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

Abstract

Let $τ_k(Ω)$ be the $k$-th eigenvalue of the Laplace operator in a bounded domain $Ω$ of the form $Ω_{\text{out}} \setminus \overline{B_α}$ under the Neumann boundary condition on $\partial Ω_{\text{out}}$ and the Robin boundary condition with parameter $h \in (-\infty,+\infty]$ on the sphere $\partial B_α$ of radius $α>0$ centered at the origin, the limiting case $h=+\infty$ being understood as the Dirichlet boundary condition on $\partial B_α$. In the case $h>0$, it is known that the first eigenvalue $τ_1(Ω)$ does not exceed $τ_1(B_β\setminus \overline{B_α})$, where $β>0$ is chosen such that $|Ω| = |B_β\setminus \overline{B_α}|$, 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 $Ω$, which can be seen as Szegő-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 $τ_{i}(B_β\setminus \overline{B_α})$ 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.