Monotone Operators in Banach Space and Nonlinear Partial Differential Equations (Mathematical Surveys and Monographs)

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 is given in the Appendix. surv49-frnt.pdf -1 Frontmatter 1 Title 1 Copyright 2 Contents 3 Preface 5 PDE Examples by Type 8 Chapter I. Linear Problems...an Introduction -1 Chapter II. Nonlinear Stationary Problems -1 Chapter III. Nonlinear Evolution Problems -1 Chapter IV. Accretive Operators and Nonlinear Cauchy Problems -1 Endmatter -1 surv49-chI.pdf 1 Frontmatter -1 Chapter I. Linear Problems...an Introduction 9 I.1 Boundary Value Problems in 1-D 9 I.2 Variational Methods in Hilbert Space 14 I.3 Applications to Stretched String Problems 20 I.4 Unbounded operators 25 I.5 The Cauchy Problem 30 I.6 Wave Equations 38 Chapter II. Nonlinear Stationary Problems -1 Chapter III. Nonlinear Evolution Problems -1 Chapter IV. Accretive Operators and Nonlinear Cauchy Problems -1 Endmatter -1 surv49-chII.pdf 1 Frontmatter -1 Chapter I. Linear Problems...an Introduction -1 Chapter II. Nonlinear Stationary Problems 42 II.1 Banach Spaces 42 II.2 Existence Theorems 44 II.3 L^p Spaces 51 II.4 Sobolev Spaces 58 II.5 Elliptic Boundary-Value Problems 66 II.6 Variational Inequalities and Quasimonotone Operators 76 II.7 Convex functions 85 II.8 Examples 92 II.9 Elliptic Equations in L^1 100 Chapter III. Nonlinear Evolution Problems -1 Chapter IV. Accretive Operators and Nonlinear Cauchy Problems -1 Endmatter -1 surv49-chIII.pdf 1 Frontmatter -1 Chapter I. Linear Problems...an Introduction -1 Chapter II. Nonlinear Stationary Problems -1 Chapter III. Nonlinear Evolution Problems 109 III.1 Vector-valued Functions 109 III.2 Linear Evolution Equations 113 III.3 Linear Degenerate Equations 120 III.4 Nonlinear Parabolic Equations 126 III.5 Abstract Difference Approximations 133 III.6 Degenerate Parabolic Equations 140 III.7 Variational Inequalities 150 Chapter IV. Accretive Operators and Nonlinear Cauchy Problems -1 Endmatter -1 surv49-chIV.pdf 1 Frontmatter -1 Chapter I. Linear Problems...an Introduction -1 Chapter II. Nonlinear Stationary Problems -1 Chapter III. Nonlinear Evolution Problems -1 Chapter IV. Accretive Operators and Nonlinear Cauchy Problems 161 IV.1 Accretive Operators in Hilbert Space 161 IV.2 Examples of m-accretive Operators 169 IV.3 The Cauchy Problem in Hilbert Space 177 IV.4 Additional Topics and Evolution Equations 185 IV.5 Parabolic Equations and Inequalities 197 IV.6 Semilinear Degenerate Evolution Equations 207 IV.7 Accretive Operators in Banach Space 216 IV.8 The Cauchy Problem in General Banach Space 224 IV.9 Evolution Equations in L^1 238 Endmatter -1 surv49-bck.pdf 1 Frontmatter -1 Chapter I. Linear Problems...an Introduction -1 Chapter II. Nonlinear Stationary Problems -1 Chapter III. Nonlinear Evolution Problems -1 Chapter IV. Accretive Operators and Nonlinear Cauchy Problems -1 Endmatter 246 Appendix: Applications 246 A.1 Heat Conduction 246 A.2 Flow in Porous Media 258 A.3 Continuum Mechanics 261 Bibliography 272 Index 276 Intends 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. Pde Examples By Type -- Ch. I. Linear Problems ... An Introduction -- Ch. Ii. Nonlinear Stationary Problems -- Ch. Iii. Nonlinear Evolution Problems -- Ch. Iv. Accretive Operators And Nonlinear Cauchy Problems. R.e. Showalter. Includes Bibliographical References And Index.