On the reverse Faber-Krahn inequalities

T. V. Anoop, K. Ashok Kumar

Published 2018-06-29Version 1

For $N\geq 2$, let $\Omega\subset \mathbb{R}^N$ be a $C^1$ smooth multiply connected bounded domain. Let $\Gamma_1$ be the outer boundary of $\Omega$ and $\Gamma_0, \gamma_1,\ldots \gamma_n$ be the boundaries of the interior holes. For $p\in (1,\infty)$, we consider the first eigenvalue $\nu_1(\Omega)$ of the $p$-Laplacian with mixed boundary conditions: $$-\Delta_p u= \nu |u|^{p-2}u \quad \text{in } \, \Omega;\qquad u=0 \text{ on } \Gamma_1;\qquad \frac{\partial{u}}{\partial{\eta}}=0 \text{ on } \Gamma_0\cup\left(\cup_{i=1}^n\gamma_i\right).$$ Let $\Omega^\#$ be the annulus centred at the origin having the same volume as $\Omega$ and the same outer surface measure as $\Gamma_1$. For $N=2,$ we show that $\nu_1$ satisfies the following reverse Faber-Krahn inequality: \begin{equation}\label{R-RFK} \nu_1(\Omega) \le \nu_1(\Omega^\#) . \end{equation} For $N\ge 3,$ we obtain the above inequality, provided $\Gamma_1$ is a sphere. In this case, we also show that if the equality holds, then $\Omega$ is a translation of $\Omega^\#$. An analogous result is proved for the case when the Dirichlet condition on $\Gamma_0$ and the Neumann condition on the rest of the boundaries.