| Self-Dual Partial Differential Systems and Their Variational Principles 2009 Edition Contributor(s): Ghoussoub, Nassif (Author) |
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ISBN: 0387848967 ISBN-13: 9780387848969 Publisher: Springer OUR PRICE: $52.24 Product Type: Hardcover - Other Formats Published: November 2008 Annotation: This text is intended for a beginning graduate course on convexity methods for PDEs. The generality chosen by the author puts this under the classification of "functional analysis." The applications, however, require a fair knowledge of classical analysis and PDEs which is needed to make judicious choices of function spaces where the self-dual variational principles need to be applied, and these choices necessarily require prior knowledge of the expected regularity of the (weak) solutions. While this text contains many new results, it is the authora (TM)s hope that this material will soon become standard for all graduate students. |
| Additional Information |
| BISAC Categories: - Mathematics | Functional Analysis - Mathematics | Differential Equations - General - Business & Economics | Information Management |
| Dewey: 515.3 |
| Series: Springer Monographs in Mathematics |
| Physical Information: 0.8" H x 6.1" W x 9.3" (1.40 lbs) 354 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: How to solve partial differential systems by completing the square. This could well have been the title of this monograph as it grew into a project to develop a s- tematic approach for associating suitable nonnegative energy functionals to a large class of partial differential equations (PDEs) and evolutionary systems. The minima of these functionals are to be the solutions we seek, not because they are critical points (i. e., from the corresponding Euler-Lagrange equations) but from also - ing zeros of these functionals. The approach can be traced back to Bogomolnyi's trick of "completing squares" in the basic equations of quantum eld theory (e. g., Yang-Mills, Seiberg-Witten, Ginzburg-Landau, etc., ), which allows for the deri- tion of the so-called self (or antiself) dual version of these equations. In reality, the "self-dual Lagrangians" we consider here were inspired by a variational - proach proposed - over 30 years ago - by Brezis and Ekeland for the heat equation and other gradient ows of convex energies. It is based on Fenchel-Legendre - ality and can be used on any convex functional - not just quadratic ones - making them applicable in a wide range of problems. In retrospect, we realized that the "- ergy identities" satis ed by Leray's solutions for the Navier-Stokes equations are also another manifestation of the concept of self-duality in the context of evolution equations. |