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].
Accepted:
DOI: 10.1051/cocv/2017065
Keywords: Variational problems, linear growth, Lipschitz minimizers, non-convex domains
@article{COCV_2018__24_4_1395_0, author = {Beck, Lisa and Bul{\'\i}\v{c}ek, Miroslav and Maringov\'a, Erika}, title = {Globally {Lipschitz} minimizers for variational problems with linear growth}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1395--1413}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017065}, zbl = {1418.35179}, mrnumber = {3922433}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017065/} }
TY - JOUR AU - Beck, Lisa AU - Bulíček, Miroslav AU - Maringová, Erika TI - Globally Lipschitz minimizers for variational problems with linear growth JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1395 EP - 1413 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017065/ DO - 10.1051/cocv/2017065 LA - en ID - COCV_2018__24_4_1395_0 ER -
%0 Journal Article %A Beck, Lisa %A Bulíček, Miroslav %A Maringová, Erika %T Globally Lipschitz minimizers for variational problems with linear growth %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1395-1413 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017065/ %R 10.1051/cocv/2017065 %G en %F COCV_2018__24_4_1395_0
Beck, Lisa; Bulíček, Miroslav; Maringová, Erika. Globally Lipschitz minimizers for variational problems with linear growth. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1395-1413. doi : 10.1051/cocv/2017065. http://archive.numdam.org/articles/10.1051/cocv/2017065/
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