Geodesic
On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian
Electronic journal of differential equations, Tome 2006 (2006)
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.
Classification : 35J50, 35J55, 35J60, 35J65, 35P30
Keywords: nonlinear elliptic equations, eigenvalue problems, p-Laplacian, nonlinear boundary condition, Steklov problem, viscosity solutions 
@article{EJDE_2006__2006__a26,
     author = {Le,  An},
     title = {On the first eigenvalue of the {Steklov} eigenvalue problem for the infinity {Laplacian}},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1128.35347},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a26/}
}
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Le,  An. On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian. Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a26/