Some new ideas for a Schiffer’s conjecture

Abstract

We discuss a Schiffer's conjecture which is a symmetry problem for an overdetermined spectral p.d.e .. We show the connection between this problem and the critical points of the eigenvalue with a volume constraint as well as the Faber-Krahn inequality. We give two original proofs of these symmetry results in the case of the first eigenvalue. Keywords Domain derivative, first eigenvalue, Faber-Krahn inequality, continuous Steiner symmetrization This conjecture (SC)' has been more intensively studied than the previous one, because K. Malanowski et al. (eds.

On characterization of balls via solutions to the Helmholtz equation

“…In conclusion, we compare Theorem 2 with the result of Ramm [15,16], concerning the so-called refined Schiffer's conjecture; the latter is similar to the celebrated Serrin's theorem [17], but involves equation (3) instead of Poisson's. Berenstein [2], p. 143, investigated this conjecture for simply connected two-dimensional domains with smooth boundary (see also [14] for another approach), whereas the original Schiffer's conjecture described in [5] is discussed in the review [4].…”