TY - INPR A1 - Calvo, D. A1 - Schulze, Bert-Wolfgang T1 - Edge symbolic structures of second generation N2 - Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions. T3 - Preprint - (2005) 18 KW - Operators on manifolds with second order singularities KW - edge quantizations KW - continuity in Sobolev spaces with double weights Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29940 ER - TY - INPR A1 - Harutjunjan, G. A1 - Schulze, Bert-Wolfgang T1 - Conormal symbols of mixed elliptic problems with singular interfaces N2 - Mixed elliptic problems are characterised by conditions that have a discontinuity on an interface of the boundary of codimension 1. The case of a smooth interface is treated in [3]; the investigation there refers to additional interface conditions and parametrices in standard Sobolev spaces. The present paper studies a necessary structure for the case of interfaces with conical singularities, namely, corner conormal symbols of such operators. These may be interpreted as families of mixed elliptic problems on a manifold with smooth interface. We mainly focus on second order operators and additional interface conditions that are holomorphic in an extra parameter. In particular, for the case of the Zaremba problem we explicitly obtain the number of potential conditions in this context. The inverses of conormal symbols are meromorphic families of pseudo-differential mixed problems referring to a smooth interface. Pointwise they can be computed along the lines [3]. T3 - Preprint - (2005) 12 KW - Corner boundary value problems KW - mixed elliptic problems KW - interfaces with conical singularities KW - Zaremba problem Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29885 ER - TY - INPR A1 - Xiaochun, Liu A1 - Schulze, Bert-Wolfgang T1 - Boundary value problems in edge representation N2 - Edge representations of operators on closed manifolds are known to induce large classes of operators that are elliptic on specific manifolds with edges, cf. [9]. We apply this idea to the case of boundary value problems. We establish a correspondence between standard ellipticity and ellipticity with respect to the principal symbolic hierarchy of the edge algebra of boundary value problems, where an embedded submanifold on the boundary plays the role of an edge. We first consider the case that the weight is equal to the smoothness and calculate the dimensions of kernels and cokernels of the associated principal edge symbols. Then we pass to elliptic edge operators for arbitrary weights and construct the additional edge conditions by applying relative index results for conormal symbols. T3 - Preprint - (2004) 14 KW - Boundary value problems KW - edge singularities KW - ellipticity in the edge calculus Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26746 ER - TY - INPR A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - The Zaremba problem with singular interfaces as a corner boundary value problem N2 - We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y , i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T− Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z \ {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions. T3 - Preprint - (2004) 26 KW - Zaremba problem KW - corner Sobolev spaces with double weights KW - pseudodifferential boundary value problems Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26855 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Differential operators on manifolds with singularities : analysis and topology : Chapter 6: Elliptic theory on manifolds with edges N2 - Contents: Chapter 6: Elliptic Theory on Manifolds with Edges Introduction 6.1. Motivation and Main Constructions 6.1.1. Manifolds with edges 6.1.2. Edge-degenerate differential operators 6.1.3. Symbols 6.1.4. Elliptic problems 6.2. Pseudodifferential Operators 6.2.1. Edge symbols 6.2.2. Pseudodifferential operators 6.2.3. Quantization 6.3. Elliptic Morphisms and the Finiteness Theorem 6.3.1. Matrix Green operators 6.3.2. General morphisms 6.3.3. Ellipticity, Fredholm property, and smoothness Appendix A. Fiber Bundles and Direct Integrals A.1. Local theory A.2. Globalization A.3. Versions of the Definition of the Norm T3 - Preprint - (2004) 15 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26757 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Volpato, A. T1 - Green operators in the edge calculus N2 - Green operators on manifolds with edges are known to be an ingredient of parametrices of elliptic (edge-degenerate) operators. They play a similar role as corresponding operators in boundary value problems. Close to edge singularities the Green operators have a very complex asymptotic behaviour. We give a new characterisation of Green edge symbols in terms of kernels with discrete and continuous asymptotics in the axial variable of local model cones. T3 - Preprint - (2004) 25 KW - operators on manifolds with edges KW - weighted spaces with asymptotics KW - Green and Mellin edge operators Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26846 ER - TY - INPR A1 - Jaiani, George A1 - Schulze, Bert-Wolfgang T1 - Some degenerate elliptic systems and applications to cusped plates N2 - The tension-compression vibration of an elastic cusped plate is studied under all the reasonable boundary conditions at the cusped edge, while at the noncusped edge displacements and at the upper and lower faces of the plate stresses are given. T3 - Preprint - (2004) 27 KW - Casped plates KW - vibration KW - degenerate elliptic systems KW - weighted spaces KW - Hardy‘s inequality KW - Korn’s weighted inequality Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26866 ER - TY - INPR A1 - Dines, Nicoleta A1 - Liu, X. A1 - Schulze, Bert-Wolfgang T1 - Edge quantisation of elliptic operators N2 - The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols. T3 - Preprint - (2004) 24 KW - Boundary value problems KW - edge singularities KW - ellipticity KW - spectral flow Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26838 ER - TY - INPR A1 - Egorov, Jurij V. A1 - Kondratiev, V. A. A1 - Schulze, Bert-Wolfgang T1 - On the completeness of root functions of elliptic boundary problems in a domain with conical points on the boundary N2 - Contents: 1 Introduction 2 Definitions 3 Rays of minimal growth 4 Proof of Theorem 2. 5 The growth of the resolvent 6 Proof of Theorem 3. 7 The completeness of root functions 8 Some generalizations T3 - Preprint - (2004) 17 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26773 ER - TY - INPR A1 - Nazaikinskii, Vladimir E. A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - On the homotopy classification of elliptic operators on manifolds with edges N2 - We obtain a stable homotopy classification of elliptic operators on manifolds with edges. T3 - Preprint - (2004) 16 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26769 ER - TY - INPR A1 - Krainer, Thomas A1 - Schulze, Bert-Wolfgang T1 - The conormal symbolic structure of corner boundary value problems N2 - Ellipticity of operators on manifolds with conical singularities or parabolicity on space-time cylinders are known to be linked to parameter-dependent operators (conormal symbols) on a corresponding base manifold. We introduce the conormal symbolic structure for the case of corner manifolds, where the base itself is a manifold with edges and boundary. The specific nature of parameter-dependence requires a systematic approach in terms of meromorphic functions with values in edge-boundary value problems. We develop here a corresponding calculus, and we construct inverses of elliptic elements. T3 - Preprint - (2004) 01 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26662 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Differential operators on manifolds with singularities : analysis and topology : Chapter 7: The index problem on manifolds with singularities N2 - Contents: Chapter 7: The Index Problemon Manifolds with Singularities Preface 7.1. The Simplest Index Formulas 7.1.1. General properties of the index 7.1.2. The index of invariant operators on the cylinder 7.1.3. Relative index formulas 7.1.4. The index of general operators on the cylinder 7.1.5. The index of operators of the form 1 + G with a Green operator G 7.1.6. The index of operators of the form 1 + G on manifolds with edges 7.1.7. The index on bundles with smooth base and fiber having conical points 7.2. The Index Problem for Manifolds with Isolated Singularities 7.2.1. Statement of the index splitting problem 7.2.2. The obstruction to the index splitting 7.2.3. Computation of the obstruction in topological terms 7.2.4. Examples. Operators with symmetries 7.3. The Index Problem for Manifolds with Edges 7.3.1. The index excision property 7.3.2. The obstruction to the index splitting 7.4. Bibliographical Remarks T3 - Preprint - (2004) 06 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26700 ER - TY - INPR A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Boundary problems with meromorphic symbols in cylindrical domains N2 - We show relative index formulas for boundary value problems in cylindrical domains and Sobolev spaces with different weigths at ±∞. The amplitude functions are meromorphic in the axial covariable and take values in the space of boundary value problems on the cross section of the cylinder. T3 - Preprint - (2004) 12 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26735 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Differential operators on manifolds with singularities : analysis and topology : Chapter 3: Eta invariant and the spectral flow N2 - Contents: Chapter 3: Eta Invariant and the Spectral Flow 3.1. Introduction 3.2. The Classical Spectral Flow 3.2.1. Definition and main properties 3.2.2. The spectral flow formula for periodic families 3.3. The Atiyah–Patodi–Singer Eta Invariant 3.3.1. Definition of the eta invariant 3.3.2. Variation under deformations of the operator 3.3.3. Homotopy invariance. Examples 3.4. The Eta Invariant of Families with Parameter (Melrose’s Theory) 3.4.1. A trace on the algebra of parameter-dependent operators 3.4.2. Definition of the Melrose eta invariant 3.4.3. Relationship with the Atiyah–Patodi–Singer eta invariant 3.4.4. Locality of the derivative of the eta invariant. Examples 3.5. The Spectral Flow of Families of Parameter-Dependent Operators 3.5.1. Meromorphic operator functions. Multiplicities of singular points 3.5.2. Definition of the spectral flow 3.6. Higher Spectral Flows 3.6.1. Spectral sections 3.6.2. Spectral flow of homotopies of families of self-adjoint operators 3.6.3. Spectral flow of homotopies of families of parameter-dependent operators 3.7. Bibliographical Remarks T3 - Preprint - (2003) 12 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26595 ER - TY - INPR A1 - Dines, Nicoleta A1 - Harutjunjan, Gohar A1 - Schulze, Bert-Wolfgang T1 - The Zaremba problem in edge Sobolev spaces N2 - Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge Sobolev spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. Our methods from the calculus of boundary value problems on a manifold with edges will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator. T3 - Preprint - (2003) 13 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26615 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - Crack theory with singularties at the boundary N2 - We investigate crack problems, where the crack boundary has conical singularities. Elliptic operators with two-sided elliptc boundary conditions on the plus and minus sides of the crack will be interpreted as elements of a corner algebra of boundary value problems. The corresponding operators will be completed by extra edge conditions on the crack boundary to Fredholm operators in corner Sobolev spaces with double weights, and there are parametrices within the calculus. T3 - Preprint - (2003) 14 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26600 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Differential operators on manifolds with singularities : analysis and topology : Chapter 4: Pseudodifferential operators N2 - Contents: Chapter 4: Pseudodifferential Operators 4.1. Preliminary Remarks 4.1.1. Why are pseudodifferential operators needed? 4.1.2. What is a pseudodifferential operator? 4.1.3. What properties should the pseudodifferential calculus possess? 4.2. Classical Pseudodifferential Operators on Smooth Manifolds 4.2.1. Definition of pseudodifferential operators on a manifold 4.2.2. Hörmander’s definition of pseudodifferential operators 4.2.3. Basic properties of pseudodifferential operators 4.3. Pseudodifferential Operators in Sections of Hilbert Bundles 4.3.1. Hilbert bundles 4.3.2. Operator-valued symbols. Specific features of the infinite-dimensional case 4.3.3. Symbols of compact fiber variation 4.3.4. Definition of pseudodifferential operators 4.3.5. The composition theorem 4.3.6. Ellipticity 4.3.7. The finiteness theorem 4.4. The Index Theorem 4.4.1. The Atiyah–Singer index theorem 4.4.2. The index theorem for pseudodifferential operators in sections of Hilbert bundles 4.4.3. Proof of the index theorem 4.5. Bibliographical Remarks T3 - Preprint - (2003) 11 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26587 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - Toeplitz operators, and ellipticity of boundary value problems with global projection conditions N2 - Ellipticity of (pseudo-) differential operators A on a compact manifold X with boundary (or with edges) Y is connected with boundary (or edge) conditions of trace and potential type, formulated in terms of global projections on Y together with an additional symbolic structure. This gives rise to operator block matrices A with A in the upper left corner. We study an algebra of such operators, where ellipticity is equivalent to the Fredhom property in suitable scales of spaces: Sobolev spaces on X plus closed subspaces of Sobolev spaces on Y which are the range of corresponding pseudo-differential projections. Moreover, we express parametrices of elliptic elements within our algebra and discuss spectral boundary value problems for differential operators. T3 - Preprint - (2003) 03 Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26510 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Savin, Anton A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Elliptic theory on manifolds with nonisolated singularities : V. Index formulas for elliptic problems on manifolds with edges N2 - For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge. T3 - Preprint - (2003) 02 KW - manifold with edge KW - elliptic problem KW - index formula KW - symmetry conditions Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26500 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - On index theorem for symplectic orbifolds N2 - We give an explicit construction of the trace on the algebra of quantum observables on a symplectic orbifold and propose an index formula. T3 - Preprint - (2003) 07 KW - star product KW - symmetry group KW - G-trace KW - G-index Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26550 ER -