Electron. J. Differential Equations, Vol. 2018 (2018), No. 07, pp. 1-9.

Uniform stability of the ball with respect to the first Dirichlet and Neumann infinity-eigenvalues

Joao Vitor da Silva, Julio D. Rossi, Ariel M. Salort

Abstract:
In this note we analyze how perturbations of a ball $ B_r \subset \mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $\infty$-eigenvalues when a volume constraint $\mathcal{L}^n(\Omega) = \mathcal{L}^n( B_r)$ 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 $B_r$. In fact, we show that, if
$$ |\lambda_{1,\infty}^D(\Omega) - \lambda_{1,\infty}^D( B_r)| = \delta_1 \quad \text{and} \quad |\lambda_{1,\infty}^N(\Omega) - \lambda_{1,\infty}^N( B_r)| = \delta_2, $$
then there are two balls such that
$$ B_{\frac{r}{\delta_1 r+1}} \subset \Omega \subset B_{\frac{r+\delta_2 r}{1-\delta_2 r}}. $$
In addition, we obtain a result concerning stability of the Dirichlet $\infty$-eigenfunctions.

Submitted September 9, 2017. Published January 6, 2018.
Math Subject Classifications: 35B27, 35J60, 35J70.
Key Words: Infinity-eigenvalues estimates; infinity-eigenvalue problem; approximation of domains.

Show me the PDF file (326 KB), TEX file for this article.

João Vitor da Silva
Departamento de Matem ática
FCEyN - Universidad de Buenos Aires, Argentina
email: jdasilva@dm.uba.ar
Julio D. Rossi
Departamento de Matem ática
FCEyN - Universidad de Buenos Aires, Argentina
email: jrossi@dm.uba.ar
Ariel M. Salort
Departamento de Matem ática
FCEyN - Universidad de Buenos Aires, Argentina
email: asalort@dm.uba.ar

Return to the EJDE web page