Volume 24, Number 4, October–December 2018
|Page(s)||1395 - 1413|
|Published online||24 October 2018|
Globally Lipschitz minimizers for variational problems with linear growth★
Institut für Mathematik,
86159 Augsburg, Germany
2 Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Prague, Czech Republic
* Corresponding author: firstname.lastname@example.org
Accepted: 5 October 2017
We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space W1,1 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].
Mathematics Subject Classification: 35A01 / 35B65 / 35J70 / 49N60
Key words: Variational problems / linear growth / Lipschitz minimizers / non-convex domains
L. Beck is grateful for the support by M.O.P.S. of the Mathematical Institute at the University of Augsburg. M. Bulíček and E. Maringová were supported by the ERC-CZ project LL1202 financed by the Ministry of Education, Youth and Sports, Czech Republic. The support of the grant SVV-2016-260335 is also acknowledged. M. Bulíček is a member of the Nečas Center for Mathematical Modeling.
© EDP Sciences, SMAI 2018
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