Alessandro Savo

The spectrum of the Laplacian, the Pompeiu problem and isoparametric foliations

Let D be a Euclidean domain homeomorphic to a ball. The classical Pompeiu problem is the following: assume that there exists a continuous function, globally defined on R^n, which integrates to zero on D and on all its congruent images: then D must be a ball.
This conjecture is open since almost a century, and is related to overdetermined PDE's via the so-called Schiffer conjecture. Scope of the talk is to discuss the Pompeiu problem on other Riemannian manifolds from the viewpoint of spectral geometry, in particular, showing that the Pompeiu conjecture fails on the sphere, and explain why with the help of the theory of isoparametric foliations.
(Joint work with Luigi Provenzano).