Bounds and monotonicity for the generalized Robin problem

Abstract.

We consider the principal eigenvalue λ ^Ω₁ (α) corresponding to Δu = λ (α) u in $\Omega, \frac{\partial u}{\partial v} = \alpha u$ on ∂Ω, with α a fixed real, and $\Omega \subset {\mathcal{R}}^n$ a C ^0,1 bounded domain. If α > 0 and small, we derive bounds for λ ^Ω₁ (α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.