Self-Adjoint Extensions of the Laplacian and Krein-Type Resolvent Formulas in Nonsmooth Domains
Fritz Gesztesy, Marius Mitrea · arXiv (Cornell University) · 2009
This paper has two main goals. First, we are concerned with the classification of self-adjoint extensions of the Laplacian − ∆ ˛ ˛ C ∞ 0 (Ω) in L2 (Ω; d n x). Here, the domain Ω 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 ∈ (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