与二次型有关的边界对
Boundary pairs associated with quadratic forms
摘要：
我们引入了一个变分形式的椭圆边值问题的抽象框架。给定希尔伯特空间中的非负二次型，边界对由有界算子（边界算子）和辅助希尔伯特空间（边界空间）组成，使得边界算子从二次型域有界到辅助希尔伯特空间。这些数据确定了一个诺依曼和狄利克雷算子，一个狄利克雷解和一个狄利克雷-诺依曼算子。我们想到的基本例子是一个有边界的带边流形，其中二次型是对平方导数的积分，边界映射是函数对流形边界（的子集）的限制。作为主要定理之一，我们导出了Neumann算子和Dirichlet算子的预解式之差与Dirichlet解和Dirichlet-to-Neumann算子的预解式之间的关系。从这一点，我们推导出一个点的谱特征是在谱的诺依曼算子的Dirichlet-to-Neumann算子的家庭。该关系也可以推广到罗宾型边界条件。我们确定的条件表示纯粹的边界对，这使我们能够将我们的概念与现有的概念，如边界三元组。我们用许多例子来说明这一理论，包括Jacobi算子，具有（非光滑）边界的空间上的Laplacian算子和Zaremba（混合边界条件）问题。
英文摘要：
We introduce an abstract framework for elliptic boundary value problems in a variational form. Given a non-negative quadratic form in a Hilbert space, a boundary pair consists of a bounded operator, the boundary operator, and an auxiliary Hilbert space, the boundary space, such that the boundary operator is bounded from the quadratic form domain into the auxiliary Hilbert space. These data determine a Neumann and Dirichlet operator, a Dirichlet solution and a Dirichlet-to-Neumann operator. The basic example we have in mind is a manifold with boundary, where the quadratic form is the integral over the squared derivative, and the boundary map is the restriction of a function to (a subset of) the boundary of the manifold. As one of the main theorems, we derive a resolvent formula relating the difference of the resolvents of the Neumann and Dirichlet operator with the Dirichlet solution and the Dirichlet-to-Neumann operator. From this, we deduce a spectral characterisation for a point being in the spectrum of the Neumann operator in terms of the family of Dirichlet-to-Neumann operators. The relation can be generalised also to Robin-type boundary conditions. We identify conditions expressed purely in terms of boundary pairs, which allow us to relate our concept with existing concepts such as boundary triples. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non-smooth) boundary and the Zaremba (mixed boundary conditions) problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
