Analytic Semigroups and Optimal Regularity in Parabolic Problems: 16 (Progress in Nonlinear Differential Equations and Their Applications) - Softcover

Lunardi, Alessandra

Softcover

ISBN 10: 3034899564 ISBN 13: 9783034899567

Publisher: Birkhäuser, 2011

Synopsis

This book gives a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and of how such a theory may be used in parabolic PDE's. It takes into account the developments of the theory during the last fifteen years, and it is focused on classical solutions, with continuous or Holder continuous derivatives. On one hand, working in spaces of continuous functions rather than in Lebesgue spaces seems to be appropriate in view of the number of parabolic problems arising in applied mathematics, where continuity has physical meaning; on the other hand it allows one to consider any type of nonlinearities (even of nonlocal type), even involving the highest order derivatives of the solution, avoiding the limitations on the growth of the nonlinear terms required by the LP approach. Moreover, the continuous space theory is, at present, sufficiently well established. For the Hilbert space approach we refer to J. L. LIONS - E. MAGENES [128], M. S. AGRANOVICH - M. l. VISHIK [14], and for the LP approach to V. A. SOLONNIKOV [184], P. GRISVARD [94], G. DI BLASIO [72], G. DORE - A. VENNI [76] and the subsequent papers [90], [169], [170]. Many books about abstract evolution equations and semigroups contain some chapters on analytic semigroups. See, e. g. , E. HILLE - R. S. PHILLIPS [100]' S. G. KREIN [114], K. YOSIDA [213], A. PAZY [166], H. TANABE [193], PH.

About this title

From the Back Cover

The book shows how the abstract methods of analytic semigroups and evolution equations in Banach spaces can be fruitfully applied to the study of parabolic problems.

Particular attention is paid to optimal regularity results in linear equations. Furthermore, these results are used to study several other problems, especially fully nonlinear ones.

Owing to the new unified approach chosen, known theorems are presented from a novel perspective and new results are derived.

The book is self-contained. It is addressed to PhD students and researchers interested in abstract evolution equations and in parabolic partial differential equations and systems. It gives a comprehensive overview on the present state of the art in the field, teaching at the same time how to exploit its basic techniques.

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This very interesting book provides a systematic treatment of the basic theory of analytic semigroups and abstract parabolic equations in general Banach spaces, and how this theory may be used in the study of parabolic partial differential equations; it takes into account the developments of the theory during the last fifteen years. (...) For instance, optimal regularity results are a typical feature of abstract parabolic equations; they are comprehensively studied in this book, and yield new and old regularity results for parabolic partial differential equations and systems.
(Mathematical Reviews)  

Motivated by applications to fully nonlinear problems the approach is focused on classical solutions with continuous or Hölder continuous derivatives.
(Zentralblatt MATH)

About the Author

Alessandra Lunardi is a professor of mathematics at the University of Parma, Italy.

Edition details

• Publisher: Birkhäuser
• Publication date: 2011
• Language: English
• ISBN 10: 3034899564
• ISBN 13: 9783034899567
• Binding: Paperback
• Number of pages: 448