Calculus of Variations and Geometric Measure Theory
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L. Beck - M. Bulíček - E. Maringová

Globally Lipschitz minimizers for variational problems with linear growth

created by beck on 09 Jan 2018


Accepted Paper

Inserted: 9 jan 2018
Last Updated: 9 jan 2018

Journal: ESAIM Control Optim. Calc. Var.
Year: 2018

ArXiv: 1609.07601 PDF


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.

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