Faber–Krahn-type inequality for the overline-∂ Robin Laplacian

Prove that for every bounded C^2 planar domain Ω and every disk D with |D|=|Ω|, if Ω is not a disk then the first eigenvalue μ_Ω(a) of the ∂-Robin Laplacian R_a, defined by R_a u=−Δu on L^2(Ω) with domain Dom(R_a)={u∈H^1(Ω): ∂_{ z}u∈H^1(Ω), 2 ν ∂_{ z}u + a u=0 on ∂Ω} and parameter a>0, satisfies μ_Ω(a)>μ_D(a) for all a>0.

Background

The authors establish a precise connection between the nonnegative spectrum of quantum dot Dirac operators and the spectrum of the ∂-Robin Laplacians R_a. Through this correspondence, the Dirac conjecture is equivalent to a Faber–Krahn-type conjecture for R_a.

This Robin-type problem asks whether the disk minimizes the first eigenvalue μ_Ω(a) among domains of fixed area, uniformly in the boundary parameter a>0, with strict inequality unless Ω is a disk.