A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains
A description of all self-adjoint extensions of the Laplacian and Kreĭn-type resolvent formulas on non-smooth domains
复制标题
非光滑域上拉普拉斯和 Kreĭn 型解析公式的所有自伴扩展的描述
DOI：
10.1007/s11854-011-0002-2
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
M. Mitrea
中科院分区：
文献类型：
--
作者：
F. Gesztesy;M. Mitrea
本文有两个主要目标。首先，我们关注L2（Ω； dnx）中拉普拉斯算子$$ - \Delta {|_{C_0^\infty (\Omega )}}$$的所有自伴随扩展的描述。这里，域Ω属于有界Lipschitz域的一个子类（我们称之为拟凸域），它包含所有的凸域以及所有C1，r类的域，对于r > 1/2。其次，建立了拟凸域上拉普拉斯算子各种自伴随扩展解的Kreĭn-type公式，研究了拉普拉斯算子边值问题的适定性以及相应的Weyl-Titchmarsh算子（或能量依赖的dirichlet - - - neumann映射）的基本性质。本文的一个重要创新是对传统意义上缺乏Sobolev正则但适合于拉普拉斯正则的函数的经典边界迹理论进行了扩展。
This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian $$ - \Delta {|_{C_0^\infty (\Omega )}}$$ in L2(Ω; dnx). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C1,r, for r > 1/2. Second, we establish Kreĭn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.