Mathematical modeling and quantitative study of biological motility (in particular, of motility at microscopic scales) is producing new biophysical insight and is offering opportunities for new discoveries at the level of both fundamental science and technology. These range from the explanation of how complex behavior at the level of a single organism emerges from body architecture, to the understanding of collective phenomena in groups of organisms and tissues, and of how these forms of swarm intelligence can be controlled and harnessed in engineering applications, to the elucidation of processes of fundamental biological relevance at the cellular and sub-cellular level. In this paper, some of the most exciting new developments in the fields of locomotion of unicellular organisms, of soft adhesive locomotion across scales, of the study of pore translocation properties of knotted DNA, of the development of synthetic active solid sheets, of the mechanics of the unjamming transition in dense cell collectives, of the mechanics of cell sheet folding in volvocalean algae, and of the self-propulsion of topological defects in active matter are discussed. For each of these topics, we provide a brief state of the art, an example of recent achievements, and some directions for future research.

}, keywords = {active matter, adhesive locomotion, cell motility, cell sheet folding, knotted DNA, topological defects, unicellular swimmers, unjamming transition}, issn = {2640-3501}, doi = {10.3934/mine.2020011}, url = {http://dx.doi.org/10.3934/mine.2020011}, author = {Daniele Agostinelli and Roberto Cerbino and Del Alamo, Juan C and Antonio DeSimone and Stephanie H{\"o}hn and Cristian Micheletti and Giovanni Noselli and Eran Sharon and Julia Yeomans} } @article {LS20b, title = {Minimizers of the prescribed mean curvature functional in a Jordan domain with no necks}, journal = {ESAIM Control Optim. Calc. Var.}, volume = {26}, year = {2020}, pages = {76}, doi = {10.1051/cocv/2020030}, author = {Leonardi, G. P. and Saracco, G.} } @article {AlzettaHeltai-2020-a, title = {Multiscale modeling of fiber reinforced materials via non-matching immersed methods}, journal = {Computers \& Structures}, year = {2020}, note = {To appear}, url = {https://arxiv.org/abs/1906.03881}, author = {Giovanni Alzetta and Luca Heltai} } @article {mukoseeva2019minimality, title = {Minimality of the ball for a model of charged liquid droplets}, journal = {arXiv preprint arXiv:1912.07092}, year = {2019}, author = {Ekaterina Mukoseeva and Vescovo, Giulia} } @article {HeltaiCaiazzo-2018-a, title = {Multiscale modeling of vascularized tissues via non-matching immersed methods}, journal = {International Journal for Numerical Methods in Biomedical Engineering}, volume = {35}, number = {12}, year = {2019}, pages = {e3264}, url = {https://doi.org/10.1002/cnm.3264}, author = {Luca Heltai and Alfonso Caiazzo} } @article {2018, title = {A minimization approach to the wave equation on time-dependent domains}, number = {SISSA;21/2018/MATE}, year = {2018}, institution = {SISSA}, abstract = {We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.}, url = {http://preprints.sissa.it/handle/1963/35318}, author = {Gianni Dal Maso and Lucia De Luca} } @article {BELLETTINI20181, title = {Minimizing movements for mean curvature flow of droplets with prescribed contact angle}, journal = {Journal de Math{\'e}matiques Pures et Appliqu{\'e}es}, volume = {117}, year = {2018}, pages = {1 - 58}, abstract = {We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the solutions constructed as a limit of an approximation algorithm of Almgren{\textendash}Taylor{\textendash}Wang and Luckhaus{\textendash}Sturzenhecker, we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results. R{\'e}sum{\'e} Nous {\'e}tudions le mouvement par courbure moyenne d{\textquoteright}une goutte qui glisse par courbure moyenne sur un hyperplan horizontal avec un angle de contact prescrit {\'e}ventuellement non constant. En utilisant les solutions construites comme limites d{\textquoteright}un algorithme d{\textquoteright}approximation d{\^u} {\`a} Almgren, Taylor et Wang et Luckhaus et Sturzenhecker, nous montrons l{\textquoteright}existence d{\textquoteright}une {\'e}volution faible, et sa compatibilit{\'e} avec une solution au sens des distributions. Nous d{\'e}montrons {\'e}galement plusieurs r{\'e}sultats de comparaison.

}, keywords = {Capillary functional, Mean curvature flow with prescribed contact angle, Minimizing movements, Sets of finite perimeter}, issn = {0021-7824}, doi = {https://doi.org/10.1016/j.matpur.2018.06.003}, url = {http://www.sciencedirect.com/science/article/pii/S0021782418300825}, author = {Giovanni Bellettini and Matteo Novaga and Shokhrukh Kholmatov} } @article {doi:10.1137/17M1159294, title = {Minimizing Movements for Mean Curvature Flow of Partitions}, journal = {SIAM Journal on Mathematical Analysis}, volume = {50}, number = {4}, year = {2018}, pages = {4117-4148}, abstract = {We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdorff convergence of the initial partitions.

}, doi = {10.1137/17M1159294}, url = {https://doi.org/10.1137/17M1159294}, author = {Giovanni Bellettini and Shokhrukh Kholmatov} } @conference {tezzele2018model, title = {Model Order Reduction by means of Active Subspaces and Dynamic Mode Decomposition for Parametric Hull Shape Design Hydrodynamics}, booktitle = {Technology and Science for the Ships of the Future: Proceedings of NAV 2018: 19th International Conference on Ship \& Maritime Research}, year = {2018}, publisher = {IOS Press}, organization = {IOS Press}, address = {Trieste, Italy}, abstract = {We present the results of the application of a parameter space reduction methodology based on active subspaces (AS) to the hull hydrodynamic design problem. Several parametric deformations of an initial hull shape are considered to assess the influence of the shape parameters on the hull wave resistance. Such problem is relevant at the preliminary stages of the ship design, when several flow simulations are carried out by the engineers to establish a certain sensibility with respect to the parameters, which might result in a high number of time consuming hydrodynamic simulations. The main idea of this work is to employ the AS to identify possible lower dimensional structures in the parameter space. The complete pipeline involves the use of free form deformation to parametrize and deform the hull shape, the full order solver based on unsteady potential flow theory with fully nonlinear free surface treatment directly interfaced with CAD, the use of dynamic mode decomposition to reconstruct the final steady state given only few snapshots of the simulation, and the reduction of the parameter space by AS, and shared subspace. Response surface method is used to minimize the total drag.}, doi = {10.3233/978-1-61499-870-9-569}, url = {http://ebooks.iospress.nl/publication/49270}, author = {Marco Tezzele and Nicola Demo and Mahmoud Gadalla and Andrea Mola and Gianluigi Rozza} } @article {doi:10.1137/17M1150591, title = {Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering}, journal = {SIAM Journal on Scientific Computing}, volume = {40}, number = {4}, year = {2018}, pages = {B1055-B1079}, doi = {10.1137/17M1150591}, url = {https://doi.org/10.1137/17M1150591}, author = {Maria Strazzullo and Francesco Ballarin and Mosetti, R. and Gianluigi Rozza} } @article {riccobelli2018morpho, title = {Morpho-elastic model of the tortuous tumour vessels}, journal = {Int. J. Non-Linear Mech.}, volume = {107}, year = {2018}, pages = {1{\textendash}9}, publisher = {Elsevier BV}, author = {Davide Riccobelli and Pasquale Ciarletta} } @article {MR3665212, title = {The Malgrange form and Fredholm determinants}, journal = {SIGMA Symmetry Integrability Geom. Methods Appl.}, volume = {13}, year = {2017}, pages = {Paper No. 046, 12}, issn = {1815-0659}, doi = {10.3842/SIGMA.2017.046}, url = {http://dx.doi.org/10.3842/SIGMA.2017.046}, author = {Marco Bertola} } @article {MR3663616, title = {Maximal amplitudes of finite-gap solutions for the focusing Nonlinear Schr{\"o}dinger Equation}, journal = {Comm. Math. Phys.}, volume = {354}, number = {2}, year = {2017}, pages = {525{\textendash}547}, issn = {0010-3616}, doi = {10.1007/s00220-017-2895-9}, url = {http://dx.doi.org/10.1007/s00220-017-2895-9}, author = {Marco Bertola and Alexander Tovbis} } @article {Michelangeli2017, title = {Mean-field quantum dynamics for a mixture of Bose{\textendash}Einstein condensates}, journal = {Analysis and Mathematical Physics}, volume = {7}, number = {4}, year = {2017}, month = {Dec}, pages = {377{\textendash}416}, abstract = {We study the effective time evolution of a large quantum system consisting of a mixture of different species of identical bosons in interaction. If the system is initially prepared so as to exhibit condensation in each component, we prove that condensation persists at later times and we show quantitatively that the many-body Schr{\"o}dinger dynamics is effectively described by a system of coupled cubic non-linear Schr{\"o}dinger equations, one for each component.

}, issn = {1664-235X}, doi = {10.1007/s13324-016-0147-3}, url = {https://doi.org/10.1007/s13324-016-0147-3}, author = {Alessandro Michelangeli and Alessandro Olgiati} } @article {1534-0392_2017_4_1427, title = {Minimizers of anisotropic perimeters with cylindrical norms}, journal = {Communications on Pure \& Applied Analysis}, volume = {16}, number = {1534-0392_2017_4_142}, year = {2017}, pages = {1427}, abstract = {We study various regularity properties of minimizers of the Φ{\textendash}perimeter, where Φ is a norm. Under suitable assumptions on Φ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

}, keywords = {anisotropic Bernstein problem;, minimal cones, Non parametric minimal surfaces, Sets of finite perimeter}, issn = {1534-0392}, doi = {10.3934/cpaa.2017068}, url = {http://aimsciences.org//article/id/47054f15-00c7-40b7-9da1-4c0b1d0a103d}, author = {Giovanni Bellettini and Matteo Novaga and Shokhrukh Kholmatov} } @inbook {ChinestaHuertaRozzaWillcox2017, title = {Model Reduction Methods}, booktitle = {Encyclopedia of Computational Mechanics Second Edition}, year = {2017}, pages = {1-36}, publisher = {John Wiley \& Sons}, organization = {John Wiley \& Sons}, chapter = {Model Reduction Methods}, abstract = {This chapter presents an overview of model order reduction {\textendash} a new paradigm in the field of simulation-based engineering sciences, and one that can tackle the challenges and leverage the opportunities of modern ICT technologies. Despite the impressive progress attained by simulation capabilities and techniques, a number of challenging problems remain intractable. These problems are of different nature, but are common to many branches of science and engineering. Among them are those related to high-dimensional problems, problems involving very different time scales, models defined in degenerate domains with at least one of the characteristic dimensions much smaller than the others, model requiring real-time simulation, and parametric models. All these problems represent a challenge for standard mesh-based discretization techniques; yet the ability to solve these problems efficiently would open unexplored routes for real-time simulation, inverse analysis, uncertainty quantification and propagation, real-time optimization, and simulation-based control {\textendash} critical needs in many branches of science and engineering. Model order reduction offers new simulation alternatives by circumventing, or at least alleviating, otherwise intractable computational challenges. In the present chapter, we revisit three of these model reduction techniques: proper orthogonal decomposition, proper generalized decomposition, and reduced basis methodologies.} preprint = {http://preprints.sissa.it/xmlui/bitstream/handle/1963/35194/ECM_MOR.pdf?sequence=1\&isAllowed=y

}, doi = {10.1002/9781119176817.ecm2110}, author = {Francisco Chinesta and Antonio Huerta and Gianluigi Rozza and Karen Willcox} } @article {2017, title = {Moduli of semistable sheaves as quiver moduli}, number = {arXiv:1709.05555}, year = {2017}, abstract = {In the 1980s Dr{\'e}zet and Le Potier realized moduli spaces of Gieseker-semistable sheaves on\ P2\ as what are now called quiver moduli spaces. We discuss how this construction can be understood using t-structures and exceptional collections on derived categories, and how it can be extended to a similar result\ on\ P1{\texttimes}P1.

}, url = {https://arxiv.org/abs/1709.05555}, author = {Andrea Maiorana} } @article {1534-0392_2017_3_1083, title = {Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities}, journal = {Communications on Pure \& Applied Analysis}, volume = {16}, number = {1534-0392_2017_3_108}, year = {2017}, pages = {1083}, abstract = {We study the second order nonlinear differential equation

\begindocument $ u{\textquoteright}{\textquoteright} + \sum\limits_i = 1^m α_ia_i(x)g_i(u) - \sum\limits_j = 1^m + 1 β_jb_j(x)k_j(u) = 0,\rm $ \enddocument

where $\alpha_i, \beta_j\>0$, $a_i(x), b_j(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen[0, L\mathclose]$, and the nonlinearities $g_i(s), k_j(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $u"+a(x)u.p = 0$, with $p\>1$.When the positive parameters $\beta_j$ are sufficiently large, we prove the existence of at least $2.m-1$positive solutions for the Sturm-Liouville boundary value problems associated with the equation.The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets.Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.

}, keywords = {Leray-Schauder topological degree;, positive solutions, Sturm-Liouville boundary conditions, Superlinear indefinite problems}, issn = {1534-0392}, doi = {10.3934/cpaa.2017052}, url = {http://aimsciences.org//article/id/1163b042-0c64-4597-b25c-3494b268e5a1}, author = {Guglielmo Feltrin} } @article {FELTRIN20174255, title = {Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree}, journal = {Journal of Differential Equations}, volume = {262}, number = {8}, year = {2017}, pages = {4255 - 4291}, abstract = {We study the periodic boundary value problem associated with the second order nonlinear differential equationu"+cu'+(a+(t)-μa-(t))g(u)=0, where g(u) has superlinear growth at zero and at infinity, a(t) is a periodic sign-changing weight, c∈R and μ\>0 is a real parameter. Our model includes (for c=0) the so-called nonlinear Hill{\textquoteright}s equation. We prove the existence of 2m-1 positive solutions when a(t) has m positive humps separated by m negative ones (in a periodicity interval) and μ is sufficiently large, thus giving a complete solution to a problem raised by G.J. Butler in 1976. The proof is based on Mawhin{\textquoteright}s coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

}, keywords = {Coincidence degree, Multiplicity results, Neumann boundary value problems, Positive periodic solutions, subharmonic solutions, Superlinear indefinite problems}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2017.01.009}, url = {http://www.sciencedirect.com/science/article/pii/S0022039617300219}, author = {Guglielmo Feltrin and Fabio Zanolin} } @article {2016, title = {A model for the quasistatic growth of cracks with fractional dimension}, number = {SISSA;10/2016/MATE}, year = {2016}, abstract = {We study a variational model for the quasistatic growth of cracks with fractional dimension in brittle materials. We give a minimal set of properties of the collection of admissible cracks which ensure the existence of a quasistatic evolution. Both the antiplane and the planar cases are treated.}, url = {http://urania.sissa.it/xmlui/handle/1963/35175}, author = {Gianni Dal Maso and Marco Morandotti} } @inbook {2016, title = {Model Order Reduction: a survey}, booktitle = {Wiley Encyclopedia of Computational Mechanics, 2016}, year = {2016}, publisher = {Wiley}, organization = {Wiley}, url = {http://urania.sissa.it/xmlui/handle/1963/35194}, author = {Francisco Chinesta and Antonio Huerta and Gianluigi Rozza and Karen Willcox} } @article {Battaglia2016, title = {Moser{\textendash}Trudinger inequalities for singular Liouville systems}, journal = {Mathematische Zeitschrift}, volume = {282}, number = {3}, year = {2016}, month = {Apr}, pages = {1169{\textendash}1190}, abstract = {In this paper we prove a Moser{\textendash}Trudinger inequality for the Euler{\textendash}Lagrange functional of general singular Liouville systems on a compact surface. We characterize the values of the parameters which yield coercivity for the functional, hence the existence of energy-minimizing solutions for the system, and we give necessary conditions for boundedness from below. We also provide a sharp inequality under assuming the coefficients of the system to be non-positive outside the diagonal. The proofs use a concentration-compactness alternative, Poho{\v z}aev-type identities and blow-up analysis.

}, issn = {1432-1823}, doi = {10.1007/s00209-015-1584-7}, url = {https://doi.org/10.1007/s00209-015-1584-7}, author = {Luca Battaglia} } @article {Cicconofri2016, title = {Motion planning and motility maps for flagellar microswimmers}, journal = {The European Physical Journal E}, volume = {39}, number = {7}, year = {2016}, month = {Jul}, pages = {72}, abstract = {We study two microswimmers consisting of a spherical rigid head and a passive elastic tail. In the first one the tail is clamped to the head, and the system oscillates under the action of an external torque. In the second one, head and tail are connected by a joint allowing the angle between them to vary periodically, as a result of an oscillating internal torque. Previous studies on these models were restricted to sinusoidal actuations, showing that the swimmers can propel while moving on average along a straight line, in the direction given by the symmetry axis around which beating takes place. We extend these results to motions produced by generic (non-sinusoidal) periodic actuations within the regime of small compliance of the tail. We find that modulation in the velocity of actuation can provide a mechanism to select different directions of motion. With velocity-modulated inputs, the externally actuated swimmer can translate laterally with respect to the symmetry axis of beating, while the internally actuated one is able to move along curved trajectories. The governing equations are analysed with an asymptotic perturbation scheme, providing explicit formulas, whose results are expressed through motility maps. Asymptotic approximations are further validated by numerical simulations.

}, issn = {1292-895X}, doi = {10.1140/epje/i2016-16072-y}, url = {https://doi.org/10.1140/epje/i2016-16072-y}, author = {Giancarlo Cicconofri and Antonio DeSimone} } @article {11966, title = {A multi-physics reduced order model for the analysis of Lead Fast Reactor single channel}, journal = {Annals of Nuclear Energy, 87, 2 (2016): pp. 198-208}, volume = {87}, number = {Annals of Nuclear Energy;87}, year = {2016}, pages = {208}, publisher = {Elsevier}, chapter = {198}, abstract = {In this work, a Reduced Basis method, with basis functions sampled by a Proper Orthogonal Decomposition technique, has been employed to develop a reduced order model of a multi-physics parametrized Lead-cooled Fast Reactor single-channel. Being the first time that a reduced order model is developed in this context, the work focused on a methodological approach and the coupling between the neutronics and the heat transfer, where the thermal feedbacks on neutronics are explicitly taken into account, in time-invariant settings. In order to address the potential of such approach, two different kinds of varying parameters have been considered, namely one related to a geometric quantity (i.e., the inner radius of the fuel pellet) and one related to a physical quantity (i.e., the inlet lead velocity). The capabilities of the presented reduced order model (ROM) have been tested and compared with a high-fidelity finite element model (upon which the ROM has been constructed) on different aspects. In particular, the comparison focused on the system reactivity prediction (with and without thermal feedbacks on neutronics), the neutron flux and temperature field reconstruction, and on the computational time. The outcomes provided by the reduced order model are in good agreement with the high-fidelity finite element ones, and a computational speed-up of at least three orders of magnitude is achieved as well.}, doi = {doi:10.1016/j.anucene.2015.09.002}, url = {http://urania.sissa.it/xmlui/handle/1963/35191}, author = {Alberto Sartori and Antonio Cammi and Lelio Luzzi and Gianluigi Rozza} } @article {2016, title = {Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan--Skornyakov type}, number = {SISSA;65/2016/MATE}, year = {2016}, abstract = {We reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan- Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Krein, Vi{\v s}iik, and Birman. We identify the explicit {\textquoteleft}Krein-Vi{\v s}ik-Birman extension param- eter{\textquoteright} as an operator on the {\textquoteleft}space of charges{\textquoteright} for this model (the {\textquoteleft}Krein space{\textquoteright}) and we come to formulate a sharp conjecture on the dimensionality of its kernel. Based on our conjecture, for which we also discuss an amount of evidence, we explain the emergence of a multiplicity of extensions in a suitable regime of masses and we re- produce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.}, url = {http://urania.sissa.it/xmlui/handle/1963/35267}, author = {Alessandro Michelangeli and Andrea Ottolini} } @mastersthesis {2015, title = {Mathematical Models of Locomotion: Legged Crawling, Snake-like Motility, and Flagellar Swimming}, year = {2015}, school = {SISSA}, keywords = {Motility}, author = {Giancarlo Cicconofri} } @article {MR3305428, title = {Meromorphic differentials with imaginary periods on degenerating hyperelliptic curves}, journal = {Anal. Math. Phys.}, volume = {5}, number = {1}, year = {2015}, pages = {1{\textendash}22}, issn = {1664-2368}, doi = {10.1007/s13324-014-0088-7}, url = {http://dx.doi.org/10.1007/s13324-014-0088-7}, author = {Marco Bertola and Alexander Tovbis} } @article {BennerOhlbergerPateraRozzaSorensenUrban2015, title = {Model order reduction of parameterized systems ({MoRePaS}): Preface to the special issue of advances in computational mathematics}, journal = {Advances in Computational Mathematics}, volume = {41}, number = {5}, year = {2015}, pages = {955{\textendash}960}, doi = {10.1007/s10444-015-9443-y}, author = {Peter Benner and Mario Ohlberger and Anthony Patera and Gianluigi Rozza and Sorensen, D.C. and Karsten Urban} } @article {CICCONOFRI2015233, title = {Motility of a model bristle-bot: A theoretical analysis}, journal = {International Journal of Non-Linear Mechanics}, volume = {76}, year = {2015}, pages = {233 - 239}, abstract = {Bristle-bots are legged robots that can be easily made out of a toothbrush head and a small vibrating engine. Despite their simple appearance, the mechanism enabling them to propel themselves by exploiting friction with the substrate is far from trivial. Numerical experiments on a model bristle-bot have been able to reproduce such a mechanism revealing, in addition, the ability to switch direction of motion by varying the vibration frequency. This paper provides a detailed account of these phenomena through a fully analytical treatment of the model. The equations of motion are solved through an expansion in terms of a properly chosen small parameter. The convergence of the expansion is rigorously proven. In addition, the analysis delivers formulas for the average velocity of the robot and for the frequency at which the direction switch takes place. A quantitative description of the mechanism for the friction modulation underlying the motility of the bristle-bot is also provided.

}, keywords = {Bristle-robots, Crawling motility, Frictional interactions}, issn = {0020-7462}, doi = {https://doi.org/10.1016/j.ijnonlinmec.2014.12.010}, url = {http://www.sciencedirect.com/science/article/pii/S0020746215000025}, author = {Giancarlo Cicconofri and Antonio DeSimone} } @mastersthesis {2015, title = {Multidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type}, year = {2015}, note = {161 pages}, school = {SISSA}, abstract = {The Poisson brackets of hydrodynamic type, also called Dubrovin-Novikov brackets, constitute the Hamiltonian structure of a broad class of evolutionary PDEs, that are ubiquitous in the theory of Integrable Systems, ranging from Hopf equation to the principal hierarchy of a Frobenius manifold. They can be regarded as an analogue of the classical Poisson brackets, defined on an infinite dimensional space of maps Σ {\textrightarrow} M between two manifolds. Our main problem is the study of Poisson-Lichnerowicz cohomology of such space when dim Σ > 1. We introduce the notion of multidimensional Poisson Vertex Algebras, generalizing and adapting the theory by A. Barakat, A. De Sole, and V. Kac [Poisson Vertex Algebras in the theory of Hamiltonian equations, 2009]; within this framework we explicitly compute the first nontrivial cohomology groups for an arbitrary Poisson bracket of hydrodynamic type, in the case dim Σ = dim M = 2. For the case of the so-called scalar brackets, namely the ones for which dim M = 1, we give a complete description on their Poisson{\textendash}Lichnerowicz cohomology. From this computations it follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.}, keywords = {Poisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems}, author = {Matteo Casati} } @article {2015, title = {Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations}, journal = {Numerische Mathematik, (2015), 36 p. Article in Press}, number = {Numerische Mathematik;}, year = {2015}, publisher = {Springer}, abstract = {In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems.}, doi = {10.1007/s00211-015-0743-4}, url = {http://urania.sissa.it/xmlui/handle/1963/34491}, author = {Gianluigi Rozza and Peng Chen and Alfio Quarteroni} } @article {2015, title = {Multiple positive solutions for a superlinear problem: a topological approach}, journal = {J. Differential Equations 259 (2015), 925{\textendash}963.}, number = {Journal of Differential Equations;volume 259; issue 3; pages 925-963;}, year = {2015}, note = {Work presented at the "Special Session 21" of the "10th AIMS Conference on Dynamical Systems, Differential Equations and Applications" (Madrid, July 7-11, 2014).}, publisher = {Elsevier}, abstract = {We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation u{\textquoteright}{\textquoteright}+f(x,u)=0. We allow x ↦ f(x,s) to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that f(x,s)/s is below λ_1 as s{\textrightarrow}0^+ and above λ_1 as s{\textrightarrow}+$\infty$. In particular, we can deal with the situation in which f(x,s) has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for u{\textquoteright}{\textquoteright} + a(x) g(u) = 0, where we prove the existence of 2^n-1 positive solutions when a(x) has n positive humps and a^-(x) is sufficiently large.

}, doi = {10.1016/j.jde.2015.02.032}, url = {http://urania.sissa.it/xmlui/handle/1963/35147}, author = {Guglielmo Feltrin and Fabio Zanolin} } @article {2014, title = {Maximal generalized solution of eikonal equation}, number = {Journal of differential equations;volume 257; issue 1; pages 231-263;}, year = {2014}, publisher = {Elsevier}, abstract = {We study the Dirichlet problem for the eikonal equation: 1/2 |∇u(x)|^2-a(x)=0 in Ω u(x)=(x) on Ω, without continuity assumptions on the map a(.). We find a class of maps a(.) contained in the space L$\infty$(Ω) for which the problem admits a (maximal) generalized solution, providing a generalization of the notion of viscosity solution.}, doi = {10.1016/j.jde.2014.04.001}, url = {http://urania.sissa.it/xmlui/handle/1963/34642}, author = {Sandro Zagatti} } @article {2014, title = {Minimal Liouville gravity correlation numbers from Douglas string equation}, number = {Journal of high energy physics;volume 2014; issue 1; article number 156;}, year = {2014}, publisher = {Springer}, abstract = {We continue the study of $(q,p)$ Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of \cite{Moore:1991ir}, \cite{Belavin:2008kv}, where Lee-Yang series $(2,2s+1)$ was studied, to $(3,3s+p_0)$ Minimal Liouville Gravity, where $p_0=1,2$. We demonstrate that there exist such coordinates $\tau_{m,n}$ on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates $\tau_{m,n}$ are related in a non-linear fashion to the natural coupling constants $\lambda_{m,n}$ of the perturbations of Minimal Lioville Gravity by the physical operators $O_{m,n}$. We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature \cite{Goulian:1990qr}, \cite{Zamolodchikov:2005sj}, \cite{Belavin:2006ex}.}, doi = {10.1007/JHEP01(2014)156}, url = {http://urania.sissa.it/xmlui/handle/1963/34588}, author = {Alexander Belavin and Boris Dubrovin and Baur Mukhametzhanov} } @article {2013, title = {A model for crack growth with branching and kinking}, journal = {Asymptotic Analysis}, volume = {89}, year = {2014}, pages = {63-110}, publisher = {SISSA}, type = {Research Article}, chapter = {63}, abstract = {We study an evolution model for fractured elastic materials in the 2-dimensional case, for which the crack path is not assumed to be known a priori. We introduce some general assumptions on the structure of the fracture sets suitable to remove the restrictions on the regularity of the crack sets and to allow for kinking and branching to develop. In addition we define the front of the fracture and its velocity. By means of a time-discretization approach, we prove the existence of a continuous-time evolution that satisfies an energy inequality and a stability criterion. The energy balance also takes into account the energy dissipated at the front of the fracture. The stability criterion is stated in the framework of Griffith{\textquoteright}s theory, in terms of the energy release rate, when the crack grows at least at one point of its front.

}, keywords = {quasistatic crack evolution, branching, kinking, Griffith\\\'s criterion}, doi = {10.3233/ASY-141233}, url = {https://content.iospress.com/articles/asymptotic-analysis/asy1233}, author = {Simone Racca} } @article {2014, title = {Model Order Reduction in Fluid Dynamics: Challenges and Perspectives}, number = {Modeling simulation and applications;volume 9; pages 235-273;}, year = {2014}, publisher = {Springer}, abstract = {This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities - which are mainly related either to nonlinear convection terms and/or some geometric variability - that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and-in the unsteady case - long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.}, doi = {10.1007/978-3-319-02090-7_9}, author = {Toni Lassila and Andrea Manzoni and Alfio Quarteroni and Gianluigi Rozza} } @article {2014, title = {A modular spectral triple for κ-Minkowski space}, number = {Journal of Geometry and Physics;volume 76; pages 136-157;}, year = {2014}, publisher = {Elsevier}, abstract = {We present a spectral triple for κ-Minkowski space in two dimensions. Starting from an algebra naturally associated to this space, a Hilbert space is built using a weight which is invariant under the κ-Poincar{\'e} algebra. The weight satisfies a KMS condition and its associated modular operator plays an important role in the construction. This forces us to introduce two ingredients which have a modular flavour: the first is a twisted commutator, used to obtain a boundedness condition for the Dirac operator, and the second is a weight replacing the usual operator trace, used to measure the growth of the resolvent of the Dirac operator. We show that, under some assumptions related to the symmetries and the classical limit, there is a unique Dirac operator and automorphism such that the twisted commutator is bounded. Then, using the weight mentioned above, we compute the spectral dimension associated to the spectral triple and find that is equal to the classical dimension. Finally we briefly discuss the introduction of a real structure.}, doi = {10.1016/j.geomphys.2013.10.023}, url = {http://urania.sissa.it/xmlui/handle/1963/34895}, author = {Marco Matassa} } @article {battaglia2013moser, title = {A Moser-Trudinger inequality for the singular Toda system}, journal = {Bull. Inst. Math. Acad. Sin.}, volume = {9}, number = {1}, year = {2014}, pages = {1{\textendash}23}, author = {Luca Battaglia and Andrea Malchiodi} } @article {Piazzalunga2014, title = {M-theory interpretation of the real topological string}, journal = {Journal of High Energy Physics}, volume = {2014}, number = {8}, year = {2014}, month = {Aug}, pages = {54}, abstract = {We describe the type IIA physical realization of the unoriented topological string introduced by Walcher, describe its M-theory lift, and show that it allows to compute the open and unoriented topological amplitude in terms of one-loop diagram of BPS M2-brane states. This confirms and allows to generalize the conjectured BPS integer expansion of the topological amplitude. The M-theory lift of the orientifold is freely acting on the M-theory circle, so that integer multiplicities are a weighted version of the (equivariant subsector of the) original closed oriented Gopakumar-Vafa invariants. The M-theory lift also provides new perspective on the topological tadpole cancellation conditions. We finally comment on the M-theory version of other unoriented topological strings, and clarify certain misidentifications in earlier discussions in the literature.

}, issn = {1029-8479}, doi = {10.1007/JHEP08(2014)054}, url = {https://doi.org/10.1007/JHEP08(2014)054}, author = {Nicol{\`o} Piazzalunga and Uranga, Angel M.} } @article {cacace_chambolle_desimone_fedeli_2013, title = {Macroscopic contact angle and liquid drops on rough solid surfaces via homogenization and numerical simulations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, volume = {47}, number = {3}, year = {2013}, pages = {837{\textendash}858}, publisher = {EDP Sciences}, doi = {10.1051/m2an/2012048}, author = {Cacace, S. and Antonin Chambolle and Antonio DeSimone and Livio Fedeli} } @article {2013, title = {Minimal partitions and image classification using a gradient-free perimeter approximation}, year = {2013}, institution = {SISSA}, abstract = {In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.}, keywords = {Image classification, deblurring, optimal partitions, perimeter approximation}, url = {http://hdl.handle.net/1963/6976}, author = {Samuel Amstutz and Nicolas Van Goethem and Antonio Andr{\'e} Novotny} } @mastersthesis {2013, title = {Minimality and stability results for a class of free-discontinuity and nonlocal isoperimetric problems}, year = {2013}, school = {SISSA}, keywords = {free-discontinuity problems}, author = {Marco Bonacini} } @article {2013, title = {Monads for framed sheaves on Hirzebruch surfaces}, number = {SISSA preprint;05/2014/mate}, year = {2013}, abstract = {We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.}, keywords = {Monads, framed sheaves, Hirzebruch surfaces}, author = {Claudio Bartocci and Ugo Bruzzo and Claudio L.S. Rava} } @article {Bianchini2013, title = {The Monge Problem for Distance Cost in Geodesic Spaces}, journal = {Communications in Mathematical Physics}, volume = {318}, number = {3}, year = {2013}, month = {Mar}, pages = {615{\textendash}673}, abstract = {We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dLis a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1-dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

}, issn = {1432-0916}, doi = {10.1007/s00220-013-1663-8}, url = {https://doi.org/10.1007/s00220-013-1663-8}, author = {Stefano Bianchini and Fabio Cavalletti} } @article {2013, title = {Multiplicity result for a nonhomogeneous Yamabe type equation involving the Kohn Laplacian}, journal = {Journal of Mathematical Analysis and Applications. Volume 399, Issue 1, 1 March 2013, Pages 333-339}, year = {2013}, note = {The article is composed of 13 pages and is recorded in PDF format}, publisher = {Elsevier}, keywords = {CR-Yamabe}, doi = {10.1016/j.jmaa.2012.10.014}, url = {http://hdl.handle.net/1963/7374}, author = {Ali Maalaoui and Vittorio Martino} } @proceedings {2012, title = {Mathematical and numerical modeling of liquid crystal elastomer phase transition and deformation}, year = {2012}, publisher = {Cambridge University Press}, abstract = {Liquid crystal (in particular, nematic) elastomers consist of cross-linked flexible polymer chains with embedded stiff rod molecules that allow them to behave as a rubber and a liquid crystal. Nematic elastomers are characterized by a phase transition from isotropic to nematic past a temperature threshold. They behave as rubber at high temperature and show nematic behavior below the temperature threshold. Such transition is reversible. While in the nematic phase, the rod molecules are aligned along the direction of the "nematic director". This molecular rearrangement induces a stretch in the polymer chains and hence macroscopic spontaneous deformations. The coupling between nematic order parameter and deformation gives rise to interesting phenomena with a potential for new interesting applications. In the biological field, the ability to considerably change their length makes them very promising as artificial muscles actuators. Their tunable optical properties make them suitable, for example, as lenses for new imaging systems. We present a mathematical model able to describe the behavior of nematic elastomers and numerical simulations reproducing such peculiar behavior. We use a geometrically linear version of the Warner and Terentjev model [1] and consider cooling experiments and stretching experiments in the direction perpendicular to the one of the director at cross-linking.}, keywords = {Artificial muscle}, isbn = {9781605113807}, doi = {10.1557/opl.2012.249}, url = {http://hdl.handle.net/1963/7020}, author = {Mariarita De Luca and Antonio DeSimone} } @article {2012, title = {Modeling and control of quantum systems: An introduction}, journal = {IEEE Transactions on Automatic Control. Volume 57, Issue 8, 2012, Article number6189035, Pages 1898-1917}, year = {2012}, publisher = {Institute of Electrical and Electronics Engineers}, abstract = {The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies}, doi = {10.1109/TAC.2012.2195830}, url = {http://hdl.handle.net/1963/6505}, author = {Claudio Altafini and Francesco Ticozzi} } @article {2011, title = {Moduli of symplectic instanton vector bundles of higher rank on projective space $\\mathbb{P}^3$}, journal = {Central European Journal of Mathematics 10, nr. 4 (2012) 1232}, number = {arXiv:1109.2292v1;}, year = {2012}, note = {14 pages}, publisher = {SISSA}, abstract = {Symplectic instanton vector bundles on the projective space $\\mathbb{P}^3$ constitute a natural generalization of mathematical instantons of rank 2. We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $\\mathbb{P}^3$ with $r\\ge2$ and second Chern class $n\\ge r,\\ n\\equiv r({\\rm mod}2)$. We give an explicit construction of an irreducible component $I^*_{n,r}$ of this space for each such value of $n$ and show that $I^*_{n,r}$ has the expected dimension $4n(r+1)-r(2r+1)$.}, doi = {10.2478/s11533-012-0062-2}, url = {http://hdl.handle.net/1963/4656}, author = {Ugo Bruzzo and Dimitri Markushevich and Alexander Tikhomirov} } @article {2012, title = {Moduli spaces of noncommutative instantons: gauging away noncommutative parameters}, journal = {Quarterly Journal of Mathematics (2012) 63 (1): 41-86}, number = {SISSA;62/2009/FM}, year = {2012}, publisher = {Oxford University Press}, abstract = {Using the theory of noncommutative geometry in a braided monoidal category, we improve upon a previous construction of noncommutative families of instantons of arbitrary charge on the deformed sphere S^4_\\\\theta. We formulate a notion of noncommutative parameter spaces for families of instantons and we explore what it means for such families to be gauge equivalent, as well as showing how to remove gauge parameters using a noncommutative quotient construction. Although the parameter spaces are a priori noncommutative, we show that one may always recover a classical parameter space by making an appropriate choice of gauge transformation.}, doi = {10.1093/qmath/haq036}, url = {http://hdl.handle.net/1963/3777}, author = {Simon Brain and Giovanni Landi} } @article {Cavalletti2012, title = {The Monge problem in Wiener space}, journal = {Calculus of Variations and Partial Differential Equations}, volume = {45}, number = {1}, year = {2012}, month = {Sep}, pages = {101{\textendash}124}, abstract = {We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.

}, issn = {1432-0835}, doi = {10.1007/s00526-011-0452-5}, url = {https://doi.org/10.1007/s00526-011-0452-5}, author = {Fabio Cavalletti} } @article {2011, title = {The matching property of infinitesimal isometries on elliptic surfaces and elasticity on thin shells}, journal = {Archive for Rational Mechanics and Analysis 200 (2011) 1023-1050}, number = {SISSA;75/2008/M}, year = {2011}, publisher = {Springer}, abstract = {Using the notion of Γ-convergence, we discuss the limiting behavior of the three-dimensional nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h β with 2 < β < 4. We establish that, for the given scaling regime, the limiting theory reduces to linear pure bending. Two major ingredients of the proofs are the density of smooth infinitesimal isometries in the space of W 2,2 first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.}, doi = {10.1007/s00205-010-0387-6}, url = {http://hdl.handle.net/1963/3392}, author = {Marta Lewicka and Maria Giovanna Mora and Mohammad Reza Pakzad} } @article {Fedeli2011, title = {Metastable equilibria of capillary drops on solid surfaces: a phase field approach}, journal = {Continuum Mechanics and Thermodynamics}, volume = {23}, number = {5}, year = {2011}, month = {Sep}, pages = {453{\textendash}471}, abstract = {We discuss a phase field model for the numerical simulation of metastable equilibria of capillary drops resting on rough solid surfaces and for the description of contact angle hysteresis phenomena in wetting. The model is able to reproduce observed transitions of drops on micropillars from Cassie{\textendash}Baxter to Wenzel states. When supplemented with a dissipation potential which describes energy losses due to frictional forces resisting the motion of the contact line, the model can describe metastable states such as drops in equilibrium on vertical glass plates. The reliability of the model is assessed by a detailed comparison of its predictions with experimental data on the maximal size of water drops that can stick on vertical glass plates which have undergone different surface treatments.

}, issn = {1432-0959}, doi = {10.1007/s00161-011-0189-6}, url = {https://doi.org/10.1007/s00161-011-0189-6}, author = {Livio Fedeli and Turco, Alessandro and Antonio DeSimone} } @article { ISI:000296627000002, title = {A MODEL FOR CRACK PROPAGATION BASED ON VISCOUS APPROXIMATION}, journal = {{MATHEMATICAL MODELS \& METHODS IN APPLIED SCIENCES}}, volume = {{21}}, number = {{10}}, year = {2011}, month = {{OCT}}, pages = {{2019-2047}}, publisher = {{WORLD SCIENTIFIC PUBL CO PTE LTD}}, type = {{Article}}, address = {{5 TOH TUCK LINK, SINGAPORE 596224, SINGAPORE}}, abstract = {{In the setting of antiplane linearized elasticity, we show the existence of quasistatic evolutions of cracks in brittle materials by using a vanishing viscosity approach, thus taking into account local minimization. The main feature of our model is that the path followed by the crack need not be prescribed a priori: indeed, it is found as the limit (in the sense of Hausdorff convergence) of curves obtained by an incremental procedure. The result is based on a continuity property for the energy release rate in a suitable class of admissible cracks.}

}, keywords = {Brittle fracture, Crack propagation, energy derivative, energy release rate, free-discontinuity problems, Griffith{\textquoteright}s criterion, local minimizers, stress intensity factor}, vanishing viscosity, {Variational models}, issn = {{0218-2025}}, doi = {{10.1142/S0218202511005647}}, author = {Giuliano Lazzaroni and Rodica Toader} } @article {2011, title = {Moduli of framed sheaves on projective surfaces}, journal = {Doc. Math. 16 (2011) 399-410}, number = {arXiv:0906.1436;}, year = {2011}, publisher = {Documenta Mathematica}, abstract = {We show that there exists a fine moduli space for torsion-free sheaves on a\\r\\nprojective surface, which have a \\\"good framing\\\" on a big and nef divisor. This\\r\\nmoduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of\\r\\nHuybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space, and discuss some examples on rational surfaces.}, url = {http://hdl.handle.net/1963/5126}, author = {Ugo Bruzzo and Dimitri Markushevich} } @conference {10.1007/978-1-4419-9554-4_10, title = {The Monge Problem in Geodesic Spaces}, booktitle = {Nonlinear Conservation Laws and Applications}, year = {2011}, pages = {217{\textendash}233}, publisher = {Springer US}, organization = {Springer US}, address = {Boston, MA}, abstract = {We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish non branching geodesic space. We show that we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce an assumption on the transport problem π which implies that the conditional probabilities of the first marginal on each geodesic are continuous. It is known that this regularity is sufficient for the construction of an optimal transport map.

}, isbn = {978-1-4419-9554-4}, author = {Stefano Bianchini and Fabio Cavalletti}, editor = {Alberto Bressan and Chen, Gui-Qiang G. and Marta Lewicka and Wang, Dehua} } @article {Mola2011, title = {Multi-physics modelling and sensitivity analysis of olympic rowing boat dynamics}, journal = {Sports Engineering}, volume = {14}, number = {2-4}, year = {2011}, month = {nov}, pages = {85{\textendash}94}, publisher = {Springer Nature}, doi = {10.1007/s12283-011-0075-2}, url = {https://doi.org/10.1007/s12283-011-0075-2}, author = {Andrea Mola and Mehdi Ghommem and Muhammad R. Hajj} } @article {de2011multiplicity, title = {Multiplicity of solutions for a mean field equation on compact surfaces}, journal = {Boll. Unione Mat. Ital.(9)}, volume = {4}, year = {2011}, pages = {245{\textendash}257}, abstract = {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 $(8\pi, 4\pi^2 )$ we show under some extra assumptions that, as conjectured in [9], the functional admits at least three saddle points other than a local minimum.

}, author = {Francesca De Marchis} } @article {2010, title = {Monotonicity, frustration, and ordered response: an analysis of the energy landscape of perturbed large-scale biological networks}, journal = {BMC Systems Biology 2010, 4:83}, year = {2010}, publisher = {BioMed Central}, abstract = {Background. \\nFor large-scale biological networks represented as signed graphs, the index of frustration measures how far a network is from a monotone system, i.e., how incoherently the system responds to perturbations.\\nResults. \\nIn this paper we find that the frustration is systematically lower in transcriptional networks (modeled at functional level) than in signaling and metabolic networks (modeled at stoichiometric level). A possible interpretation of this result is in terms of energetic cost of an interaction: an erroneous or contradictory transcriptional action costs much more than a signaling/metabolic error, and therefore must be avoided as much as possible. Averaging over all possible perturbations, however, we also find that unlike for transcriptional networks, in the signaling/metabolic networks the probability of finding the system in its least frustrated configuration tends to be high also in correspondence of a moderate energetic regime, meaning that, in spite of the higher frustration, these networks can achieve a globally ordered response to perturbations even for moderate values of the strength of the interactions. Furthermore, an analysis of the energy landscape shows that signaling and metabolic networks lack energetic barriers around their global optima, a property also favouring global order.\\nConclusion. \\nIn conclusion, transcriptional and signaling/metabolic networks appear to have systematic differences in both the index of frustration and the transition to global order. These differences are interpretable in terms of the different functions of the various classes of networks.}, doi = {10.1186/1752-0509-4-83}, url = {http://hdl.handle.net/1963/4055}, author = {Giovanni Iacono and Claudio Altafini} } @article {2010, title = {Moore-Read Fractional Quantum Hall wavefunctions and SU(2) quiver gauge theories}, number = {SISSA;14/2010/FM}, year = {2010}, abstract = {We identify Moore-Read wavefunctions, describing non-abelian statistics in fractional quantum Hall systems, with the instanton partition of N=2 superconformal quiver gauge theories at suitable values of masses and \\\\Omega-background parameters. This is obtained by extending to rational conformal field theories the SU(2) gauge quiver/Liouville field theory duality recently found by Alday-Gaiotto-Tachikawa. A direct link between the Moore-Read Hall $n$-body wavefunctions and Z_n-equivariant Donaldson polynomials is pointed out.}, url = {http://hdl.handle.net/1963/3852}, author = {Raoul Santachiara and Alessandro Tanzini} } @article {BertolaLeeMo, title = {Mesoscopic colonization in a spectral band}, journal = {J. Phys. A}, volume = {42}, number = {41}, year = {2009}, pages = {415204, 17}, issn = {1751-8113}, doi = {10.1088/1751-8113/42/41/415204}, url = {http://0-dx.doi.org.mercury.concordia.ca/10.1088/1751-8113/42/41/415204}, author = {Marco Bertola and Lee, S. Y. and Mo, M. Y.} } @article {fall2009minimal, title = {Minimal disc-type surfaces embedded in a perturbed cylinder}, journal = {Differential and Integral Equations}, volume = {22}, number = {11/12}, year = {2009}, pages = {1115{\textendash}1124}, publisher = {Khayyam Publishing, Inc.}, abstract = {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.

}, url = {https://projecteuclid.org/euclid.die/1356019407}, author = {Fall, Mouhamed Moustapha and Mercuri, Carlo} } @article {Formaggia2009, title = {A model for the dynamics of rowing boats}, journal = {International Journal for Numerical Methods in Fluids}, volume = {61}, number = {2}, year = {2009}, month = {sep}, pages = {119{\textendash}143}, publisher = {Wiley}, doi = {10.1002/fld.1940}, url = {https://doi.org/10.1002/fld.1940}, author = {L. Formaggia and Edie Miglio and Andrea Mola and Antonio Montano} } @article {2009, title = {A model for the orbifold Chow ring of weighted projective spaces}, journal = {Comm. Algebra 37 (2009) 503-514}, number = {arXiv.org;0709.4559v1}, year = {2009}, publisher = {Taylor and Francis}, abstract = {We construct an isomorphism of graded Frobenius algebras between the orbifold Chow ring of weighted projective spaces and graded algebras of groups of roots of the unity.}, doi = {10.1080/00927870802248902}, url = {http://hdl.handle.net/1963/3589}, author = {Samuel Boissiere and Etienne Mann and Fabio Perroni} } @article {Bertola:MomentTau, title = {Moment determinants as isomonodromic tau functions}, journal = {Nonlinearity}, volume = {22}, number = {1}, year = {2009}, pages = {29{\textendash}50}, author = {Marco Bertola} } @article {2009, title = {mRNA stability and the unfolding of gene expression in the long-period yeast metabolic cycle}, journal = {BMC Systems Biology (2009) 3:18}, year = {2009}, publisher = {BioMed Central}, abstract = {Background: In yeast, genome-wide periodic patterns associated with energy-metabolic oscillations have been shown recently for both short (approx. 40 min) and long (approx. 300 min) periods.\\nResults: The dynamical regulation due to mRNA stability is found to be an important aspect of the genome-wide coordination of the long-period yeast metabolic cycle. It is shown that for periodic genes, arranged in classes according either to expression profile or to function, the pulses of mRNA abundance have phase and width which are directly proportional to the corresponding turnover rates.\\nConclusion: The cascade of events occurring during the yeast metabolic cycle (and their correlation with mRNA turnover) reflects to a large extent the gene expression program observable in other dynamical contexts such as the response to stresses/stimuli.}, doi = {10.1186/1752-0509-3-18}, url = {http://hdl.handle.net/1963/3630}, author = {Nicola Soranzo and Mattia Zampieri and Lorenzo Farina and Claudio Altafini} } @article {2008, title = {Minimization of non quasiconvex functionals by integro-extremization method}, journal = {Discrete Contin. Dyn. Syst. 21 (2008) 625-641}, year = {2008}, doi = {10.3934/dcds.2008.21.625}, url = {http://hdl.handle.net/1963/2761}, author = {Sandro Zagatti} } @article {2008, title = {Minimizers of non convex scalar functionals and viscosity solutions of Hamilton-Jacobi equations}, journal = {Calc. Var. Partial Differential Equations 31 (2008) 511-519}, year = {2008}, doi = {10.1007/s00526-007-0124-7}, url = {http://hdl.handle.net/1963/2760}, author = {Sandro Zagatti} } @article {2008, title = {Morse theory and a scalar field equation on compact surfaces}, journal = {Adv. Differential Equations 13 (2008) 1109-1129}, year = {2008}, publisher = {Khayyam Publishing}, url = {http://hdl.handle.net/1963/3531}, author = {Andrea Malchiodi} } @article {2008, title = {Multiple bound states for the Schroedinger-Poisson problem}, journal = {Commun. Contemp. Math. 10 (2008) 391-404}, year = {2008}, doi = {10.1142/S021919970800282X}, url = {http://hdl.handle.net/1963/2679}, author = {Antonio Ambrosetti and David Ruiz} } @inbook {Bertola:MasslessdeSitter, title = {Massless scalar field in a two-dimensional de Sitter universe}, booktitle = {Rigorous quantum field theory}, series = {Progr. Math.}, volume = {251}, year = {2007}, pages = {27{\textendash}38}, publisher = {Birkh{\"a}user}, organization = {Birkh{\"a}user}, address = {Basel}, author = {Marco Bertola and Corbetta, Francesco and Moschella, Ugo} } @article {2007, title = {On the Maz\\\'ya inequalities: existence and multiplicity results for an elliptic problem involving cylindrical weights}, number = {SISSA;92/2007/M}, year = {2007}, url = {http://hdl.handle.net/1963/2522}, author = {Marita Gazzini and Roberta Musina} } @article {2007, title = {Metrics on semistable and numerically effective Higgs bundles}, journal = {J. Reine Angew. Math. 612 (2007) 59-79}, number = {SISSA;17/2006/FM}, year = {2007}, abstract = {We consider fibre metrics on Higgs vector bundles on compact K\\\\\\\"ahler manifolds, providing notions of numerical effectiveness and numerical flatness in terms of such metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, providing semistability criteria for Higgs bundles on projective manifolds of any dimension.}, doi = {10.1515/CRELLE.2007.084}, url = {http://hdl.handle.net/1963/1840}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2007, title = {Multi-bump solitons to linearly coupled systems of nonlinear Schr{\"o}dinger equations}, number = {SISSA;29/2006/M}, year = {2007}, doi = {10.1007/s00526-006-0079-0}, url = {http://hdl.handle.net/1963/1835}, author = {Antonio Ambrosetti and Eduardo Colorado and David Ruiz} } @article {2006, title = {Matching Procedure for the Sixth Painlev{\'e} Equation (May 2006)}, journal = {Journal of Physics A: Mathematical and General, Volume 39, Issue 39, 29 September 2006, Article numberS02, Pages 11973-12031}, number = {arXiv:1010.1952;}, year = {2006}, note = {This paper appeared in May 2006. I put it on the archive now, with more that four years of delay, for completeness sake. The paper is published in J.Phys.A: Math.Gen. 39 (2006), 11973-12031, with some modifications.}, publisher = {SISSA}, abstract = {We present a constructive procedure to obtain the critical behavior of\r\nPainleve\' VI transcendents and solve the connection problem. This procedure\r\nyields two and one parameter families of solutions, including trigonometric and\r\nlogarithmic behaviors, and three classes of solutions with Taylor expansion at\r\na critical point.}, doi = {doi:10.1088/0305-4470/39/39/S02}, url = {http://hdl.handle.net/1963/6524}, author = {Davide Guzzetti} } @article {2005, title = {Minimal surfaces in pseudohermitian geometry}, journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177.}, number = {arXiv:math/0401136;}, year = {2005}, publisher = {Scuola Normale Superiore}, abstract = {We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.}, doi = {10.2422/2036-2145.2005.1.05}, url = {http://hdl.handle.net/1963/4579}, author = {Jih-Hsin Cheng and JennFang Hwang and Andrea Malchiodi and Paul Yang} } @article {2005, title = {On the Minimum Problem for Nonconvex Scalar Functionals}, journal = {SIAM J. Math. Anal. 37 (2005) 982-995}, year = {2005}, abstract = {We study the minimum problem for scalar nonconvex functionals defined on Sobolev maps satisfying a Dirichlet boundary condition and refine well-known existence results under standard regularity assumptions.}, doi = {10.1137/040612506}, url = {http://hdl.handle.net/1963/2764}, author = {Sandro Zagatti} } @article {2005, title = {Modulation of the Camassa-Holm equation and reciprocal transformations}, journal = {Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834}, number = {SISSA;107/2004/FM}, year = {2005}, abstract = {We derive the modulation equations or Whitham equations for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.}, url = {http://hdl.handle.net/1963/2305}, author = {Simonetta Abenda and Tamara Grava} } @article {2005, title = {Multiple clustered layer solutions for semilinear Neumann problems on a ball}, journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163}, year = {2005}, publisher = {Elsevier}, doi = {10.1016/j.anihpc.2004.05.003}, url = {http://hdl.handle.net/1963/3532}, author = {Andrea Malchiodi and Wei-Ming Ni and Juncheng Wei} } @proceedings {2004, title = {On the minimal degree of a common Lyapunov function for planar switched systems}, year = {2004}, publisher = {IEEE}, abstract = {In this paper, we consider linear switched systems x(t) = Au(t)x(t), x ε Rn, u ε U, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a polynomial common Lyapunov function. Then our main result is that the degree of that the common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin.}, doi = {10.1109/CDC.2004.1428884}, url = {http://hdl.handle.net/1963/4834}, author = {Paolo Mason and Ugo Boscain and Yacine Chitour} } @article {2004, title = {Multidimensional boundary layers for a singularly perturbed Neumann problem}, journal = {Duke Math. J. 124 (2004) 105-143}, year = {2004}, publisher = {Duke University Press}, doi = {10.1215/S0012-7094-04-12414-5}, url = {http://hdl.handle.net/1963/2960}, author = {Andrea Malchiodi and Marcelo Montenegro} } @article {2004, title = {Multiplicity of periodic solutions of nonlinear wave equations}, journal = {Nonlinear Anal. 56 (2004) 1011-1046}, year = {2004}, publisher = {Elsevier}, doi = {10.1016/j.na.2003.11.001}, url = {http://hdl.handle.net/1963/2974}, author = {Massimiliano Berti and Philippe Bolle} } @article {Bertola:MixedJPA, title = {Mixed correlation functions of the two-matrix model}, journal = {J. Phys. A}, volume = {36}, number = {28}, year = {2003}, pages = {7733{\textendash}7750}, issn = {0305-4470}, author = {Marco Bertola and B. Eynard} } @article {2003, title = {Motion on submanifolds of noninvariant holonomic constraints for a kinematic control system evolving on a matrix Lie group}, journal = {Syst. Control Lett. 50 (2003) 241-250}, year = {2003}, publisher = {Elsevier}, abstract = {For a control system on a matrix Lie group with one or more configuration constraints that are not left/right invariant, finding the combinations of (kinematic) control inputs satisfying the motion constraints is not a trivial problem. Two methods, one coordinate-dependent and the other coordinate-free are suggested. The first is based on the Wei-Norman formula; the second on the calculation of the annihilator of the coadjoint action of the constraint one-form at each point of the group manifold. The results are applied to a control system on SE(3) with a holonomic inertial constraint involving the noncommutative part in a nontrivial way. The difference in terms of compactness of the result between the two methods is considerable.}, doi = {10.1016/S0167-6911(03)00168-3}, url = {http://hdl.handle.net/1963/3018}, author = {Claudio Altafini and Ruggero Frezza} } @article {2003, title = {Multi-instanton calculus and equivariant cohomology}, journal = {J.High Energy Phys. 2003,no.5,054,24 pp.}, number = {SISSA;75/2002/FM}, year = {2003}, publisher = {SISSA Library}, doi = {10.1088/1126-6708/2003/05/054}, url = {http://hdl.handle.net/1963/1645}, author = {Ugo Bruzzo and Jose F. Morales and Francesco Fucito and Alessandro Tanzini} } @article {2002, title = {A model for the quasi-static growth of a brittle fracture: existence and approximation results}, journal = {Math. Models Methods Appl. Sci., 12 (2002), no. 12, 1773}, number = {SISSA;115/00/M}, year = {2002}, publisher = {SISSA Library}, abstract = {We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.}, doi = {10.1142/S0218202502002331}, url = {http://hdl.handle.net/1963/1571}, author = {Gianni Dal Maso and Rodica Toader} } @article {2002, title = {A model for the quasi-static growth of brittle fractures based on local minimization}, journal = {Math.Models Methods Appl. Sci., 12 (2002) , p.1773-1800.}, number = {SISSA;51/2002/M}, year = {2002}, publisher = {SISSA Library}, abstract = {We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.}, doi = {10.1142/S0218202502002331}, url = {http://hdl.handle.net/1963/1621}, author = {Gianni Dal Maso and Rodica Toader} } @article {2002, title = {A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results}, journal = {Arch. Ration. Mech. Anal. 162 (2002) 101-135}, number = {SISSA;30/2001/M}, year = {2002}, publisher = {Springer}, abstract = {We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith\\\'s theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.}, doi = {10.1007/s002050100187}, url = {http://hdl.handle.net/1963/3056}, author = {Gianni Dal Maso and Rodica Toader} } @article {2002, title = {A multiplicity result for the Schrodinger-Maxwell equations with negative potential}, journal = {Ann. Pol. Math. 79 (2002) 21-30}, number = {SISSA;20/2001/M}, year = {2002}, publisher = {IMPAN}, abstract = {We prove the existence of a sequence of radial solutions with negative energy of the Schr{\"o}dinger-Maxwell equations under the action of a negative potential.}, url = {http://hdl.handle.net/1963/3053}, author = {Giuseppe Maria Coclite} } @article {2002, title = {Multiplicity results for the Yamabe problem on Sn}, journal = {Proceedings of the National Academy of Sciences of the United States of America. 2002 Nov; 99(24):15252-6}, number = {PMID:12399546;}, year = {2002}, publisher = {National Academy of Sciences}, abstract = {We discuss some results related to the existence of multiple solutions for the Yamabe problem.}, doi = {10.1073/pnas.222494199}, url = {http://hdl.handle.net/1963/5885}, author = {Antonio Ambrosetti} } @article {2001, title = {A monotonicity approach to nonlinear Dirichlet problems in perforated domains}, journal = {Adv. Math. Sci. Appl. 11 (2001) 721-751}, number = {SISSA;99/00/M}, year = {2001}, publisher = {SISSA Library}, abstract = {We study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation, which is satisfied by the weak limits of the solutions. The additional term in the limit problem depends only on the local behaviour of the holes, which can be expressed in terms of suitable nonlinear capacities associated with the monotone operator.}, url = {http://hdl.handle.net/1963/1555}, author = {Gianni Dal Maso and Igor V. Skrypnik} } @article {2001, title = {Morse properties for the minimum time function on 2-D manifolds}, journal = {J. Dynam. Control Systems 7 (2001), no. 3, 385--423}, number = {SISSA;84/00/M}, year = {2001}, publisher = {SISSA Library}, doi = {10.1023/A:1013190914234}, url = {http://hdl.handle.net/1963/1541}, author = {Ugo Boscain and Benedetto Piccoli} } @article {2001, title = {On the Multi-Instanton Measure for Super Yang-Mills Theories}, journal = {Nuclear Phys. B 611 (2001), no. 1-3, 205--226.}, number = {SISSA;74/00/FM}, year = {2001}, publisher = {SISSA Library}, doi = {10.1016/S0550-3213(01)00349-2}, url = {http://hdl.handle.net/1963/1531}, author = {Ugo Bruzzo and Francesco Fucito and Alessandro Tanzini and Gabriele Travaglini} } @article {2001, title = {Multiple positive solutions of some elliptic equations in \\\\bold R\\\\sp N}, journal = {Nonlinear Anal. 43 (2001) 159-172}, number = {SISSA;133/1998/M}, year = {2001}, publisher = {Elsevier}, doi = {10.1016/S0362-546X(99)00186-8}, url = {http://hdl.handle.net/1963/3094}, author = {Andrea Malchiodi} } @article {2001, title = {Multiplicity results for some nonlinear Schrodinger equations with potentials}, journal = {Arch. Ration. Mech. An., 2001, 159, 253}, number = {SISSA;108/00/AF}, year = {2001}, publisher = {SISSA Library}, doi = {10.1007/s002050100152}, url = {http://hdl.handle.net/1963/1564}, author = {Antonio Ambrosetti and Andrea Malchiodi and Simone Secchi} } @article {2000, title = {Minimization of functionals of the gradient by Baire{\textquoteright}s theorem}, journal = {SIAM J. Control Optim. 38 (2000) 384-399}, number = {SISSA;71/97/M}, year = {2000}, publisher = {SIAM}, abstract = {We give sufficient conditions for the existence of solutions of the minimum problem $$ {\mathcal{P}}_{u_0}: \qquad \hbox{Minimize}\quad \int_\Omega g(Du(x))dx, \quad u\in u_0 + W_0^{1,p}(\Omega,{\mathbb{R}}), $$ based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum $u_0\in W_0^{1,p}(\Omega,\mathbb{R})$.

}, doi = {10.1137/S0363012998335206}, url = {http://hdl.handle.net/1963/3511}, author = {Sandro Zagatti} } @article {2000, title = {Monodromy of certain Painlev{\'e}-VI transcendents and reflection groups}, journal = {Invent. Math. 141 (2000) 55-147}, number = {arXiv.org;math/9806056v1}, year = {2000}, publisher = {Springer}, abstract = {We study the global analytic properties of the solutions of a particular family of Painleve\\\' VI equations with the parameters $\\\\beta=\\\\gamma=0$, $\\\\delta={1\\\\over2}$ and $\\\\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve\\\' VI equation.}, doi = {10.1007/PL00005790}, url = {http://hdl.handle.net/1963/2882}, author = {Boris Dubrovin and Marta Mazzocco} } @inbook {1999, title = {The method of Poisson pairs in the theory of nonlinear PDEs}, booktitle = {Direct and inverse methods in nonlinear evolution equations : Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999 / Robert Conte ... ; Antonio M. Greco ed. - Berlin : Springer, 2003. - (Lecture Notes in Physics ; 632)}, number = {SISSA;135/99/FM}, year = {1999}, publisher = {Springer}, organization = {Springer}, abstract = {The aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known Korteweg-de Vries (KdV) equation.\\nIn these lectures we touch the following subjects:\\ni) the birth and the role of the method of Poisson pairs inside the theory of the KdV equation;\\nii) the theoretical basis of the method of Poisson pairs;\\niii) the Gel\\\'fand-Zakharevich theory of integrable systems on bi-Hamiltonian manifolds;\\niv) the Hamiltonian interpretation of the Sato picture of the KdV flows and of its linearization on an infinite-dimensional Grassmannian manifold.\\nv) the reduction technique(s) and its use to construct classes of solutions;\\nvi) the role of the technique of separation of variables in the study of the reduced systems;\\nvii) some relations intertwining the method of Poisson pairs with the method of Lax pairs.}, doi = {10.1007/b13714}, url = {http://hdl.handle.net/1963/1350}, author = {Gregorio Falqui and Franco Magri and Marco Pedroni} } @article {1999, title = {A multiplicity result for the Yamabe problem on $S\\\\sp n$}, journal = {J. Funct. Anal. 168 (1999), no. 2, 529-561}, number = {SISSA;50/99/M}, year = {1999}, publisher = {Elsevier}, abstract = {We prove a multiplicity result for the Yamabe problem on the manifold (S, g), where g is a perturbation of the standard metric g0 of Sn. Solutions are found by variational methods via an abstract perturbation result.}, doi = {10.1006/jfan.1999.3458}, url = {http://hdl.handle.net/1963/1264}, author = {Antonio Ambrosetti and Andrea Malchiodi} } @article {1998, title = {Mirror Symmetry on K3 Surfaces as a Hyper-K{\"a}hler Rotation}, journal = {Lett. Math. Phys. 45 (1998) 295-301}, number = {arXiv.org;physics/9802044v2}, year = {1998}, publisher = {Springer}, abstract = {We show that under the hypotheses of Strominger, Yau and Zaslow\\\'s paper, a mirror partner of a K3 surface $X$ with a fibration in special Lagrangian tori can be obtained by rotating the complex structure of $X$ within its hyperk\\\\\\\"ahler family of complex structures. The same hypotheses force the B-field to vanish.}, doi = {10.1023/A:1007446916202}, url = {http://hdl.handle.net/1963/2888}, author = {Ugo Bruzzo and Guido Sanguinetti} } @mastersthesis {1990, title = {Moduli Spaces and Geometrical Aspects of Two-Dimensional Conformal Field Theories}, year = {1990}, school = {SISSA}, keywords = {Algebraic curves}, url = {http://hdl.handle.net/1963/5552}, author = {Gregorio Falqui} } @article {1988, title = {Methods of stochastic stability and properties of the Gribov horizon in the stochastic quantization of gauge theories}, journal = {Stochastic processes, physics and geompetry (Ascona and Locarno, 1988), 302, World Sci.Publishing,NJ(1990)}, number = {SISSA;57/89/FM}, year = {1988}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/817}, author = {Gianfausto Dell{\textquoteright}Antonio} } @article {1985, title = {Maximal acceleration and Sakharov{\textquoteright}s limiting temperature}, journal = {Lett. Nuovo Cim. 42 (1985) 70-72}, number = {SISSA;69/84/EP}, year = {1985}, publisher = {Societ{\`a} Italiana di Fisica}, abstract = {It is shown that Sakharov{\textquoteright}s maximal temperature, derived by him from astrophysical considerations, is a straightforward consequence of the maximal acceleration introduced by us in previous works.

}, doi = {10.1007/BF02748306}, url = {http://hdl.handle.net/1963/372}, author = {Eduardo R. Caianiello and Giovanni Landi} }