Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics

ISBN-13: 9780817643522 / Angielski / Twarda / 2007 / 289 str.

Kichenassamy, Satyanad

Opis

This four-part text beautifully interweaves theory and applications in Fuchsian Reduction. Background results in weighted Sobolev and Holder spaces as well as Nash-Moser implicit function theorem are provided. Most chapters contain a problem section and notes with references to the literature. This volume can be used as a text in graduate courses in PDEs and/or Algebra, or as a resource for researchers working with applications to Fuchsian Reduction. The comprehensive approach features the inclusion of problems and bibliographic notes.

Recenzje

From the reviews:

“Fuchsian reduction is an analytical method to represent solutions to non-linear PDEs near singularities … . The book under review provides a careful and instructive introduction into this method. … At the end of most of the chapters some problems are posed, which are solved in an appendix. In total this is a highly interesting book containing a lot of original ideas and which suggests new developments.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 158 (3), November, 2009)

Spis treści

Fuchsian Reduction.- Formal Series.- General Reduction Methods.- Theory of Fuchsian Partial Di?erential Equations.- Convergent Series Solutions of Fuchsian Initial-Value Problems.- Fuchsian Initial-Value Problems in Sobolev Spaces.- Solution of Fuchsian Elliptic Boundary-Value Problems.- Applications.- Applications in Astronomy.- Applications in General Relativity.- Applications in Differential Geometry.- Applications to Nonlinear Waves.- Boundary Blowup for Nonlinear Elliptic Equations.- Background Results.- Distance Function and Hölder Spaces.- Nash–Moser Inverse Function Theorem.

Blurb

Fuchsian reduction is a method for representing solutions of nonlinear PDEs near singularities. The technique has multiple applications including soliton theory, Einstein's equations and cosmology, stellar models, laser collapse, conformal geometry and combustion. Developed in the 1990s for semilinear wave equations, Fuchsian reduction research has grown in response to those problems in pure and applied mathematics where numerical computations fail.

This work unfolds systematically in four parts, interweaving theory and applications. The case studies examined in Part III illustrate the impact of reduction techniques, and may serve as prototypes for future new applications. In the same spirit, most chapters include a problem section. Background results and solutions to selected problems close the volume.

This book can be used as a text in graduate courses in pure or applied analysis, or as a resource for researchers working with singularities in geometry and mathematical physics.