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2013 | Buch

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Geometric Partial Differential Equations proceedings

herausgegeben von: Antonin Chambolle, Matteo Novaga, Enrico Valdinoci

Verlag: Scuola Normale Superiore

Buchreihe : CRM Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

On the structure of phase transition maps for three or more coexisting phases

Abstract

We consider system (1.1) below. After stating certain general facts which do not depend on the structure of W, we focus on the phase transition case. For symmetric W’s the main issues have been resolved, and we summarize them here. Next we recall the De Giorgi conjecture in the scalar case, and we point to a Bernstein type theorem that appears appropriate for (1.1). Finally we state a result on the hierarchical structure of the equivariant solutions.

Nicholas D. Alikakos

The nonlinear multidomain model: a new formal asymptotic analysis

Abstract

We study the asymptotic analysis of a singularly perturbed weakly parabolic system of m-equations of anisotropic reaction-diffusion type. Our main result formally shows that solutions to the system approximate a geometric motion of a hypersurface by anisotropic mean curvature. The anisotropy, supposed to be uniformly convex, is explicit and turns out to be the dual of the star-shaped combination of the m original anisotropies.

Stefano Amato, Giovanni Bellettini, Maurizio Paolini

Existence and qualitative properties of isoperimetric sets in periodic media

Abstract

We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we investigate its strict convexity and differentiability properties.

Antonin Chambolle, Michael Goldman, Matteo Novaga

Minimizing movements and level set approaches to nonlocal variational geometric flows

Abstract

This contribution describes recent results on a variational approach for the geometric gradient flow of perimeter-like functionals, which include a class of non-local perimeters. In particular, the consistency of the variational approach with viscosity solutions of an appropriate level set equation is established.

Antonin Chambolle, Massimiliano Morini, Marcello Ponsiglione

Homogenization with oscillatory Neumann boundary data in general domain

Abstract

In this article we summarize recent progress on understanding averaging properties of fully nonlinear PDEs in bounded domains, when the boundary data is oscillatory. Our result on the Neumann problem is the nonlinear version of the classical result in [4] for divergence-form operators with co-normal boundary data. We also discuss the Dirichlet boundary problem.

Sunhi Choi, Inwon C. Kim

The analysis of shock formation in 3-dimensional fluids

Abstract

In this lecture I shall discuss the ideas of my monograph “The Formation of Shocks in 3-Dimensonal Fluids”. The monograph studies the relativistic Euler equations in 3 space dimensions for a perfect fluid with an arbitrary equation of state.

Demetrios Christodoulou

Regularity of the extremal solutions for the Liouville system

Abstract

We study the smoothness of the extremal solutions to the Liouville system.

Louis Dupaigne, Alberto Farina, Boyan Sirakov

On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force

Abstract

A general anisotropic curvature flow equation with singular interfacial energy and spatially inhomogeneous driving force is considered for a curve given by the graph of a periodic function. We prove that the initial value problem admits a unique global-in-time viscosity solution for a general periodic continuous initial datum. The notion of a viscosity solution used here is the same as proposed by Giga, Giga and Rybka, who established a comparison principle. We construct the global-in-time solution by careful adaptation of Perron’s method.

Mi-Ho Giga, Yoshikazu Giga, Atsushi Nakayasu

On representation of boundary integrals involving the mean curvature for mean-convex domains

Abstract

Given a mean-convex domain Ω it (ℝⁿ with boundary of class C ^2,1, we provide a representation formula for a boundary integral of the type

$$ \int_{\partial \Omega } {} f(k(x))dH^{n - 1} $$

where k ≥ 0 is the mean curvature of ∂Ω and f is non-increasing and sufficiently regular, in terms of volume integrals and defect measure on the ridge set.

Yoshikazu Giga, Giovanni Pisante

Boundary regularity for the Poisson equation in reifenberg-flat domains

Abstract

This paper is devoted to the investigation of the boundary regularity for the Poisson equation

$$ \left\{ {\begin{array}{*{20}c} { - \Delta u = f\quad in\Omega } \\ {u = 0\quad on\partial \Omega } \\ \end{array} } \right. $$

where f belongs to some L ^p (Ω) and Ω is a Reifenberg-flat domain of ℝ^N. More precisely, we prove that given an exponent α ∈ (0,1), there exists an ε > 0 such that the solution u to the previous system is locally Hölder continuous provided that Ω is (ε, r ₀)-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman’s monotonicity formula and Morrey-Campanato theorem.

Antoine Lemenant, Yannick Sire

Limiting models in condensed matter Physics and gradient flows of 1-homogeneous functional

Abstract

We survey some recent results on variational and evolution problems concerning a certain class of convex 1-homogeneous functionals for vector-valued maps related to models in phase transitions (Hele-Shaw), superconductivity (Ginzburg-Landau) and superfluidity (Gross-Ktaevskii). Minimizers and gradient flows of such functionals may be characterized as solutions of suitable non-local vectorial generalizations of the classical obstacle problem.

Matteo Novaga, Giandomenico Orlandi

Maximally localized Wannier functions: existence and exponential localization

Abstract

We describe recent results proved in [32] in collaboration with G.Panati, concerning a periodic Schrodinger operator and the maximally localized (composite) Wannier functions corresponding to a relevant family of its Bloch bands. More precisely, we discuss the minimization problem for the associated localization functional introduced in [22] and we review some rigorous results about the existence and exponential localization of its minimizers, in dimension d ≤ 3. The proof combines ideas and methods from the Calculus of Variations and the regularity theory for harmonic maps between Riemannian manifolds.

Adriano Pisante

Flows by powers of centro-affine curvature

Abstract

We consider compact, strictly convex, origin-symmetric, smooth hypersurfaces inℝⁿ⁺¹ shrinking with speed given by powers of their centro-affine curvature. We show that, as long as the support function of the evolving convex bodies is bounded from both sides, the centro-affine curvature is also bounded above and below. We prove that the flow’s singularity which appears when the support function goes to zero is a compact contained in a hyperplane of dimension (n − 1). This information is exploited in ℝ³ to show that these flows shrink any admissible surface to a point and that, up to SL(3) transformations, the rescaled images of the evolving surface converge, in the Hausdorff metric, to a ball.

Alina Stancu

Backmatter

Titel: Geometric Partial Differential Equations proceedings
herausgegeben von: Antonin Chambolle
Matteo Novaga
Enrico Valdinoci
Verlag: Scuola Normale Superiore
Electronic ISBN: 978-88-7642-473-1
Print ISBN: 978-88-7642-472-4
DOI: https://doi.org/10.1007/978-88-7642-473-1