An optimization problem with volume constrain in Orlicz spaces

Abstract

We consider the optimization problem of minimizing ${\int }_{\Omega }G(|\nabla u|)dx$ in the class of functions $W^{1,G}(\Omega )$, with a constrain on the volume of $\{u>0\}$. The conditions on the function $G$ allow for a different behavior at 0 and at $\infty$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial \{u>0\}\cap \Omega$, is smooth.

References in corpus (1)

• An optimization problem with volume constrain for a degenerate quasilinear operator

Cited by in corpus (3)

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