Bert-Wolfgang Schulze

Crack Theory and Edge Singularities

Preface. Introduction. 1: Boundary value problems with the transmission property. 1.1. Symbolic calculus and pseudo-differential operators. 1.2. Parameter-dependent boundary value problems. 1.3. General kernel cut-off constructions. 1.4. Notes and complementary remarks. 2: Operators on manifolds with conical singularities. 2.1. Mellin operators and cone asymptotics. 2.2. The cone algebra. 2.3. Analytic functionals and asymptotics. 2.4. Notes and complementary remarks. 3: Operators on manifolds with exits to infinity. 3.1. Scalar operators. 3.2. Calculus with operator-valued symbols. 3.3. Boundary value problems on manifolds with exits to infinity. 3.4. Notes and complementary remarks. 4: Boundary value problems on manifolds with edges. 4.1. Manifolds with edges and typical operators. 4.2. Weighted Sobolov spaces. 4.3. Operator conventions in the edge pseudo-differential calculus. 4.4. Operator-valued edge symbols. 4.5. The algebra of edge boundary value problems. 4.6. Further material on edge operators. 4.7. Notes and complementary remarks. 5: Crack theory. 5.1. Differential operators in crack configurations. 5.2. Parameter-dependent calculus in the model cone. 5.3. Local crack theory. 5.4. The global calculus. 5.5. Notes and complementary remarks. Bibliography. List of Symbols. Index.

Pseudo-differential operators on manifolds with edges

The pseudo-differential operators on manifolds with edges can be obtained as a calculus along the edges with operator-valued symbols acting along the model cones of corresponding wedges. The “abstract” background can be formulated independently, and it has applications also to more complicated singularities, cf. [Sc3]. Sections 9.1.1–9.1.3 are devoted to the main definitions and results that are needed below in the concrete edge theory. Proofs and further details may be found in [Sc1], [Sc2]. On the other hand, the theory of pseudo-differential operators with operator-valued symbols is a natural extension of the scalar case, and the reader will recognize the basic ideas of the ordinary calculus.

The edge algebra structure of boundary value problems

Boundary value problems for pseudodifferential operators (with orwithout the transmission property) are characterised as a substructureof the edge pseudodifferential calculus with constant discreteasymptotics. The boundary in this case is the edge and the inner normalthe model cone of local wedges. Elliptic boundary value problems fornoninteger powers of the Laplace symbol belong to the examples as wellas problems for the identity operator in the interior with a prescribednumber of trace and potential conditions. Transmission operators arecharacterised as smoothing Mellin and Green operators with meromorphicsymbols.

The Mellin Pseudo-Differential Calculus on Manifolds with Corners

It is well-known that the parametrices of elliptic partial differential equations on C ∞ manifolds (say closed compact or compact with C ∞ boundary) can be expressed by pseudo-differential operators. This implies the elliptic regularity in terms of (for instance) the standard Sobolev spaces.

Pseudo-differential operators

In this chapter we briefly state the classical theory of pseudo-differential operators. This theory has its roots in the works of Giraud, Calderon-Zygmund, Mikhlin, Agranovich-Dynin, and Vishik-Eskin, but officially it started in 1965 with the remarkable work of Kohn-Nirenberg [KN], where it was stated in a very simple and attractive manner, showing its applications to the theory of boundary-value problems.

Elliptic theory on singular manifolds

I Singular Manifolds and Differential Operators GEOMETRY OF SINGULARITIES Preliminaries Manifolds with conical singularities Manifolds with edges ELLIPTIC OPERATORS ON SINGULAR MANIFOLDS Operators on manifolds with conical singularities Operators on manifolds with edges Examples of elliptic edge operators II Analytical Tools PSEUDODIFFERENTIAL OPERATORS Preliminary remarks Classical theory Operators in sections of Hilbert bundles Operators on singular manifolds Ellipticity and finiteness theorems Index theorems on smooth closed manifolds LOCALIZATION (SURGERY) IN ELLIPTIC THEORY The index locality principle Localization in index theory on smooth manifolds Surgery for the index of elliptic operators on singular manifolds Relative index formulas on manifolds with isolated singularities III Topological Problems INDEX THEORY Statement of the problem Invariants of interior symbol and symmetries Invariants of the edge symbol Index theorems Index on manifolds with isolated singularities Supplement. Classification of elliptic symbols with symmetry and K-theory Supplement. Proof of Proposition 5.16 ELLIPTIC EDGE PROBLEMS Morphisms The obstruction to ellipticity A formula for the obstruction in topological terms Examples. Obstructions for geometric operators IV Applications and Related Topics FOURIER INTEGRAL OPERATORS ON SINGULAR MANIFOLDS Homogeneous canonical (contact) transformations Definition of Fourier integral operators Properties of Fourier integral operators The index of elliptic Fourier integral operators Application to quantized contact transformations Example RELATIVE ELLIPTIC THEORY Analytic aspects of relative elliptic theory Topological aspects of relative elliptic theory INDEX OF GEOMETRIC OPERATORS ON MANIFOLDS WITH CYLINDRICAL ENDS Operators on manifolds with cylindrical ends Index formulas HOMOTOPY CLASSIFICATION OF ELLIPTIC OPERATORS The homotopy classification problem Classification on smooth manifolds Atiyah-de Rham duality Abstract elliptic operators and analytic K-homology Classification on singular manifolds Some applications LEFSCHETZ FORMULAS Main result Proof of the theorem Contributions of conical points as sums of residues Supplement. The Lefschetz number Supplement. The Sternin-Shatalov method APPENDICES Spectral Flow Eta Invariants Index of Parameter-Dependent Elliptic Families Bibliographic Remarks Bibliography Index

On the Index of Differential Operators on Manifolds with Conical Singularities

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah–Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.

Operator Algebras with Symbol Hierarchies on Manifolds with Singularities

Problems for elliptic partial differential equations on manifoldsMwith singularitiesM’(here with piece—wise smooth geometry) are studied in terms of pseudo-differential algebras with hierarchies of symbols that consist of scalar and operator—valued components. Classical boundary value problems (with or without the transmission property) belong to the examples. They are a model for operator algebras on manifoldsMwith higher “polyhedral” singularities. The operators are block matrices that have upper left corners containing the pseudo—differential operators on the regular part M\M’ (plus certain Mellin and Green summands) and are degenerate (in stretched coordinates) in a typical way nearM’.By definitionM’is again a manifold with singularities. The same is true ofM“, and so on. The block matrices consist of trace, potential and Mellin and Green operators, acting between weighted Sobolev spaces onM()andM( 0 ),with 0 <j k <ord M; hereMO)denotesM, MO) denotesM’,etc. We generate these algebras, including their symbol hierarchies, by iterating so—called “edgifications” and “conifications” of algebras that have already been constructed, and we study ellipticity, parametrices, and Fredholm property within these algebras.

Edge operators with conditions of Toeplitz type

Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of

The Iterative Structure of the Corner Calculus

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