Asymptotics of Robin eigenvalues for non-isotropic peaks

Abstract

Let Ω⊂R3 be an open set such thatΩ∩(−δ,δ)3={(x1,x2,x3)∈R2×(0,δ):(x1x3p,x2x3q)∈(−1,1)2}⊂R3,Ω∖[−δ,δ]3 is a bounded Lipschitz domain, for some δ>0 and 1<p<q<2. If a set satisfies the first condition one says that it has a non-isotropic peak at 0. Now consider the operator QΩα acting as the Laplacian u↦−Δu on Ω with the Robin boundary condition ∂νu=αu on ∂Ω, where ∂ν is the outward normal derivative. We are interested in the strong coupling asymptotics of QΩα. We prove that for large α the jth eigenvalue Ej(QΩα) behaves as Ej(QΩα)≈Ajα22−q, where the constants Aj<0 are eigenvalues of a one dimensional Schrödinger operator which depends on p and q.