TY - THES A1 - Khalil, Sara T1 - Boundary Value Problems on Manifolds with Singularities T1 - Randwertprobleme auf Mannigfaltigkeiten mit Singularitäten N2 - In the thesis there are constructed new quantizations for pseudo-differential boundary value problems (BVPs) on manifolds with edge. The shape of operators comes from Boutet de Monvel’s calculus which exists on smooth manifolds with boundary. The singular case, here with edge and boundary, is much more complicated. The present approach simplifies the operator-valued symbolic structures by using suitable Mellin quantizations on infinite stretched model cones of wedges with boundary. The Mellin symbols themselves are, modulo smoothing ones, with asymptotics, holomorphic in the complex Mellin covariable. One of the main results is the construction of parametrices of elliptic elements in the corresponding operator algebra, including elliptic edge conditions. N2 - In der Dissertation wurden neue Quantisierungen konstruiert für pseudo-differentielle Randwertprobleme auf Mannigfaltigkeiten mit Kanten-Singularitäten. Die Gestalt der hier behandelten Operatoren ist motiviert durch Boutet de Monvels Kalkül, der auf glatten Mannigfaltigkeiten mit Rand bekannt ist. Der singuläre Fall, hier mit Kanten und Rand, ist weitaus komplizierter. Der gegenwärtige Zugang vereinfacht die operatarwertigen Symbolstrukturen unter Verwendung geeigneter Mellin-Quantisierungen auf unendlichen gestreckten Modell- Kegeln, die entsprechenden Keilen mit Rand zugeordnet sind. Die Mellin-Symbole selbst sind holomorph in der komplexen Mellin Kovariablen bis auf glättende Restglieder mit Asymptotiken. Zu den Hauptresultaten gehört die Konstruktion von Parametrices elliptischer Elemente in der erzeugten Operator-Algebra, einschließlich elliptischer Kanten-Bedingungen. KW - manifolds with singularities KW - boundary value problems KW - pseudo-differential equation KW - manifolds with edge KW - Boutet de Monvel's calculus KW - edge boundary value problems KW - Mannigfaltigkeiten mit Singularitäten KW - Randwertprobleme KW - pseudo-differentielle Gleichungen KW - Mannigfaltigkeiten mit Kante KW - Boutet de Monvels Kalkül KW - Kanten-Randwertprobleme Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-419018 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Elliptic complexes of pseudodifferential operators on manifolds with edges N2 - On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof. T3 - Preprint - (1998) 14 KW - manifolds with singularities KW - pseudodifferential operators KW - elliptic complexes KW - Hodge theory Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25257 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A remark on the index of symmetric operators N2 - We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol. T3 - Preprint - (1998) 04 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25169 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The index of higher order operators on singular surfaces N2 - The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol. T3 - Preprint - (1998) 03 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25127 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - On the index formula for singular surfaces N2 - In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators. T3 - Preprint - (1997) 31 KW - manifolds with singularities KW - differential operators KW - index KW - 'eta' invariant KW - monodromy matrix Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25116 ER - TY - INPR A1 - Fedosov, Boris A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The index of elliptic operators on manifolds with conical points N2 - For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero. T3 - Preprint - (1997) 24 KW - manifolds with singularities KW - pseudodifferential operators KW - elliptic operators KW - index Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25096 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - The Riemann-Roch theorem for manifolds with conical singularities N2 - The classical Riemann-Roch theorem is extended to solutions of elliptic equations on manifolds with conical points. T3 - Preprint - (1997) 18 KW - manifolds with singularities KW - elliptic operators KW - divisors Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25051 ER -