Globally Lipschitz minimizers for variational problems with linear growth

TL;DR

This paper investigates conditions under which Lipschitz continuous solutions exist for variational problems with linear growth, focusing on the role of radial symmetry and barrier functions to ensure solvability of boundary value problems.

Contribution

It establishes a necessary and sufficient condition on the integrand for the existence of Lipschitz solutions in radially symmetric cases, addressing a gap in the understanding of such variational problems.

Findings

Identifies conditions for solvability of the Dirichlet problem with Lipschitz solutions

Constructs barrier functions to prove existence under radial symmetry

Highlights limitations of standard methods due to lack of compactness

Abstract

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler--Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrin's paper [19].