Publication at Faculty of Mathematics and Physics |
2018
Abstract
We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space W-1,W-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.