Elliptic Partial Differential Equations: Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains

ISBN-13: 9783034605366 / Angielski / Twarda / 2011 / 642 str.

Vitaly Volpert

Opis

The theory of elliptic partial differential equations has undergone an important development over the last two centuries. Together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications, it has become one of the most richly enhanced fields of mathematics. This monograph undertakes a systematic presentation of the theory of general elliptic operators. The author discusses a priori estimates, normal solvability, the Fredholm property, the index of an elliptic operator, operators with a parameter, and nonlinear Fredholm operators. Particular attention is paid to elliptic problems in unbounded domains which have not yet been sufficiently treated in the literature and which require some special approaches. The book also contains an analysis of non-Fredholm operators and discrete operators as well as extensive historical and bibliographical comments. The selected topics and the author's level of discourse will make this book a most useful resource for researchers and graduate students working in the broad field of partial differential equations and applications.

Recenzje

From the reviews:

"In the book under review the author concentrates on the Fredholm theory for (mainly linear) boundary value problems in unbounded domains. ... Both for a researcher in the field and for a mathematical research library, this is a must-have item." (Niels Jacob, The Mathematical Gazette, Vol. 98 (541), March, 2014)

"This very interesting book presents a systematic study of elliptic problems in unbounded domains of Rⁿ. ... At the end of the book ... the author gives a comprehensive account of the large development of the elliptic theory from its birth in the XVIIIth century up to the present time. ... This historical account gives to the reader the full perspective on each topic as well as a complete and up-to-date bibliography. This is certainly an excellent starting point for any kind of further investigation." (Paolo Acquistapace, Mathematical Reviews, Issue 2012 g)

"The book is devoted to substantial and systematic presentation of the linear and nonlinear theory of elliptic partial differential equations in unbounded domains. ... The presentation of the results is mostly self-contained. Thus, the book can be recommended either for experts in the subject or for beginners." (Sergei V. Rogosin, Zentralblatt MATH, Vol. 1222, 2011)

Spis treści

Chapter 1. Introduction.- Chapter 2. Function spaces and operators.- Chapter 3. A priori estimates.- Chapter 4. Normal solvability.- Chapter 5. Fredholm property.- Chapter 6. Formally adjoint problems.- Chapter 7. Elliptic problems with a parameter.- Chapter 8. Index of elliptic operators.- Chapter 9. Problems in cylinders.- Chapter 10. Non-Fredholm operators.- Chapter 11. Nonlinear Fredholm operators.- Supplement. Discrete operators.-Historical and bibliographical comments.- Acknowledgements.- References.

Blurb

<p>The theory of elliptic partial differential equations has undergone an important development over the last two centuries. Together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications, it has become one of the most richly enhanced fields of mathematics. This monograph undertakes a systematic presentation of the theory of general elliptic operators. The author discusses a priori estimates, normal solvability, the Fredholm property, the index of an elliptic operator, operators with a parameter, and nonlinear Fredholm operators. Particular attention is paid to elliptic problems in unbounded domains which have not yet been sufficiently treated in the literature and which require some special approaches. The book also contains an analysis of non-Fredholm operators and discrete operators as well as extensive historical and bibliographical comments.</p>

<p>The selected topics and the author's level of discourse will make this book a most useful resource for researchers and graduate students working in the broad field of partial differential equations and applications.</p>