An Introduction to Sobolev Spaces and Interpolation Spaces

Luc Tartar

Springer Science & Business Media, May 26, 2007 - Mathematics - 219 pages

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.

Contents

• Historical Background: 1
• Density of Tensor Products Consequences: 26
• 8: 36
• Sobolevs Embedding Theorem N p: 42
• The Equivalence Lemma Compact Embeddings: 52
• Traces on the Boundary: 64
• LaxMilgram Lemma: 68
• The Fourier Transform: 73
• The Space Hdiv: 78
• Traces of HsRN: 83
• Background on Interpolation the Complex Method: 102
• Interpolation of L² Spaces with Weights: 117
• Interpolation Inequalities the Spaces Eo E101: 123
• Maximal Functions: 130
• Obtaining LP by Interpolation with the Exact Norm: 144
• Sobolevs Embedding Theorem for Besov Spaces: 155
• Defining Sobolev Spaces and Besov Spaces: 162
• Characterization of W8P: 173
• Shocks for QuasiLinear Hyperbolic Systems: 182
• Interpolation Spaces as Trace Spaces: 191
• Biographical Information 205: 204
• Proving that a Point is too Small: 217

About the author

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.

Bibliographic information

• Title: An Introduction to Sobolev Spaces and Interpolation Spaces; Volume 3 of Lecture Notes of the Unione Matematica Italiana
• Author: Luc Tartar
• Edition: illustrated
• Publisher: Springer Science & Business Media, 2007
• ISBN: 3540714839, 9783540714835
• Length: 219 pages
• Subjects: Mathematics / Calculus; Mathematics / Differential Equations / General; Mathematics / Functional Analysis; Mathematics / Mathematical Analysis