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.
Author(s): Luc Tartar (auth.)
Series: Lecture Notes of the Unione Matematica Italiana 3
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2007
Language: English
Pages: 219
Tags: Partial Differential Equations; Functional Analysis
Front Matter....Pages I-XXV
Historical Background....Pages 1-7
The Lebesgue Measure, Convolution....Pages 9-14
Smoothing by Convolution....Pages 15-16
Truncation; Radon Measures; Distributions....Pages 17-20
Sobolev Spaces; Multiplication by Smooth Functions....Pages 21-25
Density of Tensor Products; Consequences....Pages 27-31
Extending the Notion of Support....Pages 33-36
Sobolev's Embedding Theorem, 1 ≤ < N....Pages 37-41
Sobolev's Embedding Theorem, N ≤ p ≤ ∞....Pages 43-47
Poincaramp;#x00E9;'s Inequality....Pages 49-51
The Equivalence Lemma; Compact Embeddings....Pages 53-57
Regularity of the Boundary; Consequences....Pages 59-63
Traces on the Boundary....Pages 65-68
Green's Formula....Pages 69-71
The Fourier Transform....Pages 73-79
Traces of H s ( R N )....Pages 81-84
Proving that a Point is too Small....Pages 85-87
Compact Embeddings....Pages 89-92
Lax–Milgram Lemma....Pages 93-98
The Space H ( div ; Ω )....Pages 99-101
Background on Interpolation; the Complex Method....Pages 103-107
Real Interpolation; K -Method....Pages 109-113
Interpolation of L 2 Spaces with Weights....Pages 115-118
Real Interpolation; J -Method....Pages 119-122
Interpolation Inequalities, the Spaces ( E 0 , E 1 ) θ,1 ....Pages 123-125
The Lions–Peetre Reiteration Theorem....Pages 127-129
Maximal Functions....Pages 131-135
Bilinear and Nonlinear Interpolation....Pages 137-140
Obtaining L p by Interpolation, with the Exact Norm....Pages 141-143
My Approach to Sobolev's Embedding Theorem....Pages 145-147
My Generalization of Sobolev's Embedding Theorem....Pages 149-154
Sobolev's Embedding Theorem for Besov Spaces....Pages 155-158
The Lions–Magenes Space $H_{00}^{1/2} ( \Omega)$ ....Pages 159-161
Defining Sobolev Spaces and Besov Spaces for Ω ....Pages 163-164
Characterization of W s,p ( R N )....Pages 165-167
Characterization of W s,p ( Ω )....Pages 169-172
Variants with BV Spaces....Pages 173-176
Replacing BV by Interpolation Spaces....Pages 177-181
Shocks for Quasi-Linear Hyperbolic Systems....Pages 183-189
Interpolation Spaces as Trace Spaces....Pages 191-194
Duality and Compactness for Interpolation Spaces....Pages 195-198
Miscellaneous Questions....Pages 199-203
Biographical Information....Pages 205-207
Abbreviations and Mathematical Notation....Pages 209-212
Back Matter....Pages 213-219
