative mechanism. We consider the relaxation dynamics of fluid vesicles put in an out-of-equilibrium state, but conclusions can be drawn regarding the kinetics or power consumption in regulated shape changes in the cell. On the basis of numerical calculations, we find that the dynamics arising from the membrane viscosity are qualitatively different from the dynamics arising from the bulk viscosity. When these two dissipation mechanisms are put in competition, we find that for small vesicles the membrane dissipation dominates, with a relaxation time that scales as the size of the vesicle to the power 2. For large vesicles, the bulk dissipation dominates, and the exponent in the relaxation time vs. size relation is 3.

Semiclassical evolution of two rotating solitons for the Nonlinear Schrödinger Equation with electric potential - Alessandro Selvitella (Adv. Differential Equations, 2010)

Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials - Mouhamed Moustapha Fall; Roberta Musina (2010)
In this paper we deal with nonnegative distributional supersolutions for a class of linear\\nelliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.

Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity - Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX (2010) 253-295, 2010)
We discuss the limiting behavior (using the notion of gamma-limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like h4, h being the thickness of a shell, we derive a limiting theory which is a generalization of the von Karman theory for plates.

Stable determination of an immersed body in a stationary Stokes fluid - Andrea Ballerini (Inverse Problems, 2010)
We consider the inverse problem of the detection of a single body, immersed in a bounded container filled with a fluid which obeys the Stokes equations, from a single measurement of force and velocity on a portion of the boundary. We obtain an estimate of the stability of log–log type.

A three-dimensional model for the dynamics and hydrodynamics of rowing boats - L. Formaggia; Andrea Mola; N Parolini; M Pischiutta (Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology, 2010)
This paper proposes a new model describing the dynamics of a rowing boat for general three-dimensional motions. The complex interaction between the different components of the rowers–-oars–-boat system is analysed and reduced to a set of ordinary differential equations governing the rigid motion along the six degrees of freedom. To treat the unstable nature of the physical problem, a rather simple (but effective) control model is included, which mimics the main active control techniques adopted by the rowers during their action.

Two-dimensional almost-Riemannian structures with tangency points - Andrei A. Agrachev; Ugo Boscain; Grégoire Charlot; Roberta Ghezzi; Mario Sigalotti (Ann. Inst. H. Poincare Anal. Non Lineaire, 2010)
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 study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including 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 classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

Well-posed infinite horizon variational problems on a compact manifold - Andrei A. Agrachev (Proceedings of the Steklov Institute of Mathematics. Volume 268, Issue 1, 2010, Pages 17-31, 2010)
We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold M to admit a smooth optimal synthesis, i. e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian submanifold (well-projected to M) of the flow of extremals in the cotangent bundle T*M. The construction uses the curvature of the flow in the cotangent bundle and some ideas of hyperbolic dynamics

Biological Fluid Dynamics, Non-linear Partial Differential Equations - Antonio DeSimone; François Alouges; Aline Lefebvre (Encyclopedia of Complexity and Systems Science / Robert A. Meyers (ed.). - Springer, 2009, 548-554, 2009)

The boundary Riemann solver coming from the real vanishing viscosity approximation - Stefano Bianchini; Laura Spinolo (Arch. Ration. Mech. Anal. 191 (2009) 1-96, 2009)
We study the limit of the hyperbolic-parabolic approximation $$ \\\\begin{array}{lll} v_t + \\\\tilde{A} ( v, \\\\, \\\\varepsilon v_x ) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in R^N\\\\\\\\ \\\\tilde \\\\beta (v (t, \\\\, 0)) = \\\\bar g \\\\\\\\ v (0, \\\\, x) = \\\\bar v_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nThe function $\\\\tilde \\\\beta$ is defined in such a way to guarantee that the initial boundary value problem is well posed even if $\\\\tilde \\\\beta$ is not invertible.\\nThe data $\\\\bar g$ and $\\\\bar v_0$ are constant. When $\\\\tilde B$ is invertible, the previous problem takes the simpler form $$ \\\\left\\\\{ \\\\begin{array}{lll} v_t + \\\\tilde{A} \\\\big( v, \\\\, \\\\varepsilon v_x \\\\big) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in \\\\mathbb{R}^N\\\\\\\\ v (t, \\\\, 0) \\\\equiv \\\\bar v_b \\\\\\\\ v (0, \\\\, x) \\\\equiv \\\\bar{v}_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nAgain, the data $\\\\bar v_b$ and $\\\\bar v_0$ are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $\\\\tilde A$ can be 0.\\n Second, as pointed out before we take into account the possibility that $\\\\tilde B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.

Bubbles with prescribed mean curvature: the variational approach - Paolo Caldiroli; Roberta Musina (2009)

A connection between viscous profiles and singular ODEs - Stefano Bianchini; Laura Spinolo (Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41, 2009)

Controllability of the discrete-spectrum Schrodinger equation driven by an external field - Thomas Chambrion; Paolo Mason; Mario Sigalotti; Ugo Boscain (Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009) 329-349, 2009)
We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete nonresonant\\nspectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the\\nGalerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential controlled by suitable potentials.

Controllability on the group of diffeomorphisms - Andrei A. Agrachev; Marco Caponigro (Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 2503-2509, 2009)
Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of identity of the group of diffeomorphisms of M.

On the convergence of viscous approximations after shock interactions - Alberto Bressan; Carlotta Donadello (Discrete Contin. Dyn. Syst. 23 (2009) 29-48, 2009)
We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. We show that, after the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient.

Differential geometry of curves in Lagrange Grassmannians with given Young diagram - Igor Zelenko; Li Chengbo (Differential Geom. Appl. 27 (2009) 723-742, 2009)
Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.

Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers - Marco Cicalese; Antonio DeSimone; Caterina Ida Zeppieri (Netw. Heterog. Media 4 (2009) 667-708, 2009)
In the framework of linear elasticity, we study the limit of a class of discrete free energies modeling strain-alignment-coupled systems by a rigorous coarse-graining procedure, as the number of molecules diverges. We focus on three paradigmatic examples: magnetostrictive solids, ferroelectric crystals and nematic elastomers, obtaining in the limit three continuum models consistent with those commonly employed in the current literature. We also derive the correspondent macroscopic energies in the presence of displacement boundary conditions and of various kinds of applied external fields.

The Disintegration Theorem and Applications to Optimal Mass Transportation - Laura Caravenna (SISSA, 2009)

Existence of extremals for the Maz\\\'ya and for the Caffarelli-Kohn-Nirenberg inequalities - Roberta Musina (Nonlinear Anal. 70 (2009) 3002-3007, 2009)
This paper deals with some Sobolev-type inequalities with weights that were proved by Maz\\\'ya in 1980 and by Caffarelli-Kohn-Nirenberg in 1984.

An existence result for the Monge problem in R^n with norm cost - Laura Caravenna (2009)

On the extremality, uniqueness and optimality of transference plans - Stefano Bianchini; Laura Caravenna (Bull. Inst. Math. Acad. Sin. (N.S.) 4 (2009) 353-458, 2009)
We consider the following standard problems appearing in optimal mass transportation theory: when a transference plan is extremal; when a transference plan is the unique transference plan concentrated on a set A,; when a transference plan is optimal.

Foliations of small tubes in Riemannian manifolds by capillary minimal discs - Fall, Mouhamed Moustapha; Mercuri, Carlo (Nonlinear Analysis: Theory, Methods & Applications, 2009)
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font-size: 18px;"><span id="MathJax-Element-3-Frame" class="MathJax_SVG" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" data-mathml="Γ"><span class="MJX_Assistive_MathML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px 0px !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); user-select: none; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important; transition: none 0s ease 0s;">Γ<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;">inside the surface of revolution of <span class="math" style="box-sizing: border-box; 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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;"> around <span class="math" style="box-sizing: border-box; margin: 0px; padding: 0px; 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;"><span id="MathJax-Element-5-Frame" class="MathJax_SVG" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" data-mathml="Γ"><span class="MJX_Assistive_MathML" style="box-sizing: border-box; margin: 0px; padding: 1px 0px 0px !important; top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px); user-select: none; position: absolute !important; border: 0px !important; height: 1px !important; width: 1px !important; overflow: hidden !important; display: block !important; transition: none 0s ease 0s;">Γ<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;">, having small radius, and intersecting it with constant angles. In particular we obtain that small tubular neighborhoods can be foliated by minimal discs.

Hardy-Sobolev-Maz\\\'ja inequalities: symmetry and breaking symmetry of extremal functions - Marita Gazzini; Roberta Musina (Commun. Contemp. Math. 11 (2009) 993-1007, 2009)

A higher order model for image restoration: the one dimensional case - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni; Massimiliano Morini (SIAM J. Math. Anal. 40 (2009) 2351-2391, 2009)
The higher order total variation-based model for image restoration proposed by Chan, Marquina, and Mulet in [6] is analyzed in one dimension. A suitable functional framework in which the minimization problem is well posed is being proposed and it is proved analytically that the\\nhigher order regularizing term prevents the occurrence of the staircase effect. The generalized version of the model considered here includes, as particular cases, some curvature dependent functionals.

Homogenization of fiber reinforced brittle materials: the extremal cases - Marco Barchiesi; Gianni Dal Maso (SIAM J. Math. Anal. 41 (2009) 1874-1889, 2009)
We analyze the behavior of a fragile material reinforced by a reticulated elastic unbreakable structure in the case of antiplane shear. The microscopic geometry of this material is described by means of two small parameters: the period $\\\\varepsilon$ of the grid and the ratio $\\\\delta$ between the thickness of the fibers and the period $\\\\varepsilon$. We show that the asymptotic behavior as $\\\\varepsilon\\\\to0^+$ and $\\\\delta\\\\to0^+$ depends dramatically on the relative size of these parameters. Indeed, in the two cases considered, i.e., $\\\\varepsilon\\\\ll\\\\delta$ and $\\\\varepsilon\\\\gg\\\\delta$, we obtain two different limit models: a perfectly elastic model and an elastic model with macroscopic cracks, respectively.

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups - Andrei A. Agrachev; Ugo Boscain; Jean-Paul Gauthier; Francesco Rossi (J. Funct. Anal. 256 (2009) 2621-2655, 2009)
We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp\\\'s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation.

Jacobi Equations and Comparison Theorems for Corank 1 Sub-Riemannian structures with symmetries - Li Chengbo; Igor Zelenko (2009)
The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical\\nbundle of moving frames and the complete system of symplectic invariants, called curvature maps, for\\nparametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural\\nequation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the\\nnormal form for the Jacobi equation along this extremal and the curvature maps can be seen as the\\n\\\"coefficients\\\"of this normal form. In the case of a Riemannian metric there is only one curvature map and it is naturally related to the Riemannian sectional curvature. In the present paper we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form(a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along the sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of an uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.

Minimal disc-type surfaces embedded in a perturbed cylinder - Fall, Mouhamed Moustapha; Mercuri, Carlo (Differential and Integral Equations, 2009)
In the present note we deal with small perturbations of an infinite cylinder in the 3D euclidian space. We find minimal disc-type surfaces embedded in the cylinder and intersecting its boundary perpendicularly. The existence and localization of those minimal discs is a consequence of a non-degeneracy condition for the critical points of a functional related to the oscillations of the cylinder from the flat configuration.

A nonlinear theory for shells with slowly varying thickness - Marta Lewicka; Maria Giovanna Mora; Mohammad Reza Pakzad (C. R. Math. 347 (2009) 211-216, 2009)
We study the Γ-limit of 3d nonlinear elasticity for shells of small, variable thickness, around an arbitrary smooth 2d surface.

A note on the paper \\\"Optimizing improved Hardy inequalities\\\" by S. Filippas and A. Tertikas - Roberta Musina (J. Funct. Anal. 256 (2009) 2741-2745, 2009)

Optimal transportation under nonholonomic constraints - Andrei A. Agrachev; Paul Lee (Trans. Amer. Math. Soc. 361 (2009) 6019-6047, 2009)
We study the Monge\\\'s optimal transportation problem where the cost is given by optimal control cost. We prove the existence and uniqueness of optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures and most importantly the absent of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2-generating distribution and cost given by d2, where d is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane.

Quasistatic evolution for Cam-Clay plasticity: examples of spatially homogeneous solutions - Gianni Dal Maso; Antonio DeSimone (Math. Models Methods Appl. Sci. 19 (2009) 1643-1711, 2009)
We study a quasistatic evolution problem for Cam-Clay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behaviour and time discontinuities.\\nThe behavior of the solutions at the jump times is studied by a viscous approximation.

Quasistatic evolution problems for nonhomogeneous elastic plastic materials - Francesco Solombrino (J. Convex Anal., 2009)
The paper studies the quasistatic evolution for elastoplastic materials when the yield surface depends on the position in the reference configuration. The main results are obtained when the yield surface is continuous with respect to the space variable. The case of piecewise constant dependence is also considered. The evolution is studied in the framework of the variational formulation for rate independent problems developed by Mielke. The results are proved by adapting the arguments introduced for a constant yield surface, using some properties of convex valued semicontinuous multifunctions. A strong formulation of the problem is also obtained, which includes a pointwise version of the plastic flow rule. Some examples are considered, which show that strain concentration may occur as a consequence of a nonconstant yield surface.

Relaxation dynamics of fluid membranes - Marino Arroyo; Antonio DeSimone (Phys. Rev. E 79 (2009) 031915, 2009)
We study the effect of membrane viscosity in the dynamics of liquid membranes-possibly with free or internal boundaries-driven by conservative forces (curvature elasticity and line tension) and dragged by the bulk dissipation of the ambient fluid and the friction occurring when the amphiphilic molecules move relative to each other. To this end, we formulate a continuum model which includes a form of the governing equations for a two-dimensional viscous fluid moving on a curved, time-evolving surface. The effect of membrane viscosity has received very limited attention in previous continuum studies of the dynamics of fluid membranes, although recent coarse-grained discrete simulations suggest its importance. By applying our model to the study of vesiculation and membrane fusion in a simplified geometry, we conclude that membrane viscosity plays a dominant role in the relaxation dynamics of fluid membranes of sizes comparable to those found in eukaryotic cells, and is not negligible in many large synthetic systems of current interest.

On a Sobolev type inequality related to the weighted p-Laplace operator - Marita Gazzini; Roberta Musina (J. Math. Anal. Appl. 352 (2009) 99-111, 2009)

Solutions of the Schrödinger–Poisson problem concentrating on spheres, part I: necessary conditions - Ianni, Isabella; Giusi Vaira (Mathematical Models and Methods in Applied Sciences, 2009)
In this paper we study a coupled nonlinear Schrödinger–Poisson problem with radial functions. This system has been introduced as a model describing standing waves for the nonlinear Schrödinger equations in the presence of the electrostatic field. We provide necessary conditions for concentration on sphere for the solutions of this kind of problem extending the results already known.

Some new entire solutions of semilinear elliptic equations on Rn - Andrea Malchiodi (Adv. Math. 221 (2009) 1843-1909, 2009)

Strain-order coupling in nematic elastomers: equilibrium configurations - Pierluigi Cesana; Antonio DeSimone (Math. Models Methods Appl. Sci. 19 (2009) 601-630, 2009)
We consider models that describe liquid crystal elastomers either in a biaxial or in a uniaxial phase and in the framework of Frank\\\'s director theory. We prove existence of static equilibrium solutions in the presence of frustrations due to electro-mechanical boundary conditions and to applied loads and fields. We find explicit solutions arising in connection with special boundary conditions and the corresponding phase diagrams, leading to significant implications on possible experimental observations.

Stratos: a code for 3D free surface flows with floating constraints - Antonio DeSimone; B. Bianchi; Luca Heltai (2009)
This report presents a brief discussion of the theoretical aspects and practical implementation of STRATOS . STRATOS is a 3D code for the simulation\\nof hydrodynamic flows for incompressible fluids, in the presence of a free surface, capable of simulating the interaction between the free surface and a\\nfloating object via Lagrange multipliers......

Tools for the Solution of PDEs Defined on Curved Manifolds with deal.II - Antonio DeSimone; Luca Heltai; Cataldo Manigrasso (2009)
The deal.II finite element library was originally designed to solve partial differential equations defined on one, two or three space dimensions, mostly\\nvia the Finite Element Method. In its versions prior to version 6.2, the user could not solve problems defined on curved manifolds embedded in two or\\nthree spacial dimensions. This infrastructure is needed if one wants to solve, for example, Boundary Integral Equations.

A variational model for quasistatic crack growth in nonlinear elasticity: some qualitative properties of the solutions - Gianni Dal Maso; Alessandro Giacomini; Marcello Ponsiglione (Boll. Unione Mat. Ital. (9) 2 (2009) 371-390, 2009)

On viscosity solutions of Hamilton-Jacobi equations - Sandro Zagatti (Trans. Amer. Math. Soc. 361 (2009) 41-59, 2009)
We consider the Dirichlet problem for Hamilton-Jacobi equations and prove existence, uniqueness and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions.

Asymptotic evolution for the semiclassical nonlinear Schrödinger equation in presence of electric and magnetic fields - Alessandro Selvitella (Journal of Differential Equations, 2008)
In this paper we study the semiclassical limit for the solutions of a subcritical focusing NLS with electric and magnetic potentials. We consider in particular the Cauchy problem for initial data close to solitons and show that, when the Planck constant goes to zero, the motion shadows that of a classical particle. Several works were devoted to the case of standing waves: differently from these we show that, in the dynamic version, the Lorentz force appears crucially.

Concentrating solutions of some singularly perturbed elliptic equations - Andrea Malchiodi (Front. Math. China 3 (2008) 239-252, 2008)
We study singularly perturbed elliptic equations arising from models in physics or biology, and investigate the asymptotic behavior of some special solutions. We also discuss some connections with problems arising in differential geometry.

On concentration of positive bound states for the Schrödinger-Poisson problem with potentials - Ianni, Isabella; Giusi Vaira (Advanced nonlinear studies, 2008)
We study the existence of semiclassical states for a nonlinear Schrödinger-Poisson system that concentrate near critical points of the external potential and of the density charge function. We use a perturbation scheme in a variational setting, extending the results in [1]. We also discuss necessary conditions for concentration.

Convergence of equilibria of three-dimensional thin elastic beams - Maria Giovanna Mora; Stefan Müller (Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 873-896, 2008)
A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\\\\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.

Decomposition results for functions with bounded variation - Gianni Dal Maso; Rodica Toader (Boll. Unione Mat. Ital. (9) 1 (2008) 497-505, 2008)

Entire solutions of autonomous equations on Rn with nontrivial asymptotics - Andrea Malchiodi (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008) 65-72, 2008)
We prove existence of a new type of solutions for the semilinear equation $- \\\\D u + u = u^p$ on $\\\\R^n$, with $1 < p < \\\\frac{n+2}{n-2}$. These solutions are positive, bounded, decay exponentially to zero away from three half-lines with a common origin, and at infinity are asymptotically periodic.

An entropy based Glimm-type functional - Laura Caravenna (J. Hyperbolic Differ. Equ. 5 (2008) 643-662, 2008)

Eulerian calculus for the displacement convexity in the Wasserstein distance - Sara Daneri; Giuseppe Savarè (SIAM J. Math. Anal. 40 (2008) 1104-1122, 2008)
In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto and Westdickenberg [SIAM J. Math. Anal., 37 (2005), pp. 1227-1255] and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.

Existence of conformal metrics with constant $Q$-curvature - Zindine Djadli; Andrea Malchiodi (Ann. of Math. 168 (2008) 813-858, 2008)
Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author.

Forced Vibrations of a Nonhomogeneous String - P Baldi; Massimiliano Berti (SIAM J. Math. Anal. 40 (2008) 382-412, 2008)
We prove existence of vibrations of a nonhomogeneous string under a nonlinear time periodic forcing term in the case in which the forcing frequency avoids resonances with the vibration modes of the string (nonresonant case). The proof relies on a Lyapunov-Schmidt reduction and a Nash-Moser iteration scheme.

Fulde-Ferrell-Larkin-Ovchinnikov pairing in one-dimensional optical lattices - Matteo Rizzi; Marco Polini; Miguel A. Cazalilla; M.R. Bakhtiari; Mario P. Tosi; Rosario Fazio (Phys. Rev. B 77 (2008) 245105, 2008)
Spin-polarized attractive Fermi gases in one-dimensional (1D) optical lattices are expected to be remarkably good candidates for the observation of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We model these systems with an attractive Hubbard model with population imbalance. By means of the density-matrix renormalization-group method we compute the pairing correlations as well as the static spin and charge structure factors in the whole range from weak to strong coupling. We demonstrate that pairing correlations exhibit quasi-long range order and oscillations at the wave number expected from FFLO theory. However, we also show by numerically computing the mixed spin-charge static structure factor that charge and spin degrees of freedom appear to be coupled already for small imbalance. We discuss the consequences of this coupling for the observation of the FFLO phase, as well as for the stabilization of the quasi-long range order into long-range order by coupling many identical 1D systems, as in quasi-1D optical lattices.

A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds - Andrei A. Agrachev; Ugo Boscain; Mario Sigalotti (Discrete Contin. Dyn. Syst. 20 (2008) 801-822, 2008)
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of rank-varying sub-Riemannian structures, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular for what concerns the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.

Globally stable quasistatic evolution in plasticity with softening - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora; Massimiliano Morini (Netw. Heterog. Media 3 (2008) 567-614, 2008)
We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each\\ntime interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics - Gianni Dal Maso; Adriana Garroni (Calc. Var. Partial Differential Equations 31 (2008) 137-145, 2008)
In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.

Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces - Ugo Boscain; Francesco Rossi (SIAM J. Control Optim. 47 (2008) 1851-1878, 2008)
In this paper we study the invariant Carnot-Caratheodory metrics on SU(2) \\\' S3,\\nSO(3) and SL(2) induced by their Cartan decomposition. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric\\ngiven on SU(2) projects on the so called Lens Spaces L(p; q). Also for Lens Spaces, we compute\\nthe cut loci (globally).

Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems - Stefano Bianchini; Laura Spinolo (2008)

Limit Time Optimal Syntheses for a control-affine system on S² - Paolo Mason; Rebecca Salmoni; Ugo Boscain; Yacine Chitour (SIAM J. Control Optim. 47 (2008) 111-143, 2008)
For $\\\\alpha \\\\in ]0,\\\\pi/2[$, let $(\\\\Sigma)_\\\\alpha$ be the control system $\\\\dot{x}=(F+uG)x$, where $x$ belongs to the two-dimensional unit sphere $S^2$, $u\\\\in [-1,1]$, and $F,G$ are $3\\\\times3$ skew-symmetric matrices generating rotations with perpendicular axes and of respective norms $\\\\cos(\\\\alpha)$ and $\\\\sin(\\\\alpha)$. In this paper, we study the time optimal synthesis (TOS) from the north pole $(0,0,1)^T$ associated to $(\\\\Sigma)_\\\\alpha$, as the parameter $\\\\alpha$ tends to zero; this problem is motivated by specific issues in the control of quantum systems. We first prove that the TOS is characterized by a \\\"two-snakes\\\" configuration on the whole $S^2$, except for a neighborhood $U_\\\\alpha$ of the south pole $(0,0,-1)^T$ of diameter at most ${\\\\cal O}(\\\\alpha)$. We next show that, inside $U_\\\\alpha$, the TOS depends on the relationship between $r(\\\\alpha):=\\\\pi/2\\\\alpha-[\\\\pi/2\\\\alpha]$ and $\\\\alpha$. More precisely, we characterize three main relationships by considering sequences $(\\\\alpha_k)_{k\\\\geq 0}$ satisfying (a) $r(\\\\alpha_k)=\\\\bar{r}$, (b) $r(\\\\alpha_k)=C\\\\alpha_k$, and (c) $r(\\\\alpha_k)=0$, where $\\\\bar{r}\\\\in (0,1)$ and $C>0$. In each case, we describe the TOS and provide, after a suitable rescaling, the limiting behavior, as $\\\\alpha$ tends to zero, of the corresponding TOS inside $U_\\\\alpha$.

Minimization of non quasiconvex functionals by integro-extremization method - Sandro Zagatti (Discrete Contin. Dyn. Syst. 21 (2008) 625-641, 2008)

Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations - Sandro Zagatti (Calc. Var. Partial Differential Equations 31 (2008) 511-519, 2008)

Morse theory and a scalar field equation on compact surfaces - Andrea Malchiodi (Adv. Differential Equations 13 (2008) 1109-1129, 2008)

Multiple bound states for the Schroedinger-Poisson problem - Antonio Ambrosetti; David Ruiz (Commun. Contemp. Math. 10 (2008) 391-404, 2008)

A note on the differentiability of Lipschitz functions and the chain rule in Sobolev spaces - Massimiliano Morini (2008)

Optimal Strokes for Low Reynolds Number Swimmers: An Example - François Alouges; Antonio DeSimone; Aline Lefebvre (J. Nonlinear Sci. 18 (2008) 277-302, 2008)
Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers. This is the regime of interest for micro-organisms and micro- or nano-robots. We focus in this paper on a simple yet representative example: the three-sphere swimmer of Najafi and Golestanian (Phys. Rev. E, 69, 062901-062904, 2004). For this system, we show how to cast the problem of swimming in the language of control theory, prove global controllability (which implies that the three-sphere swimmer can indeed swim), and propose a numerical algorithm to compute optimal strokes (which turn out to be suitably defined sub-Riemannian geodesics).

Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity - Mercuri, Carlo (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl, 2008)
We deal with a weighted nonlinear Schr¨odinger-Poisson system, allowing the potentials to vanish at infinity.

Relaxation of some transversally isotropic energies and applications to smectic A elastomers - James Adams; Sergio Conti; Antonio DeSimone; Georg Dolzmann (Math. Models Methods Appl. Sci. 18 (2008) 1-20, 2008)

A second order minimality condition for the Mumford-Shah functional - Filippo Cagnetti; Maria Giovanna Mora; Massimiliano Morini (Calc. Var. Partial Differential Equations 33 (2008) 37-74, 2008)
A new necessary minimality condition for the Mumford-Shah functional is derived by means of second order variations. It is expressed in terms of a sign condition for a nonlocal quadratic form on $H^1_0(\\\\Gamma)$, $\\\\Gamma$ being a submanifold of the regular part of the discontinuity set of the critical point. Two equivalent formulations are provided: one in terms of the first eigenvalue of a suitable compact operator, the other involving a sort of nonlocal capacity of $\\\\Gamma$. A sufficient condition for minimality is also deduced. Finally, an explicit example is discussed, where a complete characterization of the domains where the second variation is nonnegative can be given.

Solitons of linearly coupled systems of semilinear non-autonomous equations on Rn - Antonio Ambrosetti; Giovanna Cerami; David Ruiz (J. Funct. Anal. 254 (2008) 2816-2845, 2008)
Using concentration compactness type arguments, we prove some results about the existence of positive ground and bound state of linearly coupled systems of nonlinear Schrödinger equations.

Stability of planar switched systems: the nondiagonalizable case - Ugo Boscain; Moussa Balde (Commun. Pure Appl. Anal. 7 (2008) 1-21, 2008)

Topological methods for an elliptic equation with exponential nonlinearities - Andrea Malchiodi (Discrete Contin. Dyn. Syst. 21 (2008) 277-294, 2008)
We consider a class of variational equations with exponential nonlinearities on compact surfaces. From considerations involving the Moser-Trudinger inequality, we characterize some sublevels of the Euler-Lagrange functional in terms of the topology of the surface and of the data of the equation. This is used together with a min-max argument to obtain existence results.

Transition layer for the heterogeneous Allen-Cahn equation - Fethi Mahmoudi; Andrea Malchiodi; Juncheng Wei (Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631, 2008)
We consider the equation $\\\\e^{2}\\\\Delta u=(u-a(x))(u^2-1)$ in $\\\\Omega$, $\\\\frac{\\\\partial u}{\\\\partial \\\\nu} =0$ on $\\\\partial \\\\Omega$, where $\\\\Omega$ is a smooth and bounded domain in $\\\\R^n$, $\\\\nu$ the outer unit normal to $\\\\pa\\\\Omega$, and $a$ a smooth function satisfying $-1<a(x)<1$ in $\\\\ov{\\\\Omega}$. We set $K$, $\\\\Omega_+$ and $\\\\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\\\\nabla a \\\\neq 0$ on $K$ and $a\\\\ne 0$ on $\\\\partial \\\\Omega$, we show that there exists a sequence $\\\\e_j \\\\to 0$ such that the above equation has a solution $u_{\\\\e_j}$ which converges uniformly to $\\\\pm 1$ on the compact sets of $\\\\O_{\\\\pm}$ as $j \\\\to + \\\\infty$.

Transport Rays and Applications to Hamilton–Jacobi Equations - Stefano Bianchini; Matteo Gloyer (Nonlinear PDE’s and Applications : C.I.M.E. Summer School, Cetraro, Italy 2008 / Stefano Bianchini, Eric A. Carlen, Alexander Mielke, Cédric Villani. Eds. Luigi Ambrosio, Giuseppe Savaré. - Berlin : Springer, 2011. - (Lecture Notes in Mathematics ; 20, 2008)
The aim of these notes is to introduce the readers to the use of the Disintegration Theorem for measures as an effective tool for reducing problems in transport equations to simpler ones. The basic idea is to partition Rd into one dimensional sets, on which the problem under consideration becomes one space dimensional (and thus much easier, hopefully).

A vanishing viscosity approach to quasistatic evolution in plasticity with softening - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora; Massimiliano Morini (Arch. Ration. Mech. Anal. 189 (2008) 469-544, 2008)
We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analize a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. This leads to a generalized formulation in terms of Young measures, developed in the first part of the paper. In the second part we apply our approach to some concrete examples.

Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy - Stefano Bianchini; Bernard Hanouzet; Roberto Natalini (Comm. Pure Appl. Math. 60 (2007) 1559-1622, 2007)
We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition.

Asymptotic variational wave equations - Alberto Bressan; Zhang Ping; Zheng Yuxi (Arch. Ration. Mech. Anal. 183 (2007) 163-185, 2007)
We investigate the equation $(u_t + (f(u))_x)_x = f\\\'\\\'(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.\\nWe establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data.

Boundary interface for the Allen-Cahn equation - Andrea Malchiodi; Juncheng Wei (J. Fixed Point Theory Appl. 1 (2007) 305-336, 2007)

Boundary-clustered interfaces for the Allen–Cahn equation - Andrea Malchiodi; Wei-Ming Ni; Juncheng Wei (Pacific Journal of Mathematics 229 (2007), No. 2, 447–468, 2007)

BV instability for the Lax-Friedrichs scheme - Paolo Baiti; Alberto Bressan; Helge Kristian Jenssen (2007)
It is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \\\\times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation.

Concentration on minimal submanifolds for a singularly perturbed Neumann problem - Fethi Mahmoudi; Andrea Malchiodi (Adv. Math. 209 (2007) 460-525, 2007)
We consider the equation $- \\\\e^2 \\\\D u + u= u^p$ in $\\\\Omega \\\\subseteq \\\\R^N$, where $\\\\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\partial \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, ..., N-2\\\\}$. We impose Neumann boundary conditions, assuming $1<p <\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.

Concentration phenomena for nonlinear Schrödinger equations: Recent results and new perspectives - Antonio Ambrosetti; Andrea Malchiodi (Perspectives in Nonlinear Partial Differential Equations: In Honor of Haïm Brezis / ed. by Henri Beresticky [et al.]. - Providence : American Mathematical Society, 2007. - (Contemporary mathematics ; 446). - p. 19-30, 2007)
We survey some results on (NLSepsilon), discussing also new perspectives and open problems.

Conformal Metrics with Constant Q-Curvature - Andrea Malchiodi (SIGMA 3 (2007) 120, 2007)
We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.

Convergence of equilibria of planar thin elastic beams - Maria Giovanna Mora; Stefan Müller; Maximilian G. Schultz (Indiana Univ. Math. J. 56 (2007) 2413-2438, 2007)
We consider a thin elastic strip of thickness h and we show that stationary points of the nonlinear elastic energy (per unit height) whose energy is of order h^2 converge to stationary points of the Euler-Bernoulli functional. The proof uses the rigidity estimate for low-energy deformations by Friesecke, James, and Mueller (Comm. Pure Appl. Math. 2002), and a compensated compactness argument in a singular geometry. In addition, possible concentration effects are ruled out by a careful truncation argument.

Critical voltages and blocking stresses in nematic gels - Antonio DeSimone; Antonio Di Carlo; Luciano Teresi (Eur. Phys. J. E 24 (2007) 303-310, 2007)
We use a recently proposed model to study the dynamics of director remodeling in nematic gels under combined electro-mechanical loading. Focusing on a model specimen, we describe the critical volt-ages that must be exceeded to achieve mesogen reorientation, and the blocking stresses that prevent alignment of the nematic mesogens with the electric field. The corresponding phase diagram shows that the dynamic thresholds defined above are different from those predicted on the sole basis of energetics. Multistep loading programs are used to explore the energy landscape of our model specimen, showing the existence of multiple local minima under the same voltage and applied stress. This leads us to conclude that hysteresis should be expected in the electro-mechanical response of nematic gels.

Derivation of a rod theory for phase-transforming materials - Maria Giovanna Mora; Stefan Müller (Calc. Var. Partial Differential Equations 28 (2007) 161-178, 2007)
A rigorous derivation is given of a rod theory for a multiphase material,starting from three-dimensional nonlinear elasticity. The stored energy density is supposed to be nonnegative and to vanish exactly on a set consisting of two copies of the group of rotations SO(3). The two potential wells correspond to the two crystalline configurations preferred by the material. We find the optimal scaling of the energy in terms of the diameter of the rod and we identify the limit, as the diameter goes to zero, in the sense of Gamma-convergence.

Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results - Irene Fonseca; Nicola Fusco; Giovanni Leoni; Massimiliano Morini (Arch. Ration. Mech. Anal. 186 (2007) 477-537, 2007)
Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavorable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup in which the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume-constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero-contact-angle condition between islands and wetting layers.

On the Euler-Lagrange equation for a variational problem - Stefano Bianchini (Discrete Contin. Dynam. Systems A 17 (2007) 449-480, 2007)

On feedback classification of control-affine systems with one and two-dimensional inputs - Andrei A. Agrachev; Igor Zelenko (SIAM J. Control Optim. 46 (2007) 1431-1460, 2007)
The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intrinsic numbers of functional moduli of each prescribed number of variables on which a classification problem depends. In order to classify affine systems with scalar input we associate with such a system the canonical frame by normalizing some structural functions in a commutative relation of the vector fields, which define our control system. Then, using this canonical frame, we introduce the canonical coordinates and find a complete system of state-feedback invariants of the system. It also gives automatically the micro-local (i.e. local in state-input space) classification of the generic non-affine n-dimensional control system with scalar input for n>2. Further we show how the problem of feedback-equivalence of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine systems with scalar input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension 4 and all abnormal extremals in dimension 5 of the time optimal problem, defined by the original control system. In each classification problem under consideration we find the intrinsic numbers of functional moduli of each prescribed number of variables according to its Poincare series.

On finite-dimensional projections of distributions for solutions of randomly forced PDE\\\'s - Andrei A. Agrachev; Sergei Kuksin; Andrey Sarychev; Armen Shirikyan (Ann. Inst. Henri Poincare-Prob. Stat. 43 (2007) 399-415, 2007)
The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier-Stokes equations perturbed by various random forces of low dimension.

Gaussian estimates for hypoelliptic operators via optimal control - Ugo Boscain; Sergio Polidoro (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 333-342, 2007)
We obtain Gaussian lower bounds for the fundamental solution of a class of hypoelliptic equations, by using repeatedly an invariant Harnack inequality. Our main result is given in terms of the value function of a suitable optimal control problem.

High-order angles in almost-Riemannian geometry - Ugo Boscain; Mario Sigalotti (2007)
Let X and Y be two smooth vector fields on a two-dimensional manifold M. If X and Y are everywhere linearly independent, then they define a Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become linearly dependent somewhere on M, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula for domains with piecewise-C2 boundary. The main feature of such formula is the presence of terms that play the role of high-order angles at the intersection points with the set of singularities.

Luther-Emery Phase and Atomic-Density Waves in a Trapped Fermion Gas - Gao Xianlong; Matteo Rizzi; Marco Polini; Rosario Fazio; Mario P. Tosi; Vivaldo L. Jr. Campo; Klaus Capelle (Phys. Rev. Lett. 98 (2007) 030404, 2007)
The Luther-Emery liquid is a state of matter that is predicted to occur in one-dimensional systems of interacting fermions and is characterized by a gapless charge spectrum and a gapped spin spectrum. In this Letter we discuss a realization of the Luther-Emery phase in a trapped cold-atom gas. We study by means of the density-matrix renormalization-group technique a two-component atomic Fermi gas with attractive interactions subject to parabolic trapping inside an optical lattice. We demonstrate how this system exhibits compound phases characterized by the coexistence of spin pairing and atomic-density waves. A smooth crossover occurs with increasing magnitude of the atom-atom attraction to a state in which tightly bound spin-singlet dimers occupy the center of the trap. The existence of atomic-density waves could be detected in the elastic contribution to the light-scattering diffraction pattern.

On the Maz\\\'ya inequalities: existence and multiplicity results for an elliptic problem involving cylindrical weights - Marita Gazzini; Roberta Musina (2007)

Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations - Antonio Ambrosetti; Eduardo Colorado; David Ruiz (Calc. Var. Partial Differential Equations 30 (2007) 85-112, 2007)

Nearly time optimal stabilizing patchy feedbacks - Fabio Ancona; Alberto Bressan (Ann. Inst. H. Poincare Anal. Non Lineaire 24 (2007) 279-310, 2007)
We consider the time optimal stabilization problem for a nonlinear control system $\\\\dot x=f(x,u)$. Let $\\\\tau(y)$ be the minimum time needed to steer the system from the state $y\\\\in\\\\R^n$ to the origin, and call $\\\\A(T)$ the set of initial states that can be steered to the origin in time $\\\\tau(y)\\\\leq T$. Given any $\\\\ve>0$, in this paper we construct a patchy feedback $u=U(x)$ such that every solution of $\\\\dot x=f(x, U(x))$, $x(0)=y\\\\in \\\\A(T)$ reaches an $\\\\ve$-neighborhood of the origin within time $\\\\tau(y)+\\\\ve$.

Necessary and sufficient conditions for the chainrule in W1,1loc(RN;Rd) and BVloc(RN;Rd) - Giovanni Leoni; Massimiliano Morini (J. Eur. Math. Soc. (JEMS) 9 (2007) 219-252, 2007)
In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in Sobolev spaces and in the space of functions of bounded variation.

A new model for contact angle hysteresis - Antonio DeSimone; Natalie Gruenewald; Felix Otto (Netw. Heterog. Media 2 (2007) 211-225, 2007)
We present a model which explains several experimental observations relating contact angle hysteresis with surface roughness. The model is based on the balance between released energy and dissipation, and it describes the stick-slip behavior of drops on a rough surface using ideas similar to those employed in dry friction, elasto-plasticity and fracture mechanics. The main results of our analysis are formulas giving the interval of stable contact angles as a function of the surface roughness. These formulas show that the difference between advancing and receding angles is much larger for a drop in complete contact with the substrate (Wenzel drop) than for one whose cavities are filled with air (Cassie-Baxter drop). This fact is used as the key tool to interpret the experimental evidence.

On a notion of unilateral slope for the Mumford-Shah functional - Gianni Dal Maso; Rodica Toader (NoDEA 13 (2007) 713-734, 2007)
In this paper we introduce a notion of unilateral slope for the Mumford-Shah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional.

Parametrized curves in Lagrange Grassmannians - Igor Zelenko; Li Chengbo (C. R. Math. 345 (2007) 647-652, 2007)

Perturbation techniques applied to the real vanishing viscosity approximation of an initial boundary value problem - Stefano Bianchini (SISSA, 2007)

Quasistatic crack growth for a cohesive zone model with prescribed crack path - Gianni Dal Maso; Chiara Zanini (Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253-279, 2007)
In this paper we study the quasistatic crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.

Quasistatic evolution problems for pressure-sensitive plastic materials - Gianni Dal Maso; Alexey Demyanov; Antonio DeSimone (Milan J. Math. 75 (2007) 117-134, 2007)
We study quasistatic evolution problems for pressure-sensitive plastic materials in the context of small strain associative perfect plasticity.

On the regularity of weak solutions to H-systems - Roberta Musina (Atti .Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 209-219, 2007)
Abstract. In this paper we prove that every weak solution to the H-surface equation is locally bounded, provided the prescibed mean curvatore H is asymptotic to a constant at infinity (with a suitable decay rate). No smoothness ssumptions are required on H. We consider also the Dirichlet problem....

Soft elasticity and microstructure in smectic C elastomers - Antonio DeSimone; James Adams; Sergio Conti (Contin. Mech. Thermodyn. 18 (2007) 319-334, 2007)
Smectic C elastomers are layered materials exhibiting a solid-like elastic response along the layer normal and a rubbery one in the plane. The set of strains minimizing the elastic energy contains a one-parameter family of simple stretches associated with an internal degree of freedom, coming from the in-plane component of the director. We investigate soft elasticity and the corresponding microstructure by determining the quasiconvex hull of the set , and use this to propose experimental tests that should make the predicted soft response observable.

Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals - Sandro Zagatti (J. Math. Anal. Appl. 335 (2007) 1143-1160, 2007)
We provide a unified approach to prove existence results for the Dirichlet problem for Hamilton-Jacobi equations and for the minimum problem for nonquasiconvex functionals of the Calculus of Variations with affine boundary data. The idea relies on the definition of integro-extremal solutions introduced in the study of nonconvex scalar variational problem.

Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions - Fethi Mahmoudi; Andrea Malchiodi; Marcelo Montenegro (2007)
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \\\\epsilon^2 \\\\Delta \\\\psi + V(x) \\\\psi = |\\\\psi|^{p-1} \\\\psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $\\\\epsilon$ a small parameter corresponding to the Planck constant. As $\\\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $\\\\epsilon^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $\\\\epsilon$. Based on these, an existence result will be proved in the second part.

Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result - Fethi Mahmoudi; Andrea Malchiodi (2007)
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- \\\\epsilon^2 \\\\Delta \\\\psi + V(x) \\\\psi = |\\\\psi|^{p-1} \\\\psi$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $\\\\epsilon$ a small parameter corresponding to the Planck constant. As $\\\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this work we identified the limit set and constructed approximate solutions, while here we give the complete proof of our main existence result.

Some existence results for the Toda system on closed surfaces - Andrea Malchiodi; Cheikh Birahim Ndiaye (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 18 (2007) 391-412, 2007)
Given a compact closed surface $\\\\Sig$, we consider the {\\\\em generalized Toda} system of equations on $\\\\Sig$: $- \\\\D u_i = \\\\sum_{j=1}^2 \\\\rho_j a_{ij} \\\\left( \\\\frac{h_j e^{u_j}}{\\\\int_\\\\Sig h_j e^{u_j} dV_g} - 1 \\\\right)$ for $i = 1, 2$, where $\\\\rho_1, \\\\rho_2$ are real parameters and $h_1, h_2$ are smooth positive functions. Exploiting the variational structure of the problem and using a new minimax scheme we prove existence of solutions for generic values of $\\\\rho_1$ and for $\\\\rho_2 < 4 \\\\pi$.

Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws - Andrea Marson; Carlotta Donadello (NoDEA Nonlinear Differential Equations Appl. 14 (2007) 569-592, 2007)

Standing waves of some coupled Nonlinear Schrödinger Equations - Antonio Ambrosetti; Eduardo Colorado (J. Lond. Math. Soc. 75 (2007) 67-82, 2007)
We deal with a class of systems of NLS equations, proving the existence of bound and ground states provided the coupling parameter is small, respectively, large.

Surfactants in Foam Stability: A Phase-Field Model - Irene Fonseca; Massimiliano Morini; Valeriy Slastikov (Arch. Rational Mech. Anal. 183 (2007) 411-456, 2007)
The role of surfactants in stabilizing the formation of bubbles in foams is studied using a phase-field model. The analysis is centered on a van der Walls-Cahn-Hilliard-type energy with an added term accounting for the interplay between the presence of a surfactant density and the creation of interfaces. In particular, it is concluded that the surfactant segregates to the interfaces, and that the prescriptionof the distribution of surfactant will dictate the locus of interfaces, what is in agreement with experimentation.

Time optimal swing-up of the planar pendulum - Mireille E. Broucke; Paolo Mason; Benedetto Piccoli (46th IEEE Conference on Decision and Control (2007) 5389 - 5394, 2007)
This paper presents qualitative and numerical results on the global structure of the time optimal trajectories of the planar pendulum on a cart.

Time-dependent systems of generalized Young measures - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora; Massimiliano Morini (Netw. Heterog. Media 2 (2007) 1-36, 2007)
In this paper some new tools for the study of evolution problems in the framework of Young measures are introduced. A suitable notion of time-dependent system of generalized Young measures is defined, which allows to extend the classical notions of total variation and absolute continuity with respect to time, as well as the notion of time derivative. The main results are a Helly type theorem for sequences of systems of generalized Young measures and a theorem about the existence of the time derivative for systems with bounded variation with respect to time.

Uniqueness and continuous dependence on boundary data for integro-extremal minimizers of the functional of the gradient - Sandro Zagatti (J. Convex Anal. 14 (2007) 705-727, 2007)
We study some qualitative properties of the integro-extremal minimizers of the functional of the gradient defined on Sobolev spaces with Dirichlet boundary conditions. We discuss their use in the non-convex case via viscosity methods and give conditions under which they are unique and depend continuously on boundary data.

Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients - Giuseppe Maria Coclite; Nils Henrik Risebro (J. Hyperbolic Differ. Equ. 4 (2007) 771-795, 2007)
We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to \\\"internal boundaries\\\". By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\\\\infty$, and that the limits are viscosity solutions.

2-d stability of the Néel wall - Antonio DeSimone; Hans Knuepfer; Felix Otto (Calc. Var. Partial Differential Equations 27 (2006) 233-253, 2006)
We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls.

4e-condensation in a fully frustrated Josephson junction diamond chain - Matteo Rizzi; Vittorio Cataudella; Rosario Fazio (Phys. Rev. B 73 (2006) 100502(R), 2006)
Fully frustrated one-dimensional diamond Josephson chains have been shown [B. Dou\\\\c{c}ot and J. Vidal, Phys. Rev. Lett. {\\\\bf 88}, 227005 (2002)] to posses a remarkable property: The superfluid phase occurs through the condensation of pairs of Cooper pairs. By means of Monte Carlo simulations we analyze quantitatively the Insulator to $4e$-Superfluid transition. We determine the location of the critical point and discuss the behaviour of the phase-phase correlators. For comparison we also present the case of a diamond chain at zero and 1/3 frustration where the standard $2e$-condensation is observed.

An artificial viscosity approach to quasistatic crack growth - Rodica Toader; Chiara Zanini (2006)
We introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $\\\\epsilon$-gradient flow of the energy functional, as the \\\"viscosity\\\" parameter $\\\\epsilon$ tends to zero.

A Birkhoff-Lewis-Type Theorem for Some Hamiltonian PDEs - Dario Bambusi; Massimiliano Berti (SIAM J. Math. Anal. 37 (2006) 83-102, 2006)
In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from resonant finite dimensional invariant tori of the fourth order normal form of the system. Besides standard nonresonance and nondegeneracy assumptions, our main result is obtained assuming a regularizing property of the nonlinearity. We apply our main theorem to a semilinear beam equation and to a nonlinear Schr\\\\\\\"odinger equation with smoothing nonlinearity.

Bound and ground states of coupled nonlinear Schrödinger equations - Antonio Ambrosetti; Eduardo Colorado (C. R. Acad. Sci. Paris, Ser. I 342 (2006) 453-458, 2006)
We prove existence of bound and ground states of some systems of coupled nonlinear Schrodinger equations.

Bound states of Nonlinear Schroedinger Equations with Potentials Vanishing at Infinity - Antonio Ambrosetti; Andrea Malchiodi; David Ruiz (J. Anal. Math. 98 (2006) 317-348, 2006)

On Bressan\\\'s conjecture on mixing properties of vector fields - Stefano Bianchini (Self-Similar Solutions of Nonlinear PDE / Ed. Piotr Biler and Grzegorz Karch. - Banach Center Publ. 74 (2006) 13-31, 2006)

A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class - Boris Doubrov; Igor Zelenko (C. R. Math. Acad. Sci. Paris 342 (2006) 589-594, 2006)
In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in R5. We solve the analogous problems for rank 2 distributions in Rn for arbitrary n > 5. Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence.

Cantor families of periodic solutions for completely resonant nonlinear wave equations - Massimiliano Berti; Philippe Bolle (Duke Math. J. 134 (2006) 359-419, 2006)
We prove the existence of small amplitude, $2\\\\pi \\\\slash \\\\om$-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions, for any frequency $ \\\\om $ belonging to a Cantor-like set of positive measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem. In spite of the complete resonance of the equation we show that we can still reduce the problem to a {\\\\it finite} dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows to deal also with nonlinearities which are not odd and with finite spatial regularity.

Classification of stable time-optimal controls on 2-manifolds - Ugo Boscain; Igor Nikolaev; Benedetto Piccoli (J. Math. Sci. 135 (2006) 3109-3124, 2006)

Common Polynomial Lyapunov Functions for Linear Switched Systems - Paolo Mason; Ugo Boscain; Yacine Chitour (SIAM J. Control Optim. 45 (2006) 226-245, 2006)
In this paper, we consider linear switched systems $\\\\dot x(t)=A_{u(t)} x(t)$, $x\\\\in\\\\R^n$, $u\\\\in U$, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching ({\\\\bf UAS} for short). We first prove that, given a {\\\\bf UAS} system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the {\\\\bf UAS} systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.

Compactness of solutions to some geometric fourth-order equations - Andrea Malchiodi (J. Reine Angew. Math. 594 (2006) 137-174, 2006)
We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics.

Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem - Fethi Mahmoudi; Andrea Malchiodi (Atti Accad. Naz Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 17 (2006) 279-290, 2006)
We consider the equation $- \\\\e^2 \\\\D u + u = u^p$ in $\\\\O \\\\subseteq \\\\R^N$, where $\\\\O$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\pa \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, \\\\dots, N-2\\\\}$. We impose Neumann boundary conditions, assuming $1<\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.

Conservative Solutions to a Nonlinear Variational Wave Equation - Alberto Bressan; Zheng Yuxi (Comm. Math. Phys. 266 (2006) 471-497, 2006)
We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\\\\cdot)$ is any smooth function with uniformly positive bounded values.

The Dirichlet problem for H-systems with small boundary data: blowup phenomena and nonexistence results - Paolo Caldiroli; Roberta Musina (Arch. Ration. Mech. Anal. 181 (2006) 1-42, 2006)

An estimation of the controllability time for single-input systems on compact Lie Groups - Andrei A. Agrachev; Thomas Chambrion (ESAIM Control Optim. Calc. Var. 12 (2006) 409-441, 2006)
Geometric control theory and Riemannian techniques are used to describe the reachable set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to estimate the minimal time needed to reach any point from identity. This method provides an effective way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum system with a drift from a given initial position to a given final position. The bounds include diameters of the flag manifolds; the latter are also explicitly computed in the paper.

Experimental and modeling studies of desensitization of P2X3 receptors. - Elena Sokolova; Andrei Skorinkin; Igor Moiseev; Andrei A. Agrachev; Andrea Nistri; Rashid Giniatullin (Molecular pharmacology. 2006 Jul; 70(1):373-82, 2006)
The function of ATP-activated P2X3 receptors involved in pain sensation is modulated by desensitization, a phenomenon poorly understood. The present study used patch-clamp recording from cultured rat or mouse sensory neurons and kinetic modeling to clarify the properties of P2X3 receptor desensitization. Two types of desensitization were observed, a fast process (t1/2 = 50 ms; 10 microM ATP) following the inward current evoked by micromolar agonist concentrations, and a slow process (t1/2 = 35 s; 10 nM ATP) that inhibited receptors without activating them. We termed the latter high-affinity desensitization (HAD). Recovery from fast desensitization or HAD was slow and agonist-dependent. When comparing several agonists, there was analogous ranking order for agonist potency, rate of desensitization and HAD effectiveness, with 2-methylthioadenosine triphosphate the strongest and beta,gamma-methylene-ATP the weakest. HAD was less developed with recombinant (ATP IC50 = 390 nM) than native P2X3 receptors (IC50 = 2.3 nM). HAD could also be induced by nanomolar ATP when receptors seemed to be nondesensitized, indicating that resting receptors could express high-affinity binding sites. Desensitization properties were well accounted for by a cyclic model in which receptors could be desensitized from either open or closed states. Recovery was assumed to be a multistate process with distinct kinetics dependent on the agonist-dependent dissociation rate from desensitized receptors. Thus, the combination of agonist-specific mechanisms such as desensitization onset, HAD, and resensitization could shape responsiveness of sensory neurons to P2X3 receptor agonists. By using subthreshold concentrations of an HAD-potent agonist, it might be possible to generate sustained inhibition of P2X3 receptors for controlling chronic pain.

Forced vibrations of wave equations with non-monotone nonlinearities - Massimiliano Berti; Luca Biasco (Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006) 439-474, 2006)
We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz.

Fundamental form and Cartan tensor of (2,5)-distributions coincide - Igor Zelenko (J. Dyn. Control Syst. 12 (2006) 247-276, 2006)
In our previous paper for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 5) we constructed a special differential invariant, the fundamental form. In the case n=5 this differential invariant has the same algebraic nature, as the covariant binary biquadratic form, constructed by E.Cartan in 1910, using his ``reduction- prolongation\\\'\\\' procedure (we call this form Cartan\\\'s tensor). In the present paper we prove that our fundamental form coincides (up to constant factor -35) with Cartan\\\'s tensor. This result explains geometric reason for existence of Cartan\\\'s tensor (originally this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for Cartan\\\'s tensor in terms of structural functions of any frame naturally adapted to the distribution.

On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1 - Igor Zelenko (J. Math. Sci. 135 (2006) 3168-3194, 2006)
The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way first we obtain a new elementary proof of classical Levi-Civita\\\'s Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics. Secondly we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally we make the classification of all pairs of geodesically equivalent Riemannian metrics on a surface, which proportional in an isolated point. This is the simplest case, which was not covered by Levi-Civita\\\'s Theorem.

Glimm interaction functional for BGK schemes - Stefano Bianchini (2006)

Infinite Horizon Noncooperative Differential Games - Alberto Bressan; Fabio Simone Priuli (J. Differential Equations 227 (2006) 230-257, 2006)
For a non-cooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite-horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value\\nfunctions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.