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Showalter R.E. - Monotone Operators in Banach Spase and Nonlinear Partial Differential Equations

Название: Monotone Operators in Banach Spase and Nonlinear Partial Differential Equations
Автор: Showalter R.E.
Категория: Математика
Тип: Книга
Дата: 31.12.2008 00:08:10
Скачано: 20
Описание: The objectives in 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 the presentation will be the large number and variety of examples which are introduced to illustrate this 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 of 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. This is the place to begin study of a particular problem. Once a proper setting has been established for a given problem to be well-posed, one can then proceed to investigate those properties of solutions which distinguish the problem, such as regularity, asymptotic behavior, numerical analysis, stability, special properties, controllability,.... None of these topics will be discussed here. The objective is rather to develop for the reader an instinct for the right place (or places) to look for a solution and the right techniques to use to establish existence-uniqueness in appropriate function spaces for a broad class of problems. Much attention has been devoted to develop the connection between the abstract theory and the specific problems for partial differential equations. The work is arranged in four chapters and an appendix, and each of these is divided into numbered sections. All results are formally stated as Theorems, Propositions, Lemmas, or Corollaries which are independently numbered in the respective category by their section number and order within that section. Thus, in Section 4 of Chapter I the second Proposition is called Proposition 4.2, and it is referenced that way within Chapter I. From any other chapter it will be recalled as Proposition 1.4.2. Most examples are named alphabetically within their section, so the third example of Section 1.4 is Example 4.C, and from outside Chapter I it is called Example I.4.C.
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