TY - INPR A1 - Schulze, Bert-Wolfgang T1 - The iterative structure of corner operators N2 - We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27 - 31, 2008, at the MSRI, University of Berkeley. T3 - Preprint - (2008) 08 KW - Categories of stratified spaces KW - ellipticity of corners operators KW - principal symbolic hierarchies KW - boundary value problems Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30353 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Surgery and the relative index theorem for families of elliptic operators N2 - We prove a theorem describing the behaviour of the relative index of families of Fredholm operators under surgery performed on spaces where the operators act. In connection with additional conditions (like symmetry conditions) this theorem results in index formulas for given operator families. By way of an example, we give an application to index theory of families of boundary value problems. T3 - Preprint - (2002) 11 KW - elliptic operators KW - index theory KW - surgery KW - relative index KW - boundary value problems Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26300 ER - TY - INPR A1 - Nazaikinskii, Vladimir E. A1 - Sternin, Boris T1 - Surgery and the relative index in elliptic theory N2 - We prove a general theorem on the local property of the relative index for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions) this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities as well as for elliptic boundary value problems with a symmetry condition for the conormal symbol. T3 - Preprint - (1999) 17 KW - elliptic operators KW - index theory KW - surgery KW - relative index KW - manifold with singularities KW - boundary value problems Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25538 ER - TY - GEN A1 - Bandara, Menaka Lashitha A1 - Rosén, Andreas T1 - Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe N2 - On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 758 KW - boundary value problems KW - Dirac operator KW - functional calculus KW - real-variable harmonic analysis KW - Riesz continuity KW - spectral flow Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-434078 SN - 1866-8372 IS - 758 SP - 1253 EP - 1284 ER - TY - JOUR A1 - Gil, Juan B. A1 - Krainer, Thomas A1 - Mendoza, Gerardo A. T1 - Resolvents of elliptic cone operators JF - Journal of functional analysis N2 - We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent. KW - resolvents KW - manifolds with conical singularities KW - spectral theory KW - parametrices KW - boundary value problems Y1 - 2006 U6 - https://doi.org/10.1016/j.jfa.2006.07.010 SN - 0022-1236 VL - 241 IS - 1 SP - 1 EP - 55 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Chang, D. -C. A1 - Viahmoudi, M. Hedayat A1 - Schulze, Bert-Wolfgang T1 - PSEUDO-DIFFERENTIAL ANALYSIS WITH TWISTED SYMBOLIC STRUCTURE JF - Journal of nonlinear and convex analysis : an international journal N2 - This paper is devoted to pseudo-differential operators and new applications. We establish necessary extensions of the standard calculus to specific classes of operator-valued symbols occurring in principal symbolic hierarchies of operators on manifolds with singularities or stratified spaces. KW - Pseudo-differential operators KW - boundary value problems KW - operator valued symbols KW - Fourier transform Y1 - 2016 SN - 1345-4773 SN - 1880-5221 VL - 17 SP - 1889 EP - 1937 PB - Yokohama Publishers CY - Yokohama ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Elliptic operators in odd subspaces N2 - An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established. T3 - Preprint - (1999) 11 KW - index of elliptic operators in subspaces KW - K-theory KW - eta invariant KW - Atiyah-Patodi-Singer theory KW - boundary value problems Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25478 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Elliptic operators in even subspaces N2 - An elliptic theory is constructed for operators acting in subspaces defined via even pseudodifferential projections. Index formulas are obtained for operators on compact manifolds without boundary and for general boundary value problems. A connection with Gilkey's theory of η-invariants is established. T3 - Preprint - (1999) 10 KW - index of elliptic operators in subspaces KW - K-theory KW - eta invariant KW - Atiyah-Patodi-Singer theory KW - boundary value problems Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25461 ER - TY - INPR A1 - Grudsky, Serguey A1 - Tarkhanov, Nikolai Nikolaevich T1 - Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary N2 - We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)10 KW - singular integral equations KW - nonsmooth curves KW - boundary value problems Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57745 ER - 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 - Harutyunyan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Boundary value problems in weighted edge spaces N2 - We study elliptic boundary value problems in a wedge with additional edge conditions of trace and potential type. We compute the (difference of the) number of such conditions in terms of the Fredholm index of the principal edge symbol. The task will be reduced to the case of special opening angles, together with a homotopy argument. T3 - Preprint - (2006) 09 KW - Elliptic operators in domains with edges KW - boundary value problems KW - weighted edge spaces Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30104 ER - TY - INPR A1 - Rabinovich, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Boundary value problems in cuspidal wedges N2 - The paper is devoted to pseudodifferential boundary value problems in domains with cuspidal wedges. Concerning the geometry we even admit a more general behaviour, namely oscillating cuspidal wedges. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to edges. T3 - Preprint - (1998) 24 KW - pseudodifferential operators KW - boundary value problems KW - manifolds with edges Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25363 ER - TY - INPR A1 - Rabinovich, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - A calculus of boundary value problems in domains with Non-Lipschitz Singular Points N2 - The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points. T3 - Preprint - (1997) 09 KW - pseudodifferential operators KW - boundary value problems KW - manifolds with cusps Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-24957 ER -