Strictly Convex Steklov-Isospectral Plane Domains

Abstract

We construct pairs of noncongruent bounded Euclidean plane domains with identical Steklov spectra, including multiplicities. This gives a negative answer to the planar Steklov analogue of Kac's question ``Can one hear the shape of a drum?'' The domains are simply connected and strictly convex, have real-analytic boundaries, and may be chosen arbitrarily close to a disk in the $C^{\infty }$ topology.

arXiv comment13 pages. v2 contains minor revisions to the Introduction; this is the submitted version