Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint

Abstract

In this paper we study qualitative geometric properties of optimal configurations to a variational problem with free boundary, under suitable assumptions on a fixed boundary. More specifically, we study the problem of minimizing the flow of heat given by ${\int }_{\partial D}\Gamma (u_{\mu })d\sigma$, where $D$ is a fixed domain and $u$ is the potential of a domain $\Omega \supset \partial D$, with a prescribed volume on $\Omega \setminus D$. Our main goal is to establish uniqueness and symmetry results when $\partial D$ has a given geometric property. Full regularity of the free boundary is obtained under these symmetry conditions imposed on the fixed boundary.