Geodesic
On the first eigenvalue of the Steklov eigenvalue problem for the infinity Laplacian
Electronic Journal of Differential Equations, Tome 2006 (2006).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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},
     publisher = {mathdoc},
     volume = {2006},
     year = {2006},
     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/