Annalisa Malusa

THE DISTANCE FUNCTION FROM THE BOUNDARY IN A MINKOWSKI SPACE

Let the space be endowed with a Minkowski structure (that is, is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class ), and let be the (asymmetric) distance associated to . Given an open domain of class , let be the Minkowski distance of a point from the boundary of . We prove that a suitable extension of to (which plays the role of a signed Minkowski distance to ) is of class in a tubular neighborhood of , and that is of class outside the cut locus of (that is, the closure of the set of points of nondifferentiability of in ). In addition, we prove that the cut locus of has Lebesgue measure zero, and that can be decomposed, up to this set of vanishing measure, into geodesics starting from and going into along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point outside the cut locus the pair , where denotes the (unique) projection of on , and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

Existence Results for Noncoerecive Variational Problems

The aim of this paper is to give an existence result for a class of one-dimensional, nonconvex, noncoercive problems in the calculus of variations. The main tools for the proof are an existence theorem in the convex case and the closure of the convex hull of the epigraph of functions strictly convex at infinity.

On a system of partial differential equations of Monge–Kantorovich type

We consider a system of PDEs of Monge–Kantorovich type arising from models in granular matter theory and in electrodynamics of hard superconductors. The existence of a solution of such system (in a regular open domain Ω⊂Rn), whose construction is based on an asymmetric Minkowski distance from the boundary of Ω, was already established in [G. Crasta, A. Malusa, The distance function from the boundary in a Minkowski space, Trans. Amer. Math. Soc., submitted for publication]. In this paper we prove that this solution is essentially unique. A fundamental tool in our analysis is a new regularity result for an elliptic nonlinear equation in divergence form, which is of some interest by itself.

Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

Given an elliptic operator~

A Variational Approach to the Macroscopic Electrodynamics of Anisotropic Hard Superconductors

L

A sharp uniqueness result for a class of variational problems solved by a distance function

on a bounded domain~

Existence and regularity results for relaxed Dirichlet problems with measure data

\Omega \subseteq {\bf R}^n

A nonhomogeneous boundary value problem in mass transfer theory

, and a positive Radon measure~

On the Finsler metrics obtained as limits of chessboard structures

\mu

Crystalline evolutions with rapidly oscillating forcing terms

on~