Monotone operators in Banach space and nonlinear partial differential equations / R.E. Showalter.

Format:

Book

Author/Creator:

Showalter, R. E. (Ralph Edwin), 1942- author.

Series:

Mathematical surveys and monographs ; Volume 49.

Mathematical surveys and monographs, 0076-5376 ; Volume 49

Language:

English

Subjects (All):

Monotone operators.

Semigroups.

Banach spaces.

Differential equations, Nonlinear.

Physical Description:

1 online resource (295 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 1997.

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""PDE Examples by Type""; ""Chapter I. Linear Problems... an Introduction""; ""I.1 Boundary-Value Problems in 1-D""; ""I.2 Variational Method in Hilbert Space""; ""I.3 Applications to Stretched String Problems""; ""I.4 Unbounded Operators""; ""I.5 The Cauchy Problem""; ""I.6 Wave Equations""; ""Chapter II. Nonlinear Stationary Problems""; ""II.1 Banach Spaces""; ""II.2 Existence Theorems""; ""II.3 L[sup(P)] Spaces""; ""II.4 Sobolev Spaces""; ""II.5 Elliptic Boundary-Value Problems""; ""II.6 Variational Inequalities and Quasimonotone Operators""

""II.7 Convex Functions""""II.8 Examples""; ""II.9 Elliptic Equations in L[sup(1)]""; ""Chapter III. Nonlinear Evolution Problems""; ""III.1 Vector-valued Functions""; ""III.2 Linear Evolution Equations""; ""III.3 Linear Degenerate Equations""; ""III.4 Nonlinear Parabolic Equations""; ""III.5 Abstract Difference Approximations""; ""III.6 Degenerate Parabolic Equations""; ""III.7 Variational Inequalities""; ""Chapter IV. Accretive Operators and Nonlinear Cauchy Problems""; ""IV.1 Accretive Operators in Hilbert Space""; ""IV.2 Examples of m-accretive Operators""

""IV.3 The Cauchy Problem in Hilbert Space""""IV.4 Additional Topics and Evolution Equations""; ""IV.5 Parabolic Equations and Inequalities""; ""IV.6 Semilinear Degenerate Evolution Equations""; ""IV.7 Accretive Operators in Banach Space""; ""IV.8 The Cauchy Problem in General Banach Space""; ""IV.9 Evolution Equations in L[sup(1)]""; ""Appendix: Applications""; ""A.1 Heat Conduction""; ""A.2 Flow in Porous Media""; ""A.3 Continuum Mechanics""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""

""T""""U""; ""V""; ""W""; ""Y""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-1280-2