Electron. J. Diff. Eqns., Vol. 2006(2006), No. 111, pp. 1-9.

On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian

An Le

Abstract:
Let $\Lambda_p^p$ be the best Sobolev embedding constant of $W^{1,p}(\Omega )\hookrightarrow L^p(\partial\Omega)$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$. We prove that as $p \to \infty$ the sequence $\Lambda_p$ converges to a constant independent of the shape and the volume of $\Omega$, namely 1. Moreover, for any sequence of eigenfunctions $u_p$ (associated with $\Lambda_p$), normalized by $\| u_p \|_{L^\infty(\partial\Omega)}=1$, there is a subsequence converging to a limit function $u_\infty$ which satisfies, in the viscosity sense, an $\infty$-Laplacian equation with a boundary condition.

Submitted August 4, 2006. Published September 18, 2006.
Math Subject Classifications: 35J50, 35J55, 35J60, 35J65, 35P30.
Key Words: Nonlinear elliptic equations; eigenvalue problems; p-Laplacian; nonlinear boundary condition; Steklov problem; viscosity solutions.

Show me the PDF file (220K), TEX file, and other files for this article.

An Lê
Department of Mathematics and Statistics
Utah State University
Logan, Utah 84322, USA
email: anle@cc.usu.edu

Return to the EJDE web page