On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains
Nanosistemy: fizika, himiâ, matematika, Tome 4 (2013) no. 4, pp. 474-483
Cet article a éte moissonné depuis la source Math-Net.Ru
Let $\Omega\subset\mathbb{R}^2$ be a domain having a compact boundary $\Sigma$ which is Lipschitz and piecewise $C^4$ smooth, and let $\nu$ denote the inward unit normal vector on $\Sigma$. We study the principal eigenvalue $E(\beta)$ of the Laplacian in $\Omega$ with the Robin boundary conditions $\partial f/\partial\nu+\beta f=0$ on $\Sigma$, where $\beta$ is a positive number. Assuming that $\Sigma$ has no convex corners, we show the estimate $E(\beta)=-\beta^2-\gamma_{\max}\beta+O(\beta^{2/3})$ as $\beta\to+\infty$, where $\gamma_{\max}$ is the maximal curvature of the boundary.
Keywords:
eigenvalue, Laplacian, Robin boundary condition, curvature, asymptotics.
@article{NANO_2013_4_4_a2,
author = {Konstantin Pankrashkin},
title = {On the asymptotics of the principal eigenvalue for a {Robin} problem with a large parameter in planar domains},
journal = {Nanosistemy: fizika, himi\^a, matematika},
pages = {474--483},
year = {2013},
volume = {4},
number = {4},
language = {en},
url = {http://geodesic.mathdoc.fr/item/NANO_2013_4_4_a2/}
}
TY - JOUR AU - Konstantin Pankrashkin TI - On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains JO - Nanosistemy: fizika, himiâ, matematika PY - 2013 SP - 474 EP - 483 VL - 4 IS - 4 UR - http://geodesic.mathdoc.fr/item/NANO_2013_4_4_a2/ LA - en ID - NANO_2013_4_4_a2 ER -
%0 Journal Article %A Konstantin Pankrashkin %T On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains %J Nanosistemy: fizika, himiâ, matematika %D 2013 %P 474-483 %V 4 %N 4 %U http://geodesic.mathdoc.fr/item/NANO_2013_4_4_a2/ %G en %F NANO_2013_4_4_a2
Konstantin Pankrashkin. On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains. Nanosistemy: fizika, himiâ, matematika, Tome 4 (2013) no. 4, pp. 474-483. http://geodesic.mathdoc.fr/item/NANO_2013_4_4_a2/