Vitaly Volpert

Elliptic Partial Differential Equations (eBook, PDF)

Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains

• Format: PDF

Produktbeschreibung

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.

Produktdetails

• Verlag: Springer Basel
• Erscheinungstermin: 3. März 2011
• Englisch
• ISBN-13: 9783034605373
• Artikelnr.: 37364270

Autorenporträt

Vitaly Volpert started his scientific career in Russia and continued it in the USA and in France. He works on partial differential equations and on mathematical modelling in chemical physics, biology and medicine. He is an author of more than 200 scientific publications including three monographs.

Inhaltsangabe

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.

---

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.

---

I. Introduction to the theory of reaction-diffusion equations.- Chapter 1. Reaction-diffusion processes, models and applications.- Chapter 2. Methods of analysis.- Chapter 3. Reaction-diffusion problems in bounded domains.- Chapter 4. Reaction-diffusion problems on the whole axis.- II. Reaction-diffusion waves in cylinders.- Chapter 5. Monotone systems.- Chapter 6. Reaction-diffusion problems with convection.- Chapter 7. Reaction-diffusion systems with different diffusion coefficients.- Chapter 8. Nonlinear boundary conditions.- Chapter 9. Nonlocal reaction-diffusion equations.- Chapter 10. Multi-scale models in biology.- Bibliographical comments.- Concluding remarks.- Acknowledgements.- References.- Index.

Rezensionen

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)