Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$begin{aligned} Q^Omega _alpha u=-Delta u, quad dfrac{partial u}{partial n}=alpha u text { on } partial Omega , end{aligned}$$

in a class of bounded smooth domains (Omega in mathbb {R}^nu ); here (n) is the outward unit normal and (alpha >0) is a constant. We show that for each (jin mathbb {N}) and (alpha rightarrow +infty ), the (j)th eigenvalue (E_j(Q^Omega _alpha )) has the asymptotics

$$begin{aligned} E_j(Q^Omega _alpha )=-alpha ^2 -(nu -1)H_mathrm {max}(Omega ),alpha +{mathcal O}(alpha ^{2/3}), end{aligned}$$

where (H_mathrm {max}(Omega )) is the maximum mean curvature at (partial Omega ). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of (H_mathrm {max}). In particular, we show that the ball is the strict minimizer of (H_mathrm {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 (jin mathbb {N}) we have (E_j(Q^B_alpha )>E_j(Q^Omega _alpha )) for large (alpha ). An open question concerning a larger class of domains is formulated.