6 problems

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain. Let $\mu_n(\Omega)$ be the Neumann eigenvalues of the Laplacian, enumerated with multiplicity, and define … With … wh…

Among planar simply connected domains with fixed Lebesgue measure, let $\mu_1(\Omega)$ denote the first nonzero eigenvalue of the classical Neumann Laplacian. Kornhauser–Stakgold c…

Let $D\subset\mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Let $\mu_2$ be the first nonzero eigenvalue of the Neumann Laplacian on $D$, and let $\varphi$ be an eigenfu…

Let $D\subset\mathbb{R}^d$ be a bounded, connected Euclidean domain with sufficiently regular boundary, and let $\varphi$ be an eigenfunction corresponding to the first nonzero eig…

Hyperbolic-metric dependence conjecture. The eigenvalues of the Neumann–Laplace operator depend on the hyperbolic metrics of plane domains.

Let $G$ be a compact subgroup of $O(n)$. A bounded, regular region $\Omega\subset\mathbb{R}^n$ is $G$-symmetric when it is invariant under the natural action of $G$. For an eigenva…