Distributions and Operators

ISBN-13: 9781441927439 / Angielski / Miękka / 2010 / 464 str.

Gerd Grubb

Opis

This textbook gives an introduction to distribution theory with emphasis on applications using functional analysis. In more advanced parts of the book, pseudodi?erential methods are introduced. Distributiontheoryhasbeen developedprimarilytodealwithpartial(and ordinary) di?erential equations in general situations. Functional analysis in, say, Hilbert spaces has powerful tools to establish operators with good m- ping properties and invertibility properties. A combination of the two allows showing solvability of suitable concrete partial di?erential equations (PDE). When partial di?erential operators are realized as operators in L (?) for 2 n anopensubset?ofR, theycomeoutasunboundedoperators.Basiccourses infunctionalanalysisareoftenlimitedtothestudyofboundedoperators, but we here meet the necessityof treating suitable types ofunbounded operators; primarily those that are densely de?ned and closed. Moreover, the emphasis in functional analysis is often placed on selfadjoint or normal operators, for which beautiful results can be obtained by means of spectral theory, but the cases of interest in PDE include many nonselfadjoint operators, where diagonalizationbyspectraltheoryisnotveryuseful.Weincludeinthisbooka chapter on unbounded operatorsin Hilbert space (Chapter 12), where classes of convenient operators are set up, in particular the variational operators, including selfadjoint semibounded cases (e.g., the Friedrichs extension of a symmetric operator), but with a much wider scope. Whereas the functional analysis de?nition of the operators is relatively clean and simple, the interpretation to PDE is more messy and complicated.

Recenzje

From the reviews:

"The book is directed at graduate students and 'researchers interested in its special topics' ... . Distribution and Operators is split into five parts ... . Well-written and scholarly, and equipped with many exercises of considerable pedagogical importance ... Grubb's book is a fine contribution to the literature and should soon occupy a solid place among graduate texts for aspiring hard analysts specializing, e.g., in PDE." (Michael Berg, MAA Online, December, 2008)

"This textbook gives a very clear introduction to distribution theory with emphasis on applications using functional analysis. In more advanced parts of the book, pseudodifferential methods are introduced, specially tailored for the study of boundary value problems. ... The whole book is carefully and nicely written and each chapter ends with a number of exercises ... . perfect for a Ph.D. course ... . most of the material has been used frequently at the University of Copenhagen for several graduate courses." (Fabio Nicola, Mathematical Reviews, Issue 2010 b)

"The book under review is a complete self-contained introduction to classical distribution theory with applications to the study of linear partial differential operators and, in particular, of (elliptic) boundary value problems, an area where the author has much experience ... . very clear and precise. Almost all the results given in the book are proved, and the proofs give plenty of details and are easy to follow. There are a lot of examples and exercises ... ." (David Jornet, Zentralblatt MATH, Vol. 1171, 2009)

Spis treści

Distributions and derivatives.- Motivation and overview.- Function spaces and approximation.- Distributions. Examples and rules of calculus.- Extensions and applications.- Realizations and Sobolev spaces.- Fourier transformation of distributions.- Applications to differential operators. The Sobolev theorem.- Pseudodifferential operators.- Pseudodifferential operators on open sets.- Pseudodifferential operators on manifolds, index of elliptic operators.- Boundary value problems.- Boundary value problems in a constant-coefficient case.- Pseudodifferential boundary operators.- Pseudodifferential methods for boundary value problems.- Topics on Hilbert space operators.- Unbounded linear operators.- Families of extensions.- Semigroups of operators.

Blurb

This book gives an introduction to distribution theory, in the spirit of Laurent Schwartz. Additionally, the aim is to show how the theory is combined with the study of operators in Hilbert space by methods of functional analysis, with applications to partial and ordinary differential equations. Here, the author provides an introduction to unbounded operators in Hilbert space, including a complete theory of extensions of operators, and applications using contraction semigroups.

In more advanced parts of the book, the author shows how distribution theory is used to define pseudodifferential operators on manifolds, and gives a detailed introduction to the pseudodifferential boundary operator calculus initiated by Boutet de Monvel, which allows a modern treatment of elliptic boundary value problems.

 

This book is aimed at graduate students, as well as researchers interested in its special topics, and as such, the author provides careful explanations along with complete proofs, and a bibliography of relevant books and papers. Each chapter has been enhanced with many exercises and examples.

Unique topics include:

* the interplay between distribution theory and concrete operators;

* families of extensions of nonselfadjoint operators;

* an illustration of the solution maps between distribution spaces by a fully worked out constant-coefficient case;

* the pseudodifferential boundary operator calculus;

* the Calderón projector and its applications.

 

Gerd Grubb is Professor of Mathematics at University of Copenhagen.