Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 111, pp. 1-9.

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

Author: An Le (Utah State Univ., Logan, Utah, USA)
	 
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.