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.