Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions

TL;DR

This paper extends reverse Faber-Krahn and Szego-Weinberger inequalities to annular domains with Robin-Neumann boundary conditions, providing new bounds for eigenvalues and analyzing eigenfunction shapes, including counterexamples to existing conjectures.

Contribution

It generalizes known eigenvalue inequalities to all Robin boundary parameters and explores eigenfunction shapes, including counterexamples to the Payne nodal line conjecture.

Findings

Established reverse Faber-Krahn inequalities for all Robin parameters.

Provided Szeg"H{o}-Weinberger type estimates under geometric assumptions.

Discovered eigenfunction nonradiality and counterexamples to the Payne conjecture.

Abstract

Let $\tau_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator in a bounded domain $\Omega$ of the form $\Omega_{\text{out}} \setminus \overline{B_{\alpha}}$ under the Neumann boundary condition on $\partial \Omega_{\text{out}}$ and the Robin boundary condition with parameter $h \in (-\infty,+\infty]$ on the sphere $\partial B_\alpha$ of radius $\alpha>0$ centered at the origin, the limiting case $h=+\infty$ being understood as the Dirichlet boundary condition on $\partial B_\alpha$. In the case $h>0$, it is known that the first eigenvalue $\tau_1(\Omega)$ does not exceed $\tau_1(B_\beta \setminus \overline{B_\alpha})$, where $\beta>0$ is chosen such that $|\Omega| = |B_\beta \setminus \overline{B_\alpha}|$, which can be regarded as a reverse Faber-Krahn type inequality. We establish this result for any $h \in (-\infty,+\infty]$. Moreover, we provide related estimates for higher eigenvalues under additional geometric assumptions on $\Omega$, which can be seen as Szeg\H{o}-Weinberger type inequalities. A few counterexamples to the obtained inequalities for domains violating imposed geometric assumptions are given. As auxiliary information, we investigate shapes of eigenfunctions associated with several eigenvalues $\tau_{i}(B_\beta \setminus \overline{B_\alpha})$ and show that they are nonradial at least for all positive and all sufficiently negative $h$ when $i \in \{2,\ldots,N+2\}$. At the same time, we give numerical evidence that, in the planar case $N=2$, already second eigenfunctions can be radial for some $h<0$. The latter fact provides a simple counterexample to the Payne nodal line conjecture in the case of the mixed boundary conditions.