In this paper we establish an existence and regularity result for solutions to the problem minimize∫ Ω L(|∇ u(x)|),dx on u: u-u -0 in W1,1-0(Ω) for boundary data that are constant on each connected component of the boundary of Ω. The Lagrangean L belongs to a class that contains both extended valued Lagrangeans and Lagrangeans with linear growth. Regularity means that the solution tilde w is Lipschitz continuous and that, in addition, |L′(∇ ̃ w||-∞ is bounded. © 2008 Springer-Verlag.
Cellina, A., Vornicescu, M. (2009). On the existence of solutions to a special variational problem. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 35(2), 263-270 [10.1007/s00526-008-0211-4].
On the existence of solutions to a special variational problem
CELLINA, ARRIGO;
2009
Abstract
In this paper we establish an existence and regularity result for solutions to the problem minimize∫ Ω L(|∇ u(x)|),dx on u: u-u -0 in W1,1-0(Ω) for boundary data that are constant on each connected component of the boundary of Ω. The Lagrangean L belongs to a class that contains both extended valued Lagrangeans and Lagrangeans with linear growth. Regularity means that the solution tilde w is Lipschitz continuous and that, in addition, |L′(∇ ̃ w||-∞ is bounded. © 2008 Springer-Verlag.
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