Schiffer problem and isoparametric hypersurfaces

The Schiffer Problem as originally stated for Euclidean spaces (and later for some symmetric spaces) is the following: Given a bounded connected open set n with a regular boundary and such that the complement of its closure is connected, does the existence of a solution to the Overdetermined Neumann Problem (N) imply that n is a ball? The same question for the Overdetermined Dirichlet Problem (D). We consider the generalization of the Schiffer Problem to an arbitrary Riemannian manifold and also the possibility of replacing the condition on the domain to be a ball by more general condition: to have a homogeneous boundary (i.e., boundary, admitting a transitive group of isometries). We prove that if n has a homogeneous boundary, then (N) and (D) always admit solutions (in fact, for infinitely many eigenvalues), but the converse statement is not always true. We show that in a number of spaces (symmetric and non-symmetric), many domains such that their boundaries are isoparametric hypersurfaces have eigenfunctions for (N) and (D) but fail the Schiffer Conjecture or even its generalization. These ideas can be extended to other (essentially more complicated overdetermined boundary value problems, including higher order equations and ..