Meromorphic symbolic structures for boundary value problems on manifolds with edges

Abstract

Key words Boundary problems on corner manifolds, symbolic structures with asymptotics, kernel cut-off and holomorphic symbols MSC (2000) 35J70, 35S05, 58J40We investigate the ideal of Green and Mellin operators with asymptotics for a manifold with edge-corner singularities and boundary which belongs to the structure of parametrices of elliptic boundary value problems on a configuration with corners whose base manifolds have edges. IntroductionParametrices of elliptic boundary value problems for differential operators on a manifold with smooth boundary belong to a pseudo-differential calculus with a symbolic hierarchy (interior and boundary symbols) and with typical contributions from the boundary (Green, trace, and potential operators), cf. Boutet de Monvel [1]. Similar structures may be obtained for the case of manifolds with geometric singularities, e.g., conical points, edges, corners, etc., as they are natural in a number of applications. Such problems belong to the analysis of operators on stratified and non-compact spaces. However, as is known from the analysis for conical and edge-singularities, answers are far from being straightforward. This concerns, in particular, the regularity and asymptotics of solutions near the singularities, the nature of extra conditions along lower-dimensional skeleta (including topological obstructions for their existence), and the description of analogues of Green functions (Green operators). Another experience from the known scenario for conical singularities is that meromorphic operator functions operating on the base of corners play a crucial role, both as the conormal symbolic structure and for evaluating the asymptotics of solutions and relative indices under changing of weights. In the case of corner singularities, where the base itself has conical points or edges, this has to be combined with an edge symbolic calculus. These ingredients contribute to Green (plus Mellin) operators. The program of this paper is to characterize the corresponding algebra of Green (plus Mellin) operators for the case of boundary value problems on a manifold with edges and corners.The geometry near the corner points is that of a local cone where the base is a manifold with edges and boundary (that means, our configuration has edges with conical singularities and in addition a boundary). Similar structures for the case without boundary have been investigated in Schulze [26]. Other simpler special cases are manifolds with smooth edges and boundary; this is studied systematically in the monograph of Kapanadze and Schulze [9], motivated by applications in crack theory. In particular, in such models it is interesting to analyse the mechanism of how solutions to elliptic boundary value problems "acquire" asymptotics close to the singularities. By Kondratyev's work [10] this is a famous story for conical singularities with smooth base manifolds. Later on many other special cases have been studied, cf. the references in [26] or [9].For corner singularities the asymptotic information is complicated...

The Zaremba Problem with Singular Interfaces as a Corner Boundary Value Problem

“…⊕ H s−3,δ−3 (Z, C j + ) (4) by additional operators of trace and potential type T and K, respectively, and an operator Q in the cone algebra on Z (as a manifold with conical singularities), where H s,δ (Z, C j ) are weighted cone Sobolev spaces, cf. Definition 2.3 (iii) below, such that (4) is Fredholm up to a discrete set of exceptional corner weights δ, cf.…”

“…Kernel cut-off in corner degenerate situations for the case without boundary are studied in [7]. In the version of boundary value problems as is employed here the details may be found in [13] and [4]. The process in general is as follows.…”

“…The proof of Theorem 2.14 is based on the kernel cut-off technique combined with a Mellin operator convention as is developed in the present variant in [7], see also [4] and [13] for the case of boundary value problems.…”

“…Similar asymptotics should be established for the case with boundary, cf. [4]; this requires a refinement of the calculus with a corresponding characterisation of parametrices, together with the meromorphic structure of edge and corner conormal symbols.…”

Mellin and Green operators of the corner calculus

“…To be more specific, the basics of the edge algebra are given in Sect. 2 Background materials and other topics related to this paper can be found in the references [1,3,5,[8][9][10]12,25,27,28].…”