Boundary value problems in edge representation

Liu, Xiaochun; Schulze, Bert-Wolfgang

doi:10.1002/mana.200610504

Cited by 9 publications

(3 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More details on the relationship between standard Sobolev spaces and embedded submanifolds interpreted as edges may be found in [3], [12], and also in [6], with applications to mixed elliptic problems.…”

Section: Branching Asymptotics 21 Weighted Edge Spaces With Asymptoticsmentioning

confidence: 99%

Branching asymptotics on manifolds with edge

Schulze¹,

Volpato²

2010

J. Pseudo-Differ. Oper. Appl.

View full text Add to dashboard Cite

We study pseudo-differential operators on a wedge with continuous and variable discrete branching asymptotics. Our approach relies on the fact that such asymptotics for r → 0 in the image under the Mellin transform turn to poles of corresponding meromorphic functions, depending on the edge variable y. Those represent families of analytic functionals that reproduce the asymptotics as values on the holomorphic function r −z . Variable and branching means that we admit variable multiplicities of these poles. We show that the corresponding y-wise discrete asymptotics are preserved under the action of Green and Mellin operators of the edge calculus.

show abstract

Section: Branching Asymptotics 21 Weighted Edge Spaces With Asymptoticsmentioning

confidence: 99%

Branching asymptotics on manifolds with edge

Schulze¹,

Volpato²

2010

J. Pseudo-Differ. Oper. Appl.

View full text Add to dashboard Cite

show abstract

“…Let us also point out that here we refer to algebras of singular operators with a control of asymptotics of solutions to elliptic equations close to the singularities. More material and applications may be found in [4], [19], [13], [8], [17], [27], [26], [11], [28].…”

Section: Introductionmentioning

confidence: 99%

The Mellin-Edge Quantisation for Corner Operators

Schulze

Wei

2013

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold M with second order singularities. The typical ingredients come from the "most singular" stratum of M which is a second order edge where the infinite transversal cone has a base B that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over B. In this respect our result is formally analogous to a quantisation rule of [9] for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over B that tend to infinity.for any cut-off function ω. Similarly as (1.3) we have K 0,0 (X ∧ ) = r −n/2 L 2 (R + × X). Moreover, for every s, γ, e ∈ R we form the spaces K s,γ;e (X ∧ ) := r −e K s,γ (X ∧ ).For purposes below we introduce a family of isomorphisms κ δ : K s,γ (X ∧ ) → K s,γ (X ∧ ), (κ δ u)(r, x) := δ n+1 2 u(δr, x), where n := dim X. Below we will also employ subspaces K s,γ Θ (X ∧ ) ⊂ K s,γ (X ∧ ) of flatness −ϑ relative to γ for some −∞ ≤ ϑ < 0 and the half-open weight interval Θ = (ϑ, 0]. More precisely, we set K s,γ Θ (X ∧ ) = lim ←− ε>0 K s,γ−ϑ−ε (X ∧ ) in the respective Fréchet topology. In the case Θ = (−∞, 0] the space K s,γ Θ (X ∧ ) = K s,∞ (X ∧ ) is independent of γ.

show abstract

“…in simples cases, the technique of [7], or [15]. Large classes of explicit examples of elliptic boundary value problems on manifolds with singular geometries follow by applying a similar strategy as in Liu [16].…”

Section: Introductionmentioning

confidence: 99%

Meromorphic symbolic structures for boundary value problems on manifolds with edges

Donno

Schulze

2006

Mathematische Nachrichten

View full text Add to dashboard Cite

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...

show abstract

Boundary value problems in edge representation

Cited by 9 publications

References 28 publications

Branching asymptotics on manifolds with edge

Branching asymptotics on manifolds with edge

The Mellin-Edge Quantisation for Corner Operators

Meromorphic symbolic structures for boundary value problems on manifolds with edges

Contact Info

Product

Resources

About