This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between functional analytic insights and calculus-type estimates within the context of Sobolev spaces. Treatment of all topics is complete and self-contained. The book's wide scope and clear exposition make it a suitable text for a graduate course in PDEs.
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"...Highly recommend the book for students as well as for lectures in PDE." -- Monatshefte fur Mathematik
"...entirely possible that the ...text could eventually become the benchmark. ...extremely well written..." -- SIAM Review
"For a student wishing to specialise in the theory of PDEs it provides a very solid foundation." -- The Mathematical Gazette
"Highly recommend the book for students as well as for lectures in PDE." -- Monatshefte fr Mathematik
"It is a standard treatment with good notation. It is extremely well written, with a very attractive format." -- SIAM Review
"This excellent textbook [is recommended] as the first textbook for anyone who wants to learn the theory of [PDE]" -- European Mathematical Society Newsletter
"Well written; proofs are given in full detail and pictures are inserted when needed; moreover, the exposition is perfectly self-contained, the development within each part and section being rigorous and complete." -- Mathematical Reviews
For a student wishing to specialise in the theory of PDEs it provides a very solid foundation. -- The Mathematical Gazette
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