Education
Sogge graduated from the University of Chicago in 1982, and earned a doctorate in mathematics from Princeton University in 1985 under the supervision of Elias M. Stein.
(This work presents three types of problems in the theory ...)
This work presents three types of problems in the theory of nonlinear wave equations that have varying degrees of non-trivial overlap with harmonic analysis. The author discusses results including existence for certain quasilinear wave equations and for semilinear wave equations.
http://www.amazon.com/gp/product/1571460322/?tag=2022091-20
(Fourier Integrals in Classical Analysis is an advanced tr...)
Fourier Integrals in Classical Analysis is an advanced treatment of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. Using microlocal analysis, the author in particular studies problems involving maximal functions and Riesz means using the so-called half-wave operator. This self-contained book starts with a rapid review of important topics in Fourier analysis. The author then presents the necessary tools from microlocal analysis, and goes on to give a proof of the sharp Weyl formula which he then modifies to give sharp estimates for the size of eigenfunctions on compact manifolds. Finally, the tools that have been developed are used to study the regularity properties of Fourier integral operators, culminating in the proof of local smoothing estimates and their applications to singular maximal theorems in two and more dimensions.
http://www.amazon.com/gp/product/0521434645/?tag=2022091-20
(This much-anticipated revised second edition of Christoph...)
This much-anticipated revised second edition of Christopher Sogge's 1995 work provides a self-contained account of the basic facts concerning the linear wave equation and the methods from harmonic analysis that are necessary when studying nonlinear hyperbolic differential equations. Sogge examines quasilinear equations with small data where the Klainerman-Sobolev inequalities and weighted space-time estimates are introduced to prove global existence results. New simplified arguments are given in the current edition that allow one to handle quasilinear systems with multiple wave speeds. The next topic concerns semilinear equations with small initial data. John's existence theorem for R1+3 is discussed with blow-up problems and some results for the spherically symmetric case. After this, general Strichartz estimates are treated. A proof of the endpoint Strichartz estimates of Keel and Tao and the Christ-Kiselev lemma are given, the material being new in this edition. Using the Strichartz estimates, the critical wave equation in R1+3 is studied. Table of contents Chapter 1: Background and groundwork Chapter 2: Quasilinear equations with small data Chapter 3: Semilinear equations with small data Chapter 4: General Strichartz estimates Chapter 5: Global existence for semilinear equations with large data Appendix: Some tools from classical analysis
http://www.amazon.com/gp/product/1571461736/?tag=2022091-20
( Based on lectures given at Zhejiang University in Hangz...)
Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.
http://www.amazon.com/gp/product/0691160783/?tag=2022091-20
(This revised second edition of Christopher Sogge's 1995 w...)
This revised second edition of Christopher Sogge's 1995 work provides a self-contained account of the basic facts concerning the linear wave equation and the methods from harmonic analysis that are necessary when studying nonlinear hyperbolic differential equations. Sogge examines quasilinear equations with small data where the Klainerman-Sobolev inequalities and weighted space-time estimates are introduced to prove global existence results. New simplified arguments are given in the current edition that allow one to handle quasilinear systems with multiple wave speeds. The next topic concerns semilinear equations with small initial data. John's existence theorem for R1+3 is discussed with blow-up problems and some results for the spherically symmetric case. After this, general Strichartz estimates are treated. A proof of the endpoint Strichartz estimates of Keel and Tao and the Christ-Kiselev lemma are given, the material being new in this edition. Using the Strichartz estimates, the critical wave equation in R1+3 is studied.
http://www.amazon.com/gp/product/1571462791/?tag=2022091-20
mathematician Professor of Mathematics
Sogge graduated from the University of Chicago in 1982, and earned a doctorate in mathematics from Princeton University in 1985 under the supervision of Elias M. Stein.
He is the J. J. Sylvester Professor of Mathematics at Johns Hopkins University and the editor-in-chief of the American Journal of Mathematics. His research concerns Fourier analysis and partial differential equations. He taught at the University of Chicago from 1985 to 1989 and the University of California, Los Angeles from 1989 to 1996 before moving to Johns Hopkins.
(This much-anticipated revised second edition of Christoph...)
(This revised second edition of Christopher Sogge's 1995 w...)
(This work presents three types of problems in the theory ...)
( Based on lectures given at Zhejiang University in Hangz...)
(Fourier Integrals in Classical Analysis is an advanced tr...)
("This is the International Edition. The content is in Eng...)