In this post we would like to show the impossibility of having one certain lower bound for the first eigenvalue $\nu_1(\Omega)$ of the following problem:

\left\{ \begin{aligned} -\Delta u &= \nu u, &&x \in \Omega,\\ u&=0, &&x \in \Gamma_0,\\ \frac{\partial u}{\partial n}&=0, &&x \in \partial \Omega \setminus \Gamma_0. \end{aligned} \right.

Here $\Omega \subset \mathbb{R}^N$ is a bounded domain whose boundary $\partial \Omega$ consists of two connected components: the outer one - $\Gamma_0$, and the inner one - $\partial \Omega \setminus \Gamma_0$.

Let $\Omega^\#$ be the spherical shell (i.e., difference of two balls centred at the same point) of the same measure as $\Omega$ and such that the outer boundary of $\Omega^\#$ has the same $(N-1)$-measure as $\Gamma_0$.

It was proved by Payne & Weinberger [1] that in the planar case $N=2$, $\nu_1(\Omega)$ satisfies the following reversed Faber-Krahn inequality:

$\nu_1(\Omega) \leq \nu_1(\Omega^\#).$

This result was generalized to the $N$-dimensional case, as well as to the $p$-Laplacian version of the problem, by Anoop & Kumar [2], under the assumption that $\Gamma_0$ is a sphere.

In Section 4 of the same paper [2], it is asked whether there exists a universal constant $C>0$ such that the following lower bound for $\nu_1(\Omega)$ takes place:

$C \nu_1(\Omega^\#) \leq \nu_1(\Omega).$

Let us show that such $C$ does not exists, in general. To this end, we will construct a sequence of equimeasurable domains $\Omega_n$ with equimeasurable outer boundary, such that $\nu_1(\Omega_n) \to 0$.

We construct $\Omega_n$ as follows. Let the outer boundary $\Gamma_0$ of $\Omega_n$ be a sphere $\partial B$. Now we consider a smaller concentric spherical shell $\Theta_n$ with a cylindrical hole $H_n$ of radius $1/n$ in it, as depicted in the following figure:

Clearly, the boundary of $\Theta_n \setminus H_n$ is connected. Moreover, we assume that the inner boundary of $\Theta_n$ is of a fixed radius. We set $\Omega_n = B \setminus (\Theta_n \setminus H_n)$, and we assume the width of $\Theta_n$ to be such that $\Omega_n$ is of a constant measure with respect to $n$.

Note that $\nu_1(\Omega)$ has the following variational characterization:

$\nu_1(\Omega) = \inf\left\{ \frac{\int_\Omega |\nabla u|^2 \, dx}{\int_\Omega |u|^2 \, dx}: u \in W^{1,2}(\Omega) \setminus \{0\},~ u=0 ~\text{on}~ \Gamma_0 \right\}.$

Let us take the following admissible function for this minimization problem. We set $u_n = 0$ in the “outer” part of $\Omega_n$, and $u_n=C=const$ in the “inner” part of $\Omega_n$. In the cylinder $H_n$ we assume $u_n$ to be, e.g., a linear function like $u_n(x_1,\dots,x_N) = ax_1+b$, where $a$ and $b$ are adjusted in such a way that $u_n$ is continuous. See the figure below.

Thus, we get

$\nu_1(\Omega_n) \leq \frac{\int_{\Omega_n} |\nabla u_n|^2 \, dx}{\int_{\Omega_n} |u_n|^2 \, dx} = \frac{a^2 |H_n|}{\int_{H_n} |ax_1 + b|^2 \, dx + C^2 \times (\text{measure of the "inner" part of}~ \Omega_n)}.$

Clearly, the numerator goes to zero, while the denominator is bounded away from zero. Thus, the desired result follows.

Bibliography

1. Payne, L. E., & Weinberger, H. F. (1961). Some isoperimetric inequalities for membrane frequencies and torsional rigidity. Journal of Mathematical Analysis and Applications, 2(2), 210-216.

2. Anoop, T. V., & Kumar, K. A. (2020). On reverse Faber-Krahn inequalities. Journal of Mathematical Analysis and Applications, 485(1), 123766.  2