Functional Methods in Differential Equations: (Chapman & Hall/CRC Research Notes in Mathematics Series)

In recent years, functional methods have become central to the study of theoretical and applied mathematical problems. As demonstrated in this Research Note, functional methods can not only provide more generality, but they can also unify results and techniques and lead to better results than those obtained by classical methods. Presenting entirely original results, the authors use functional methods to explore a broad range of elliptic, parabolic, and hyperbolic boundary value problems and various classes of abstract differential and integral equations. They show that while it is crucial to choose an appropriate functional framework, this approach can lead to mathematical models that better describe concrete physical phenomena. In particular, they reach a concordance between the physical sense and the mathematical sense for the solutions of some special models. Beyond its importance as a survey of the primary techniques used in the area, the results illuminated in this volume will prove valuable in a wealth of interesting applicationsReadership: This volume will be of interest to pure and applied mathematicians, engineers, and physicists.

Table of Contents:
INTRODUCTION Function and Distribution Spaces Monotone Operators, Convex Functions, and Subdifferentials Some Elements of Spectral Theory Linear Evolution Equations and Semigroups Nonlinear Evolution Equations ELLIPTIC BOUNDARY VALUE PROBLEMS Non-Degenerate Elliptic Boundary Value Problems Degenerate Elliptic Boundary Value Problems PARABOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC BOUNDARY CONDITIONS Homogeneous Boundary Conditions Nonhomogeneous Boundary Conditions Higher Regularity of Solutions PARABOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC-DIFFERENTIAL BOUNDARY CONDITIONS Homogeneous Boundary Conditions Nonhomogeneous Boundary Conditions Higher Regularity of Solutions HYPERBOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC BOUNDARY CONDITIONS Existence, Uniqueness and Long-Time Behavior of Solutions Higher Regularity of Solutions HYPERBOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC-DIFFERENTIAL BOUNDARY CONDITIONS Existence, Uniqueness and Long-Time Behavior of Solutions Higher Regularity of Solutions THE FOURIER METHOD FOR ABSTRACT DIFFERENTIAL EQUATIONS First Order Linear Equations Semilinear First Order Equations Second Order Linear Equations Semilinear Second Order Equations THE SEMIGROUP APPROACH FOR ABSTRACT DIFFERENTIAL EQUATIONS Semilinear Fist Order Equations Hyperbolic Partial Differential Systems with Nonlinear Boundary Conditions NONLINEAR NON-AUTONOMOUS ABSTRACT DIFFERENTIAL EQUATIONS First Order Differential and Functional Equations Containing Subdifferentials An Application IMPLICIT NONLINEAR ABSTRACT DIFFERENTIAL EQUATIONS Existence of Solution Uniqueness of Solution Continuous Dependence of Solution Existence of Periodic Solutions