Elliptic differential operators on Lipschitz domains and abstract boundary value problems

TL;DR

This paper develops an abstract framework for elliptic boundary value problems on non-smooth Lipschitz domains, extending boundary maps and characterizing all self-adjoint Laplacian realizations with resolvent formulas and spectral analysis.

Contribution

It introduces a generalized notion of quasi boundary triples applicable to non-smooth domains and fully characterizes self-adjoint Laplacian extensions on Lipschitz domains.

Findings

Extended boundary maps to dual spaces for non-smooth domains

Provided Kren-type resolvent formulas for Laplacians

Characterized spectral properties via Dirichlet-to-Neumann maps

Abstract

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kre\u{\i}n type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results from Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples.