Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint
Eduardo V. O. Teixeira · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2007
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.