Globally Lipschitz minimizers for variational problems with linear growth^★

¹ Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159 Augsburg, Germany
² Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Prague, Czech Republic

^* Corresponding author: lisa.beck@math.uni-augsburg.de

Received: 3 October 2016
Accepted: 5 October 2017

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 [J. Serrin, Philos. Trans. R. Soc. Lond., Ser. A 264 (1969) 413–496].