Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (Q2849108)

From the authors' introduction: The aim of the present paper is to give an introduction to and an overview of the properties of quasi boundary triples, associated \(\gamma\)-fields and Weyl functions, and to demonstrate how conveniently this technique can be applied to boundary value and spectral problems for elliptic operators. For the general case, we refer the reader to [the authors, J. Funct. Anal. 243, No. 2, 536--565 (2007; Zbl 1132.47038)]. In Section 1.2, we start by recalling the notion of boundary triples and Weyl functions and collect some well-known properties of these objects. Furthermore, we show in examples how boundary triples can be applied to ordinary, as well as elliptic differential operators. The notion of quasi boundary triples, their\(\gamma\)-fields and Weyl functions is reviewed in Section 1.3. We also provide a full proof of Krein's formula, which is difficult to find in the literature in this form. Furthermore, we give some sufficient criteria for self-adjointness of the extensions of the underlying symmetric operator. In Section 1.4, the quasi boundary triple concept is then applied to the elliptic differential expression. We stress that the essential idea here is to use the Dirichlet and Neumann boundary mappings and to identify the corresponding Weyl function with the Dirichlet-to-Neumann map from the theory of elliptic differential equations. Furthermore, we compare and connect the quasi boundary triple and its Weyl function with the regularized (ordinary) boundary triple from Section 1.2 and the associated Weyl function.NEWLINENEWLINEFor the entire collection see [Zbl 1269.47001].