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

Let $Ω\subset \RR^2$ be a domain having a compact boundary $Σ$ which is Lipschitz and piecewise $C^4$ smooth, and let $ν$ denote the inward unit normal vector on $Σ$. We study the principal eigenvalue $E(β)$ of the Laplacian in $Ω$ with the Robin boundary conditions $\partial f/\partialν+βf=0$ on $Σ$, where $β$ is a positive number. Assuming that $Σ$ has no convex corners we show the estimate $E(β)=-β^2- γ_\mxβ+ O\big(β^\{2}{3}\big)$ as $β\to+\infty$, where $γ_\mx$ is the maximal curvature of the boundary.

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