Isospectral convex domains in Euclidean space
The authors deal with the question whether the eigenvalue spectrum of the Laplace operator on a domain in the Euclidean plane determines the domain up to congruence. H. Urakawa constructed examples of isospectral domains in \(\mathbb{R}^n\) \((n\geq 4)\). Thus far, all examples of noncongruent isospectral domains in Euclidean space are nonconvex. Thus the following question has remained open: are there pairs of convex domains in Euclidean space which are isospectral but not congruent? In this paper, the authors have shown that one can modify slightly Urakawa's constructon to obtain a pair of convex domains in Euclidean \(n\)-space for \(n\geq 4\) which are noncongruent yet isospectral for both Dirichlet and Neumann boundary conditions. The proof of Neumann isospectrality uses very recent results of Hubert Pesce.
isospectral convex domains in \(\mathbb{R}^ n\) \((n\geq 4)\)
scientific article; zbMATH DE number 817722
scientific article