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

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.