Geometric Wave Equations

by Shatah, Jalal M. Ihsan

• ISBN13: 9780821827499
• ISBN10: 0821827499
• Format: Paperback
• Copyright: 2000-10-01
• Publisher: Amer Mathematical Society

Summary

This volume contains notes of the lectures given at the Courant Institute and a DMV-Seminar at Oberwolfach. The focus is on the recent work of the authors on semilinear wave equations with critical Sobolev exponents and on wave maps in two space dimensions. Background material and references have been added to make the notes self-contained. The book is suitable for use in a graduate-level course on the topic. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Table of Contents

Preface	xi
The Wave Equation	1	(6)
Basic Notations and Concepts from Geometry	1	(2)
Semilinear Problems	3	(1)
Wave Maps	3	(4)
Conservation Laws	7	(12)
Variational Formulation	7	(3)
Noether's Theorem	10	(3)
Method of Multipliers	13	(2)
Geometric Invariance	15	(4)
Function Spaces	19	(18)
Lebesgue and Sobolev Spaces	19	(7)
Besov Spaces	26	(2)
Interpolation Theory	28	(3)
H1 and BMO	31	(6)
The Linear Wave Equation	37	(18)
Representation Formulas and Duhamel's Principle	37	(7)
Fourier Transform Representation	44	(1)
Pointwise Estimates	45	(3)
Strichartz-Type Estimates	48	(7)
Well-Posedness	55	(8)
Well-Posedness and Local Existence	55	(3)
Dimensional Analysis and Critical Equations	58	(1)
Effects of Lorentz Transformation	59	(4)
Semilinear Wave Equations	63	(18)
Lipschitz Nonlinearities	63	(2)
Segal's Theorem	65	(4)
Jorgens' Theorem	69	(3)
Critical Growth	72	(9)
Wave Maps	81	(28)
Basic Questions	81	(1)
Geometric Structure	82	(7)
Analytic Structure	89	(2)
Algebraic Structure	91	(11)
Singularities and Nonuniqueness	102	(7)
Wave Maps with Symmetry	109	(20)
Equivariant Maps	109	(9)
The Radial Wave Equation on R2+1	118	(5)
Regularity of Radial Wave Maps in 2 +1 Dimensions	123	(6)
Notes	129	(6)
Bibliography	135