Partial differential equations.

*(English)*Zbl 0902.35002
Graduate Studies in Mathematics. 19. Providence, RI: American Mathematical Society (AMS). xvii, 662 p. (1998).

This monograph is a wide-ranging textbook for graduate and higher-level undergraduate students. The author treats linear and nonlinear equations in any space dimension, with particular emphasis on various modern approaches. Although he also makes use of abstract formulations of certain classes of equations in terms of operators between Banach spaces, his approach is by no means reduced to the functional-analytic setting. The author presents essential ideas in very clear settings, avoiding technicalities required by sharp versions of the theorems. Each part of the book is largely self-contained. Each chapter of the book is complemented by problems and references to research articles and advanced monographs.

The book is divided into three parts: representation formulae for solutions, linear and nonlinear PDE. The first part consists of three chapters devoted to four important linear PDE (transport, Laplace’s, heat and wave equation), nonlinear first-order PDE (complete integrals, characteristics, Hamilton-Jacobi equations, conservation laws) and other ways to represent solutions (separation of variables, similarity solutions, transform methods, converting nonlinear into linear PDE, asymptotics, power series). The three chapters of the second part (linear PDE) deal with Sobolev spaces, second-order elliptic equations (existence of weak solutions, regularity, maximum principles, eigenvalues and eigenfunctions) and linear evolution equations (second-order parabolic and hyperbolic equations, hyperbolic systems of first-order equations, semigroup theory). The last part (nonlinear PDE) is subdivided into four chapters dealing with variational methods (first and second variation, existence of minimizers for convex and polyconvex functionals, regularity, constraints, Mountain Pass Theorem), nonvariational techniques (monotonicity and fixed point methods, sub- and supersolutions, nonexistence results, geometric properties of solutions, gradient flows), Hamilton-Jacobi equations (viscosity solutions, uniqueness, control theory, dynamic programming) and systems of conservation laws (Riemann’s problem, systems of two conservation laws, entropy criteria).

The book is complemented by five appendices describing basic facts concerning notation, inequalities, calculus, linear functional analysis and measure theory. This book is definitely one of the best textbooks in the area of PDE.

The book is divided into three parts: representation formulae for solutions, linear and nonlinear PDE. The first part consists of three chapters devoted to four important linear PDE (transport, Laplace’s, heat and wave equation), nonlinear first-order PDE (complete integrals, characteristics, Hamilton-Jacobi equations, conservation laws) and other ways to represent solutions (separation of variables, similarity solutions, transform methods, converting nonlinear into linear PDE, asymptotics, power series). The three chapters of the second part (linear PDE) deal with Sobolev spaces, second-order elliptic equations (existence of weak solutions, regularity, maximum principles, eigenvalues and eigenfunctions) and linear evolution equations (second-order parabolic and hyperbolic equations, hyperbolic systems of first-order equations, semigroup theory). The last part (nonlinear PDE) is subdivided into four chapters dealing with variational methods (first and second variation, existence of minimizers for convex and polyconvex functionals, regularity, constraints, Mountain Pass Theorem), nonvariational techniques (monotonicity and fixed point methods, sub- and supersolutions, nonexistence results, geometric properties of solutions, gradient flows), Hamilton-Jacobi equations (viscosity solutions, uniqueness, control theory, dynamic programming) and systems of conservation laws (Riemann’s problem, systems of two conservation laws, entropy criteria).

The book is complemented by five appendices describing basic facts concerning notation, inequalities, calculus, linear functional analysis and measure theory. This book is definitely one of the best textbooks in the area of PDE.

Reviewer: P.Quittner (Bratislava)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |