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

This paper has two main goals. First, we are concerned with the classification of self-adjoint extensions of the Laplacian $-\mathrm{\Delta }{\mid }_{C_0^{\mathrm{\infty }}(\mathrm{\Omega })}$ in $L^2(\mathrm{\Omega };d^{n}x)$. Here, the domain $\mathrm{\Omega }$ 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 $r\in (1\mathrm{/}2,1)$. 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.