Some results about Schiffer's conjectures

Abstract

We study two overdetermined problems in spectral theory, about the Laplace operator. These problems are known as Schiffer's conjectures and are related to the Pompeiu problem. We show the connection between these problems and the critical points of the functional eigenvalue with a volume constraint. We use this fact, together with the continuous Steiner symmetrization, to give another proof of Serrin's result for the first Dirichlet eigenvalue. In two dimensions and for a general simple eigenvalue, we obtain different integral identities and a new overdetermined boundary value problem.

Unique continuation property near a corner and its fluid-structure controllability consequences

“…The same method developed in the proof of Theorem 1.1 can be used to obtain some local unique continuation properties for the Laplace operator in domains with corners, which correspond in fact to a kind of local Schiffer's conjecture (see for instance [1]). …”

“…But, there is another characterization that admits a local version. Indeed, in the case of simple eigenvalues, we can rewrite the local Schiffer property of Neumann type as an extremal Neumann eigenvalue problem under volume constraint as in [1]. That is, if λ n is a simple Neumann eigenvalue of (1.17)-(1.18) associated to a normalized eigenvector w n in L 2 (Ω), then…”

Symmetry problem 1

“…Let g(x, y, k) := e ik|x−y| 4π|x−y| . If problem (1) has a solution then this solution is unique by the uniqueness of the solution to the Cauchy problem for elliptic equation (1). The solution to equation 1by Green's formula is:…”

“…Symmetry problems for PDE were studied in many publications by many authors, see, for example, [1]. In this paper a new method is given for a study of symmetry problems for PDE.…”

Symmetry problem

“…A similar problem (Schiffer's conjecture) is also unsolved (see also [4]): A similar problem (Schiffer's conjecture) is also unsolved (see also [4]):…”