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

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.

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

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