**Mathematical Surveys and Monographs**

Volume: 49;
1997;
278 pp;
Softcover

MSC: Primary 47; 35;
Secondary 34

Print ISBN: 978-0-8218-9397-5

Product Code: SURV/49.S

List Price: $91.00

Individual Member Price: $72.80

**Electronic ISBN: 978-1-4704-1280-7
Product Code: SURV/49.S.E**

List Price: $91.00

Individual Member Price: $72.80

# Monotone Operators in Banach Space and Nonlinear Partial Differential Equations

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*R E Showalter*

The objectives of this monograph are to present some topics from the theory of monotone operators and nonlinear semigroup theory which are directly applicable to the existence and uniqueness theory of initial-boundary-value problems for partial differential equations and to construct such operators as realizations of those problems in appropriate function spaces.

A highlight of this presentation is the large number and variety of examples introduced to illustrate the connection between the theory of nonlinear operators and partial differential equations. These include primarily semilinear or quasilinear equations of elliptic or of parabolic type, degenerate cases with change of type, related systems and variational inequalities, and spatial boundary conditions of the usual Dirichlet, Neumann, Robin or dynamic type.

The discussions of evolution equations include the usual initial-value problems as well as periodic or more general nonlocal constraints, history-value problems, those which may change type due to a possibly vanishing coefficient of the time derivative, and other implicit evolution equations or systems including hysteresis models. The scalar conservation law and semilinear wave equations are briefly mentioned, and hyperbolic systems arising from vibrations of elastic-plastic rods are developed. The origins of a representative sample of such problems are given in the appendix.

#### Readership

Advanced graduate students, research mathematicians, and engineers interested in numerical analysis, applied mathematics, control theory, or dynamical systems.

#### Reviews & Endorsements

The book is extremely clear and well written, and so it is very readable. It contains a large variety of examples and applications and consequently it will prove quite useful not only to mathematicians, but also to engineers and physicists.

-- Mathematical Reviews

The completeness and the way of presentation makes the text understandable to anybody who can be interested in existence and uniqueness theory for initial-boundary-value problems.

-- European Mathematical Society Newsletter

#### Table of Contents

# Table of Contents

## Monotone Operators in Banach Space and Nonlinear Partial Differential Equations

- Contents vii8 free
- Preface ix10 free
- PDE Examples by Type xiii14 free
- Chapter I. Linear Problems... an Introduction 116 free
- Chapter II. Nonlinear Stationary Problems 3550
- II.1 Banach Spaces 3550
- II.2 Existence Theorems 3752
- II.3 L[sup(P)] Spaces 4459
- II.4 Sobolev Spaces 5166
- II.5 Elliptic Boundary-Value Problems 5974
- II.6 Variational Inequalities and Quasimonotone Operators 6984
- II.7 Convex Functions 7893
- II.8 Examples 85100
- II.9 Elliptic Equations in L[sup(1)] 93108

- Chapter III. Nonlinear Evolution Problems 103118
- Chapter IV. Accretive Operators and Nonlinear Cauchy Problems 155170
- IV.1 Accretive Operators in Hilbert Space 155170
- IV.2 Examples of m-accretive Operators 163178
- IV.3 The Cauchy Problem in Hilbert Space 171186
- IV.4 Additional Topics and Evolution Equations 179194
- IV.5 Parabolic Equations and Inequalities 191206
- IV.6 Semilinear Degenerate Evolution Equations 201216
- IV.7 Accretive Operators in Banach Space 210225
- IV.8 The Cauchy Problem in General Banach Space 218233
- IV.9 Evolution Equations in L[sup(1)] 232247

- Appendix: Applications 241256
- Bibliography 267282
- Index 271286