Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities

TL;DR

This paper characterizes optimal boundary operators for nonlocal Robin Laplacians on Lipschitz domains and extends eigenvalue inequalities between Robin and Dirichlet Laplacians, building on Filonov's approach.

Contribution

It introduces a class of boundary operators for self-adjoint Robin Laplacians and generalizes eigenvalue inequalities to nonlocal Robin cases on Lipschitz domains.

Findings

Characterization of boundary operators for Robin Laplacians.

Extension of Friedlander's eigenvalue inequalities.

Application of Filonov's approach to nonlocal Robin problems.

Abstract

The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems.