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 , where is a fixed domain and is the potential of a domain , with a prescribed volume on . Our main goal is to establish uniqueness and symmetry results when has a given geometric property. Full regularity of the free boundary is obtained under these symmetry conditions imposed on the fixed boundary.
Cite this article
Eduardo V. Teixeira, Uniqueness, symmetry and full regularity of free boundary in optimization problems with volume constraint. Interfaces Free Bound. 9 (2007), no. 1, pp. 133–148DOI 10.4171/IFB/159