Abstract

Second order elliptic equations on a domain in which both Dirichlet and Neumann conditions are prescribed at the boundary of the domains constitute a class of overdetermined problems. To deal with these problems, we are led to find two unknowns: solution and the domain. They appear in many physical questions such as fluid and solid mechanics. In addition, they appear when minimizing domain-dependent energy functionals such as Sobolev norms and eigenvalues. While a lot of progress is being made, there still remains challenging open problems, e.g. the Schiffer conjecture: which states that if a nontrivial eigenfunction of the Neumann eigenvalue problem, on a bounded domain, has a constant Dirichlet boundary condition then the domain must be a ball. In this talk, we provide an overview on recent results on overdetermined and discuss new results on the Schiffer problem on some manifolds.