An elliptic variational problem involving level surface area on Riemannian manifolds
Eduardo V. O. Teixeira, Lei Zhang · Revista Matemática Iberoamericana · 2012
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 .