The ∂ˉ-Robin Laplacian

Abstract

We study the family of operators {ℛ_a}_(a∈[0,+∞))associated to the Robin-type problems in a bounded domain Ω⊂R^2 {︄−Δu=f in Ω, 2ν¯∂_(z¯)u+au=0 on ∂Ω, and their dependency on the boundary parameter a as it moves along [0,+∞). In this regard, we study the convergence of such operators in a resolvent sense. We also describe the eigenvalues of such operators and show some of their properties, both for all fifixed aand as functions of the parameter a. As shall be seen in more detail in the paper [23], the eigenvalues of these operators characterize the positive eigenvalues of quantum dot Dirac operators.