Uniform stability of the ball with respect to the first Dirichlet and Neumann $\infty -$eigenvalues

Abstract

In this note we analyze how perturbations of a ball ${\mathfrak{B}}_r\subset {\mathbb{R}}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty -$eigenvalues when a volume constraint $mathscr{L}^n(\Omega )={\mathcal{L}}^n({\mathfrak{B}}_r)$ is imposed. Our main result states that $\Omega$ is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume ${\mathfrak{B}}_r$. In fact, we show that, if $$|{\lambda }_{1,\infty }^D(\Omega )-{\lambda }_{1,\infty }^D({\mathfrak{B}}_r)|={\delta }_1\text{and}|{\lambda }_{1,\infty }^N(\Omega )-{\lambda }_{1,\infty }^N({\mathfrak{B}}_r)|={\delta }_2,$$ then there are two balls such that $${\mathfrak{B}}_{\frac{r}{{\delta }_{1}r+1}}\subset \Omega \subset {\mathfrak{B}}_{\frac{r+{\delta }_{2}r}{1-{\delta }_{2}r}}.$$ In addition, we also obtain a result concerning stability of the Dirichlet $\infty -$eigen-functions.

10 pages, 1 figure