On the asymptotics of the principal eigenvalue for a Robin problem with a large parameter in planar domains

Abstract

Let $Ω\subset \RR^2$ 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 $E(β)=-β^2- γ_\mxβ+ O\big(β^\{2}{3}\big)$ as $\beta \to +\infty$, where $γ_\mx$ is the maximal curvature of the boundary.

12 pages, contribution to the proceedings of the conference "Mathematical Challenge of Quantum Transport in Nanosystems" (Pierre Duclos Workshop), Saint-Petersburg, Russia, March 13-15, 2013

References in corpus (3)

• Strong coupling asymptotics for Schrödinger operators with an interaction supported by an open arc in three dimensions
• Strong coupling asymptotics for a singular Schroedinger operator with an interaction supported by an open arc
• Hardy inequalities for Robin Laplacians

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