Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions, Q(alpha)(Omega) u = -Delta u, partial derivative u/partial derivative n = alpha u on partial derivative Omega, in a class of bounded smooth domains Omega is an element of R-nu; here n is the outward unit normal and alpha > 0 is a constant. We show that for each j is an element of N and alpha -> +infinity, the jth eigenvalue E-j (Q(alpha)(Omega)) has the asymptotics E-j (Q(alpha)(Omega)) = -alpha(2) - (nu - 1)H-max(Omega) alpha + O(alpha(2/3)), where H-max(Omega) is the maximum mean curvature at partial derivative Omega. The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of H-max. In particular, we show that the ball is the strict minimizer of H-max among the smooth star- shaped domains of a given volume, which leads to the following result: if B is a ball and Omega is any other star- shaped smooth domain of the same volume, then for any fixed j is an element of N we have E-j (Q(alpha)(B)) > E-j (Q(alpha)(Omega)) for large alpha. An open question concerning a larger class of domains is formulated.