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