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

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 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 ${\tau }_i(B_{\beta }\setminus \overline{B_{\alpha }})$ and show that they are nonradial at least for all positive and all sufficiently negative $h$ when $i\in \{2,\dots ,N+2\}$. 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.

36 pages, 6 figures