Parabolicity, Volterra Calculus, and Conical Singularities

Partial differential equations constitute an integral part of mathematics. They lie at the interface of areas as diverse as differential geometry, functional analysis, or the theory of Lie groups and have numerous applications in the applied sciences. A wealth of methods has been devised for their analysis. Over the past decades, operator algebras in connection with ideas and structures from geometry, topology, and theoretical physics have contributed a large variety of particularly useful tools. One typical example is the analysis on singular configurations, where elliptic equations have been studied successfully within the framework of operator algebras with symbolic structures adapted to the geometry of the underlying space. More recently, these techniques have proven to be useful also for studying parabolic and hyperbolic equations. Moreover, it turned out that many seemingly smooth, noncompact situations can be handled with the ideas from singular analysis. The three papers at the beginning of this volume highlight this aspect. They deal with parabolic equations, a topic relevant for many applications. The first article prepares the ground by presenting a calculus for pseudo differential operators with an anisotropic analytic parameter. In the subsequent paper, an algebra of Mellin operators on the infinite space-time cylinder is constructed. It is shown how timelike infinity can be treated as a conical singularity.

Auteur
Dipl.-Kaufmann Michael Demuth ist Geschäftsführer der Vermögensberatungsgesellschaft Creative Capital GmbH. 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.

Texte du rabat

This volume highlights the analysis on noncompact and singular manifolds within the framework of the cone calculus with asymptotics. The three papers at the beginning deal with parabolic equations, a topic relevant for many applications. The first article presents a calculus for pseudodifferential operators with an anisotropic analytic parameter. The subsequent paper develops an algebra of Mellin operators on the infinite space-time cylinder. It is shown how timelike infinity can be treated as a conical singularity. In the third text - the central article of this volume - the authors use these results to obtain precise information on the long-time asymptotics of solutions to parabolic equations and to construct inverses within the calculus. There follows a factorization theorem for meromorphic symbols: It is proven that each of these can be decomposed into a holomorphic invertible part and a smoothing part containing all the meromorphic information. It is expected that this result will be important for applications in the analysis of nonlinear hyperbolic equations. The final article addresses the question of the coordinate invariance of the Mellin calculus with asymptotics.

Contenu
Volterra Families of Pseudodifferential Operators.- 1. Basic notation and general conventions.- 2. General parameter-dependent symbols.- 3. Parameter-dependent Volterra symbols.- 4. The calculus of pseudodifferential operators.- 5. Ellipticity and parabolicity.- References.- The Calculus of Volterra Mellin Pseudodifferential Operators with Operator-valued Symbols.- 1. Preliminaries on function spaces and the Mellin transform.- 2. The calculus of Volterra symbols.- 3. The calculus of Volterra Mellin operators.- 4. Kernel cut-off and Mellin quantization.- 5. Parabolicity and Volterra parametrices.- References.- On the Inverse of Parabolic Systems of Partial Differential Equations of General Form in an Infinite Space-Time Cylinder.- 1. Preliminary material.- 2. Abstract Volterra pseudodifferential calculus.- 3. Parameter-dependent Volterra calculus on a closed manifold.- 4. Weighted Sobolev spaces.- 5. Calculi built upon parameter-dependent operators.- 6. Volterra cone calculus.- 7. Remarks on the classical theory of parabolic PDE.- References.- On the Factorization of Meromorphic Mellin Symbols.- 1. Introduction.- 2. Preliminaries.- 3. Logarithms of pseudodifferential operators.- 4. The kernel cut-off technique.- 5. Proof of the main theorem.- References.- Coordinate Invariance of the Cone Algebra with Asymptotics.- 1. Cone operators on the half-axis.- 2. Operators on higher-dimensional cones.- References.