Title Geometric partial differential equations / edited by Antonin Chambolle, Matteo Novaga, Enrico Valdinoci

Published Pisa : Edizioni Della Normale, 2013

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Description 1 online resource Series CRM series ; 15 CRM series in mathematical physics ; 15. Contents 880-01 On the structure of phase transition maps for three or more coexisting phases / Nicholas D. Alikakos -- The nonlinear multidomain model: a new formal asymptotic analysis / Stefano Amato, Giovanni Bellettini, Maurizio Paolini -- Existence and qualitative properties of isoperimetric sets in periodic media / Antonin Chambolle, Michael Goldman, Matteo Novaga -- Minimizing movements and level set approaches to nonlocal variational geometric flows / Antonin Chambolle, Massimiliano Morini, Marcello Ponsiglione -- Homogenization with oscillatory Neumann boundary data in general domain / Sunhi Choi, Inwon C. Kim -- The analysis of shock formation in 3-dimensional fluids / Demetrios Christodoulou -- Regularity of the extremal solutions for the Liouville system / Louis Dupaigne, Alberto Farina, Boyan Sirakov -- On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force / Mi-Ho Giga, Yoshikazu Giga, Atsushi Nakayasu -- On representation of boundary integrals involving the mean curvature for mean-convex domains / Yoshikazu Giga, Giovanni Pisante -- Boundary regularity for the Poisson equation in reifenberg-flat domains / Antoine Lemenant, Yannick Sire -- Limiting models in condensed matter Physics and gradient flows of 1-homogeneous functional / Matteo Novaga, Giandomenico Orlandi -- Maximally localized Wannier functions: existence and exponential localization / Adriano Pisante -- Flows by powers of centro-affine curvature / Alina Stancu 880-01/(S Machine generated contents note: On the structure of phase transition maps for three or more coexisting phases / Nicholas D. Alikakos -- 1. Introduction -- 2. basics for general potentials -- 3. Symmetric phase transition potentials Existence of equivariant connection maps -- 4. related Bernstein-type theorem -- Background -- 4.1. Minimizing partitions -- 4.2. Flat chains with coefficients in a group -- 4.3. Flat chains of top dimension -- 4.4. group of surface tension coefficients -- 4.5. Basics on minimizing chains -- 5. related Bernstein-type theorem -- Statements and proofs -- 6. hierarchical structure of equivariant connection maps -- References -- nonlinear multidomain model: a new formal asymptotic analysis / Maurizio Paolini -- 1. Introduction -- 2. Star-shaped combination of star bodies and of anisotropies -- 2.1. On the hessian of the combined anisotropy -- 3. bidomain model -- 4. nonlinear multidomain model -- 5. Formal asymptotics of the multidomain model -- 5.1. Outer expansion -- 5.2. Inner expansion -- References -- Existence and qualitative properties of isoperimetric sets in periodic media / Matteo Novaga -- 1. Introduction -- 2. Plane-like minimizers -- 3. Strict convexity and differentiability properties of the stable norm -- 4. Existence and asymptotic behavior of isoperimetric sets -- References -- Minimizing movements and level set approaches to nonlocal variational geometric flows / Marcello Ponsiglione -- 1. Introduction -- 2. Generalized perimeters and curvatures -- 2.1. Notion of generalized perimeter -- 2.2. Definition of the curvature -- 2.3. Assumptions on the curvature κ -- 2.4. Semi-continuous extensions -- 3. Examples of generalized curvatures -- 3.1. Euclidean perimeter -- 3.2. fractional perimeter -- 3.3. pre-Minkowski content -- 4. curvature flow: notion of viscosity solutions -- 4.1. curvature flow -- 4.2. Definition of the viscosity solutions -- 4.3. Existence of viscosity solutions -- 5. Variational curvature flows -- 5.1. time-discrete scheme -- 5.2. Level by level minimizing movements -- 6. Shrinking zebras -- References -- Homogenization with oscillatory Neumann boundary data in general domain / Inwon C. Kim -- 1. Introduction -- 1.1. Main results and the discussion of main ideas -- 2. problem in the strip domain -- 2.1. Preliminaries -- 2.2. problem in the strip domain -- 3. continuity of the homogenized slope -- 3.1. Description of the perturbation of boundary data and a sketch of the proof -- 4. Perturbed test function method -- References -- analysis of shock formation in 3-dimensional fluids / Demetrios Christodoulou -- References -- Regularity of the extremal solutions for the Liouville system / Boyan Sirakov -- References -- On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force / Atsushi Nakayasu -- 1. Introduction -- 2. Definitions of generalized solutions -- 2.1. Faceted functions -- 2.2. Nonlocal curvature with a nonuniform driving force -- 2.3. Admissible functions and definition of a generalized solution -- 3. Effective region and canonical modification -- 4. Perron type existence theorem -- 5. Existence theorem for periodic initial data -- References -- On representation of boundary integrals involving the mean curvature for mean-convex domains / Giovanni Pisante -- 1. Introduction -- 2. Preliminaries -- 3. measure -- div([ƒ](k)Δd) -- 4. Main result -- 5. anisotropic case -- References -- Boundary regularity for the Poisson equation in reifenberg-flat domains / Yannick Sire -- 1. monotonicity Lemma -- 2. elementary computation -- 3. Interior estimate -- 4. Boundary estimate -- 5. Global decay result -- 6. Conclusion and main result -- References -- Limiting models in condensed matter Physics and gradient flows of 1-homogeneous functionals / Giandomenico Orlandi -- 1. Introduction -- 1.1. Some examples -- 1.2. Gradient flows -- 1.3. Formulation for differential forms -- 2. Gradient flow of Jk -- 2.1. Dual formulation -- 2.2. Non local obstacle-type problems -- 2.3. Some properties of the gradient flow of Jn-1 -- 3. functional I1 -- 3.1. Asymptotics for the Gross-Pitaevskii model -- 3.2. Rotational symmetry and weighted TV minimization -- 3.3. Contact Curves and vortex curves -- References -- Maximally localized Wannier functions: existence and exponential localization / Adriano Pisante -- 1. Introduction -- 2. Wannier functions and Bloch bundles -- 3. Marzari-Vanderbilt localization functional -- 4. Regularity of minimizers and exponential localization -- 5. Harmonic maps into U(m) and Liouville theorem -- References -- Flows by powers of centro-affine curvature / Alina Stancu -- 1. Introduction -- 2. Gauss curvature bounds -- 3. Analysis of the singularity -- 4. Asymptotic behavior in dimension three N. Alikakos: On the structure of phase transition maps for three or more coexisting phases -- S. Amato, G. Bellettini, M. Paolini: The nonlinear multidomain model: a new formal asymptotic analysis -- A. Chambolle, M. Goldman, M. Novaga: Existence and qualitative properties of isoperimetric sets in periodic media -- A. Chambolle, M. Morini, M. Ponsiglione: Minimizing movements and level set approach to geometric flow of nonlocal perimeters -- S. Choi, I. Kim: Homogenization with oscillatory Neumann boundary data in general domain -- D. Christodoulou: The Analysis of Shock Formation in 3-Dimensional Fluids -- L. Dupaigne, A. Farina, B. Sirakov: Regularity of the extremal solutions for the Liouville system -- M.-H. Giga, Y. Giga, A. Nakayasu: On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force -- Y. Giga, G. Pisante: On representation of boundary integrals involving the mean curvature for mean-convex domains -- A. Lemenant, Y. Sire: Elliptic problem in nonsmooth domain, Reifenberg-flat domains, Regularity -- A. Pisante: Maximally localized wannier functions: existence and exponential localization -- A. Stancu: Flows by powers of centro-affine curvature Summary This book is the outcome of a conference held at the Centro De Giorgi of the Scuola Normale of Pisa in September 2012. The aim of the conference was to discuss recent results on nonlinear partial differential equations, and more specifically geometric evolutions and reaction-diffusion equations. Particular attention was paid to self-similar solutions, such as solitons and travelling waves, asymptotic behaviour, formation of singularities and qualitative properties of solutions. These problems arise in many models from Physics, Biology, Image Processing and Applied Mathematics in general, and have attracted a lot of attention in recent years Analysis calculus optimalisatie optimization wiskunde mathematics partial differential equations optimalisatiemethoden optimization methods Mathematics (General) Wiskunde (algemeen) Bibliography Includes bibliographical references Notes Print version record Subject Differential equations, Partial -- Congresses Differential equations, Nonlinear -- Congresses MATHEMATICS -- Calculus. MATHEMATICS -- Mathematical Analysis. Ecuaciones en derivadas parciales -- Congresos Ecuaciones diferenciales no lineales -- Congresos Differential equations, Nonlinear Differential equations, Partial Genre/Form proceedings (reports) Conference papers and proceedings Conference papers and proceedings. Actes de congrès. Form Electronic book Author Chambolle, Antonin, editor Novaga, Matteo, editor Valdinoci, Enrico, editor ISBN 9788876424731 8876424733

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