Globally Lipschitz minimizers for variational problems with linear growth
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1395-1413.

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].

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017065
Classification : 35A01, 35B65, 35J70, 49N60
Mots clés : Variational problems, linear growth, Lipschitz minimizers, non-convex domains
Beck, Lisa 1 ; Bulíček, Miroslav 1 ; Maringová, Erika 1

1
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     title = {Globally {Lipschitz} minimizers for variational problems with linear growth},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1395--1413},
     publisher = {EDP-Sciences},
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Beck, Lisa; Bulíček, Miroslav; Maringová, Erika. Globally Lipschitz minimizers for variational problems with linear growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 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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