Extension theory for elliptic partial differential operators with pseudodifferential methods (Q2849110)

The author of the paper under review presents a survey on the connection between general extension theories and the study of realizations of elliptic operators \(A=\sum_{ |\alpha |\leq m}a_{\alpha(x)}D^{\alpha}\) on smooth domains in \(\mathbb R^n\), \(n\geq 2\) (\(\alpha\in\mathbb N_0^n,|\alpha|=\alpha_1+\cdots+\alpha_n\), \(D^\alpha=D_1^{\alpha_1}\cdots D_n^{\alpha_n}\), \(D_j=-\frac{\imath\partial}{\partial x_j}\), \(m>0\)). After basic issues of elliptic boundary value problems, pseudodifferential operators [\textit{L. Boutet de Monvel}, Acta Math. 126, 11--51 (1971; Zbl 0206.39401)] as well as pseudodifferential value operators are considered. A general abstract extension theory is implemented for realizations of \(A\), resolvent formulas that can be obtained via general theory are observed. The lower boundedness question and the question of Weyl-type spectral asymptotics formulas for differences between resolvents are studied as applications of the theory of the pseudodifferential boundary problems. A detailed review of the literature with historical comments is included.NEWLINENEWLINEFor the entire collection see [Zbl 1269.47001].