An optimization problem with volume constrain in Orlicz spaces

TL;DR

This paper studies a volume-constrained optimization problem in Orlicz spaces, proving existence, regularity, and smoothness of solutions and free boundaries under general conditions on the function G.

Contribution

It introduces a penalization approach for volume constraints in Orlicz spaces and establishes regularity and smoothness results for solutions and free boundaries.

Findings

Solutions are locally Lipschitz continuous.

Free boundary is smooth.

Existence of solutions under penalization.

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.