An elliptic variational problem involving level surface area on Riemannian manifolds

Abstract

We study a variational problem involving a Dirichlet integral and the area of a level surface on arbitrary $n$-dimensional Riemannian manifolds. We prove optimal regularity results for minimizers and derive a jump condition along the level surface. We also obtain smoothness of the interface up to a small singular set of Hausdorff dimension less then or equal to $n-8$.