The stability of some eigenvalue estimates
Let \(\Omega\) be a bounded convex domain in \(\mathbb{R}^ n\), and consider the Laplacian on \(L^ 2(\Omega)\) with Dirichlet boundary condition. Let also \(D\) denote a ball of \(\mathbb{R}^ n\) with same volume as \(\Omega\). The author first shows that if the first eigenvalue associated to \(\Omega\) is close enough to the one associated to \(D\), then \(\Omega\) is close to a ball in the sense of Hausdorff distance. Rather precise estimates are given, and the same result is also obtained if the quotient of the first two eigenvalues associated to \(\Omega\) is close enough to the corresponding quantity associated to \(D\). Finally, the first result is extended to nonconvex domains in two dimensions. The proofs are based on isoperimetric inequalities.
scientific article; zbMATH DE number 147455