Crack Theory and Edge Singularities

ISBN-13: 9789048163847 / Angielski / Miękka / 2010 / 485 str.

D. V. Kapanadze; Bert-Wolfgang Schulze

Opis

The book studies boundary value problems connected with geometric singularities and models of the crack theory. New and interesting phenomena on the behaviour of solutions (regularity in weighted spaces, asymptotics) are analysed by means of parametrices obtained by inverting corresponding scalar and operator-valued symbols. Compared with other expositions in the field of crack theory and analysis on configurations with singularities the present book systematically develops for the first time an approach in terms of algebras of (pseudo-differential) boundary value problems. The calculus is decomposed into a number of simpler structures, namely boundary value problems (Chapter 1) and edge problems near the crack boundary (Chapter 4). Necessary tools on parameter-dependent cone operators (Chapter 2) and operators on spaces with conical exits to infinity (Chapter 3) are developed as theories of independent interest. The crack theory (Chapter 5) then appears as an application of the edge calculus.
The book is addressed to mathematicians and physicists interested in boundary value problems, geometric singularities, asymptotic analysis, as well as to specialists in the field of crack theory and other singular models.

Spis treści

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.

O autorze

Bert-Wolfgang Schulze ist emeritierter Professor am Institut für Mathematik an der Universität Potsdam, Deutschland. Vor der politischen Wende war er Professor am Karl-Weierstrass-Institut in Berlin, 1984 Euler-Medaille der Akademie der Wisenschaften in Berlin. 1992-96 war er Leiter der Max-Planck-Arbeitsgruppe 'Partielle Differentialgleichungen und Komplexe Analysis' in Potsdam. Nach anfänglichem Studium in Geophysik erhielt er sein Universitätsdiplom in Mathematik in Leipzig 1968. Die Promotion zum Dr. rer.nat. 1970 und die Habilitation in Mathematik 1974 erfolgten an der Universität Rostock. Seine wissenschaftlichen Aktivitäten umfassen Potentialtheorie, Randwert-Probleme, pseudo-differentielle Algebren und Index-Theorie auf berandeten Mannigfaltgikeiten und Räumen mit Singularitäten, darunterTransmissions- und Riss Probleme, Asymptotik von Lösungen, Randwert-Theorie mit globalen Projektionsbedingungen.