Resolvents of elliptic boundary problems on conic manifolds
Thomas Krainer
TL;DR
The paper extends resolvent analysis of elliptic operators from boundaryless conic manifolds to manifolds with boundary, addressing boundary value problems under parameter-dependent ellipticity conditions similar to the Shapiro–Lopatinsky condition; it proves the existence of sectors of minimal growth for realizations and provides a resolvent decomposition via a parametrix plus a finite-dimensional projection, i.e., $(A_T-\nabla)^{-1} = \mathcal{B}_T(\nabla) + (A_T-\nabla)^{-1}\Pi_T(\nabla)$. The authors develop a framework on compact conic manifolds with boundary, introducing totally characteristic ($b$) and cone ($c$) operators with invariant principal and boundary symbols to set up model problems. For cone operators, the analysis combines $c$-ellipticity with boundary conditions to yield Fredholm properties and index relations, using a model cone to link the manifold and model-domain behavior. A parametrix is constructed by combining Boutet de Monvel calculus away from the singularity with cone-edge analysis near the boundary, refined by generalized Green remainders to obtain a Fredholm inverse under model-spectral injectivity; under a sector-of-minimal-growth assumption for the model wedge problem, the same sector holds for the original operator, yielding the resolvent representation with a finite-dimensional projection.
Abstract
This paper is a continuation of the investigation of resolvents of elliptic operators on conic manifolds from math.AP/0410178 and math.AP/0410176 to the case of manifolds with boundary and realizations of operators under boundary conditions. We prove the existence of sectors of minimal growth for realizations of boundary value problems for cone operators under natural ellipticity conditions on the symbols. Special attention is devoted to the clarification of the analytic structure of the resolvent.