An Introduction to Sobolev Spaces and Interpolation Spaces

ISBN-13: 9783540714828 / Angielski / Miękka / 2007 / 218 str.

Luc Tartar

Opis

After publishing an introduction to the Navier-Stokes equation and oceanography (Vol. 1 of this series), Luc Tartar follows with another set of lecture notes based on a graduate course in two parts, as indicated by the title. A draft has been available on the internet for a few years. The author has now revised and polished it into a text accessible to a larger audience.

Recenzje

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. Cerdà, Mathematical Reviews, Issue 2008 g)

Spis treści

1.Historical background.- 2.The Lebesgue measure, convolution.- 3.Smoothing by convolution.- 4.Truncation, Radon measures, distributions.- 5.Sobolev spaces, multiplication by smooth functions.- 6.Density of tensor products, consequences.- 7.Extending the notion of support.- 8.Sobolev’s embedding theorem, 1 \leq p < N.- 9.Sobolev’s embedding theorem, N \leq p \leq \infty.- 10.Poincar´e’s inequality.-11.The equivalence lemma, compact embeddings.- 12.Regularity of the boundary, consequences.- 13.Traces on the boundary.- 14.Green’s formula.-15.The Fourier transform.- 16.Traces of Hs(RN).- 17.Proving that a point is too small.- 18.Compact embeddings.- 19.Lax–Milgram lemma.- 20.The space H(div; \Omega).- 21.Background on interpolation, the complex method.- 22.Real interpolation: K-method.- 23.Interpolation of L2 spaces with weights.- 24.Real interpolation: J-method.- 25.Interpolation inequalities, the spaces (E0,E1)\theta,1.- 26.The Lions–Peetre reiteration theorem.- 27.Maximal functions.- 28.Bilinear and nonlinear interpolation.- 29.Obtaining Lp by interpolation, with the exact norm.- 30.My approach to Sobolev’s embedding theorem.- 31.My generalization of Sobolev’s embedding theorem.- 32.Sobolev’s embedding theorem for Besov spaces.- 33.The Lions–Magenes space H001/2(\Omega ).- 34.Defining Sobolev spaces and Besov spaces for \Omega.- 35.Characterization of Ws,p(RN).- 36.Characterization of Ws,p (\Omega).- 37.Variants with BV spaces.- 38.Replacing BV by interpolation spaces.- 39.Shocks for quasi-linear hyperbolic systems.- 40.Interpolation spaces as trace spaces.- 41.Duality and compactness for interpolation spaces.- 42.Miscellaneous questions.-43.Biographical information.- 44.Abbreviations and mathematical notation.- References.- Index.

O autorze

Luc Tartar studied at Ecole Polytechnique in Paris, France, 1965-1967, where he was taught by Laurent Schwartz and Jacques-Louis Lions in mathematics, and by Jean Mandel in continuum mechanics.

He did research at Centre National de la Recherche Scientifique, Paris, France, 1968-1971, working under the direction of Jacques-Louis Lions for his thèse d'état, 1971.

He taught at Université Paris IX-Dauphine, Paris, France, 1971-1974, at University of Wisconsin, Madison, WI, 1974-1975, at Université de Paris-Sud, Orsay, France, 1975-1982.

He did research at Commissariat à l'Energie Atomique, Limeil, France, 1982-1987.

In 1987, he was elected Correspondant de l'Académie des Sciences, Paris, in the section Mécanique.

Since 1987 he has been teaching at Carnegie Mellon University, Pittsburgh, PA, where he has been University Professor of Mathematics since 1994.

Partly in collaboration with François Murat, he 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.