An Introduction to Sobolev Spaces and Interpolation Spaces

By: Luc Tartar

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Paperback | 6 June 2007

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248 Pages

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This volume appears parallel to the conference in honor of Luc Tartar on the occasion of his 60th birthday held in Paris, July 2007 at the CMAP of the Ecole Polytechnique.

Partly in collaboration with Fran ois Murat, Luc Tartar has specialized in the development of new mathematical tools for solving the partial differential equations of continuum mechanics (homogenization, compensated compactness, H-measures), pioneering the study of microstructures compatible with the partial differential equations describing the physical balance laws, and the constitutive relations.

He likes to point out the defects of many of the models which are used, as a natural way to achieve the goal of improving our understanding of mathematics and of continuum mechanics.

Industry Reviews

From the reviews:

"This book is based on a set of lecture notes prepared by the author from a graduate course ... . The main themes are Sobolev spaces and interpolation theory. ... The book contains 42 chapters, each intended to contain the amount of material which would be suitable for a graduate lecture. ... As well as being an excellent source of material for a graduate course on topics ... this book contains a great deal which will be of interest to the seasoned researcher." (W. D. Evans, Zentralblatt MATH, Vol. 1126 (3), 2008)

"This is a book that has grown out of a graduate course taught by the author in 2000. It keeps the structure of a set of lectures ... . Many interesting remarks are given along the text, and by means of a large number of footnotes the author explains many anecdotes and personal experiences related with people associated to the development of the topics included in the text. This book can be useful not only as a source in graduate courses, but also for researchers." (Joan L. Cerda, Mathematical Reviews, Issue 2008 g)

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Historical background
The Lebesgue measure, convolution
Smoothing by convolution
Truncation, Radon measures, distributions
Sobolev spaces, multiplication by smooth functions
Density of tensor products, consequences
Extending the notion of support
Sobolev's embedding theorem, 1 ≤ p < N
Sobolev's embedding theorem, N ≤ p ≤ ∞
Poincar'e's inequality
The equivalence lemma, compact embeddings
Regularity of the boundary, consequences
Traces on the boundary
Green's formula
The Fourier transform
Traces of Hs(RN)
Proving that a point is too small
Compact embeddings
Lax-Milgram lemma
The space H(div; ¿)
Background on interpolation, the complex method
Real interpolation: K-method
Interpolation of L2 spaces with weights
Real interpolation: J-method
Interpolation inequalities, the spaces (E0,E1)¿,1
The Lions-Peetre reiteration theorem
Maximal functions
Bilinear and nonlinear interpolation
Obtaining Lp by interpolation, with the exact norm
My approach to Sobolev's embedding theorem
My generalization of Sobolev's embedding theorem
Sobolev's embedding theorem for Besov spaces
The Lions-Magenes space H001/2(¿ )
Defining Sobolev spaces and Besov spaces for ¿
Characterization of Ws,p(RN)
Characterization of Ws,p (¿)
Variants with BV spaces
Replacing BV by interpolation spaces
Shocks for quasi-linear hyperbolic systems
Interpolation spaces as trace spaces
Duality and compactness for interpolation spaces
Miscellaneous questions
Biographical information
Abbreviations and mathematical notation
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

An Introduction to Sobolev Spaces and Interpolation Spaces

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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