An Introduction to Sobolev Spaces and Interpolation Spaces (Lecture Notes of the Unione Matematica Italiana, 3)

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. 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