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This paper has two main goals. First, we are concerned with theclassification 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 termquasi-convex domains), which contain all convex domains, as well as all domainsof class $C^{1,r}$, for $r\in(1/2,1)$. Second, we establish Krein-type formulasfor the resolvents of the various self-adjoint extensions of the Laplacian inquasi-convex domains and study the properties of the correspondingWeyl--Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps).  One significant technical innovation in this paper is an extension of theclassical boundary trace theory for functions in spaces which lack Sobolevregularity in a traditional sense, but are suitably adapted to the Laplacian.

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