A Description of All Self-Adjoint Extensions of the Laplacian and Krein-Type Resolvent Formulas on Nonsmooth Domains

TL;DR

This paper classifies all self-adjoint extensions of the Laplacian on certain nonsmooth domains and derives Krein-type resolvent formulas, extending boundary trace theory to less regular function spaces.

Contribution

It provides a comprehensive classification of self-adjoint Laplacian extensions on quasi-convex domains and establishes Krein formulas, advancing spectral analysis on nonsmooth domains.

Findings

Classified all self-adjoint extensions of the Laplacian on quasi-convex domains.

Derived Krein-type resolvent formulas for these extensions.

Extended boundary trace theory to non-Sobolev regularity spaces.

Abstract

This paper has two main goals. First, we are concerned with the classification of self-adjoint extensions of the Laplacian $-\Delta\big|_{C^\infty_0(\Omega)}$ in $L^2(\Omega; d^n x)$. Here, the domain $\Omega$ belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), which contain all convex domains, as well as all domains of class $C^{1,r}$, for $r\in(1/2,1)$. Second, we establish Krein-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasi-convex domains and study the properties of the corresponding Weyl--Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant technical innovation in this paper is an extension of the classical boundary trace theory for functions in spaces which lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.