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Pseudo-differential operators belong to the most powerful tools in the analysis of partial differential equations. Basic achievements in the early sixties have initiated a completely new understanding of many old and important problems in analy sis and mathematical physics. The standard calculus of pseudo-differential and Fourier integral operators may today be considered as classical. The development has been continuous since the early days of the first essential applications to ellip ticity, index theory, parametrices and propagation of singularities for non-elliptic operators, boundary-value problems, and spectral theory. The basic ideas of the calculus go back to Giraud, Calderon, Zygmund, Mikhlin, Agranovich, Dynin, Vishik, Eskin, and Maslov. Subsequent progress was greatly stimulated by the classical works of Kohn, Nirenberg and Hormander. In recent years there developed a new vital interest in the ideas of micro local analysis in connection with analogous fields of applications over spaces with singularities, e.g. conical points, edges, corners, and higher singularities. The index theory for manifolds with singularities became an enormous challenge for analysists to invent an adequate concept of ellipticity, based on corresponding symbolic structures. Note that index theory was another source of ideas for the later development of the theory of pseudo-differential operators. Let us mention, in particular, the fundamental contributions by Gelfand, Atiyah, Singer, and Bott.
List of contents
1 Sobolev spaces.- 1.1 Fourier transform.- 1.2 The first definition of the Sobolev space.- 1.3 General definition of Sobolev spaces in ?n.- 1.4 Representation of a linear functional over Hs.- 1.5 Embedding theorems.- 1.6 Sobolev spaces in a domain.- 2 Pseudo-differential Operators.- 2.1 The algebra of differential operators.- 2.2 Basic properties of pseudo-differential operators.- 2.3 Calculus of pseudo-differential operators.- 2.4 Pseudo-differential operators on closed manifolds.- 2.5 Gårding inequality.- 3 Elliptic pseudo-differential operators.- 3.1 Parametrices of the elliptic operators.- 3.2 Elliptic operators on a manifold.- 4 Elliptic boundary value problems.- 4.1 Model elliptic boundary value problems.- 4.2 Elliptic boundary value problems in a domain.- 5 Kondratiev's theory.- 5.1 A model problem.- 5.2 The general problem.- 5.3 The boundary value problem in an infinite cone for operators with constant coefficients.- 5.4 Equations with variable coefficients in an infinite cone.- 5.5 The boundary value problem in a bounded domain.- 6 Non-elliptic operators; propagation of singularities.- 6.1 Canonical transformations and Fourier integral operators.- 6.2 Wave fronts of distributions.- 6.3 Wave fronts and Fourier integral operators.- 6.4 Propagation of singularities.- 6.5 The Cauchy problem for a strongly hyperbolic equation.- 7 Pseudo-differential operators on manifolds with conical and edge singularities; motivation and technical preparations.- 7.1 The general background.- 7.2 Parameter-dependent pseudo-differential operators and operator-valued Mellin symbols.- 8 Pseudo-differential operators on manifolds with conical singularities.- 8.1 The cone algebra with asymptotics.- 8.2 The algebra on the infinite cone.- 9 Pseudo-differential operators on manifoldswith edges.- 9.1 Pseudo-differential operators with operator-valued symbols.- 9.2 The edge symbolic calculus.- 9.3 Edge pseudo-differential operators.- 9.4 Applications, examples and remarks.
About the author
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.
