Geodesic
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.

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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},
     publisher = {mathdoc},
     volume = {4},
     number = {4},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/NANO_2013_4_4_a2/}
}
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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/