Nonlinear partial differential equations with applications.
2nd ed.

*(English)*Zbl 1270.35005
ISNM. International Series of Numerical Mathematics 153. Basel: Birkhäuser (ISBN 978-3-0348-0512-4/hbk; 978-3-0348-0513-1/ebook). xx, 476 p. (2013).

The monograph is concerned with the study of various classes of nonlinear elliptic and parabolic partial differential equations with many applications. The book is organized in twelve chapters, the bibliography and an index. The author presents in the first chapter the auxiliary results used in the rest of the book, related to functional analysis, function spaces, Nemytskii mappings, Green formula, Bochner spaces and some ordinary differential equations. Then, he divides the monograph into two parts: the first one “Steady-state problems” which contains five chapters and the second one “Evolution problems” with six chapters.

The prototype tasks addressed in Part I are boundary value problems for quasilinear equations of the type \[ -\text{div}(a(u,\nabla u))+c(u,\nabla u)=g, \] various generalizations of it, in particular variational inequalities, and some systems of such equations. The author discusses the pseudomonotone and weakly continuous mappings (Chapter 2), accretive mappings (Chapter 3), potential problems – the smooth case (Chapter 4), nonsmooth problems and variational inequalities (Chapter 5) and some systems of equations with applications in (thermo)mechanics of solids and fluids, in electrical devices, engineering, chemistry and biology (Chapter 6).

The second part of the book is devoted to the evolution variants of the boundary-value problems studied in Part I, of the form \[ \displaystyle\frac{\partial u}{\partial t}-\text{div}(a(u,\nabla u))+c(u,\nabla u)=g, \] subject to boundary conditions and initial or periodic conditions. The author studies some special auxiliary tools (Chapter 7), evolution by pseudomonotone or weakly continuous mappings (Chapter 8), some evolution problems governed by accretive mappings (Chapter 9), evolution governed by certain set-valued mappings (Chapter 10), doubly nonlinear problems (Chapter 11), and some particular examples of systems of equations which are real-world applications (Chapter 12). There are investigated the existence, uniqueness, regularity and continuous dependence of the solutions of these problems, by using various methods, such as: indirect methods (involving the construction of auxiliary problems easier to solve, a-priori estimates and a limit passage), direct methods (which use abstract theoretical results) and iterational methods (Banach or Schauder’s fixed point theorems).

This new edition of the book published in 2005 [Zbl 1087.35002] contains some extensions and changes in many chapters, new and expanded exercises, and new applications. We remark that each chapter is finalized by a section entitled “Bibliographical remarks” which presents some comments and the available literature on that topic.

The book is very well written, the presentation is clear and rigorous, and it contains a comprehensive bibliography. This monograph will be useful to all persons who are interested in nonlinear partial differential equations or systems and their applications.

The prototype tasks addressed in Part I are boundary value problems for quasilinear equations of the type \[ -\text{div}(a(u,\nabla u))+c(u,\nabla u)=g, \] various generalizations of it, in particular variational inequalities, and some systems of such equations. The author discusses the pseudomonotone and weakly continuous mappings (Chapter 2), accretive mappings (Chapter 3), potential problems – the smooth case (Chapter 4), nonsmooth problems and variational inequalities (Chapter 5) and some systems of equations with applications in (thermo)mechanics of solids and fluids, in electrical devices, engineering, chemistry and biology (Chapter 6).

The second part of the book is devoted to the evolution variants of the boundary-value problems studied in Part I, of the form \[ \displaystyle\frac{\partial u}{\partial t}-\text{div}(a(u,\nabla u))+c(u,\nabla u)=g, \] subject to boundary conditions and initial or periodic conditions. The author studies some special auxiliary tools (Chapter 7), evolution by pseudomonotone or weakly continuous mappings (Chapter 8), some evolution problems governed by accretive mappings (Chapter 9), evolution governed by certain set-valued mappings (Chapter 10), doubly nonlinear problems (Chapter 11), and some particular examples of systems of equations which are real-world applications (Chapter 12). There are investigated the existence, uniqueness, regularity and continuous dependence of the solutions of these problems, by using various methods, such as: indirect methods (involving the construction of auxiliary problems easier to solve, a-priori estimates and a limit passage), direct methods (which use abstract theoretical results) and iterational methods (Banach or Schauder’s fixed point theorems).

This new edition of the book published in 2005 [Zbl 1087.35002] contains some extensions and changes in many chapters, new and expanded exercises, and new applications. We remark that each chapter is finalized by a section entitled “Bibliographical remarks” which presents some comments and the available literature on that topic.

The book is very well written, the presentation is clear and rigorous, and it contains a comprehensive bibliography. This monograph will be useful to all persons who are interested in nonlinear partial differential equations or systems and their applications.

Reviewer: Rodica Luca Tudorache (Iaşi)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J60 | Nonlinear elliptic equations |

35K55 | Nonlinear parabolic equations |

35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |

35K86 | Unilateral problems for nonlinear parabolic equations and variational inequalities with nonlinear parabolic operators |

47H04 | Set-valued operators |

47H05 | Monotone operators and generalizations |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |