This paper has two main goals. First, we are concerned with theclassification of self-adjoint extensions of the Laplacian$-Deltaig|_{C^infty_0(Omega)}$ 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 $rin(1/2,1)$. 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.
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机译：本文有两个主要目标。首先，我们关注Laplacian $- Delta big | _ {C ^ infty_0（ Omega）} $在$ L ^ 2（ Omega; d ^ n x）$中的自伴随扩展的分类。在此，域$ Omega $属于有界Lipschitz域的子类（我们称其为准凸域），其中包含所有凸域以及$ r的所有类$ C ^ {1，r} $ in（1 / 2,1）$。其次，我们为Laplacian拟凸域的各种自伴随扩展的解析子建立Krein型公式，并研究相应的Weyl-Titchmarsh算子的性质（或能量依赖的Dirichlet-Neumann映射）。本文的一项重大技术创新是经典边界迹线理论的扩展，适用于在传统意义上缺乏Sobolevregularity但适合于Laplacian的空间中的函数。
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