An optimization problem with volume constrain for a degenerate quasilinear operator

TL;DR

This paper studies an optimization problem involving a degenerate quasilinear operator with a volume constraint, proving existence, regularity of solutions, and smoothness of the free boundary.

Contribution

It introduces a penalization approach for the volume constraint and establishes regularity and smoothness results for solutions and their free boundaries.

Findings

Solutions are locally Lipschitz continuous.

The free boundary is smooth.

Existence of solutions for small penalization parameters.

Abstract

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. 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.