Regularity of the minimizers in the composite membrane problem in R^2
  We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator with potential equal to a fixed multiple of the characteristic function of a subset D of omega, with measure A). We show that for minimizers, the boundary of D is analytic. 
Sagun Chanillo, Carlos E.Kenig and Tung TO