An introduction to Sobolev spaces and interpolation spaces / Tartar L. - Berlin; New York: Springer, 2007. - 218 p. - (Lecture notes of the Unione Matematica Italiana; Vol. 3). - ISSN 1862-9113; ISBN 978-3540-71482-8  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

1. Historical Background ....................................... 1 2. The Lebesgue Measure, Convolution ........................... 9 3. Smoothing by Convolution ................................... 15 4. Truncation; Radon Measures; Distributions .................. 17 5. Sobolev Spaces; Multiplication by Smooth Functions ......... 21 6. Density of Tensor Products; Consequences ................... 27 7. Extending the Notion of Support ............................ 33 8. Sobolev's Embedding Theorem, 1 ≤ p < N ..................... 37 9. Sobolev's Embedding Theorem, N ≤ p ≤ ∞ ..................... 43 10. Poincare's Inequality ...................................... 49 11. The Equivalence Lemma; Compact Embeddings .................. 53 12. Regularity of the Boundary; Consequences ................... 59 13. Traces on the Boundary ..................................... 65 14. Green's Formula ............................................ 69 15. The Fourier Transform ...................................... 73 16. Traces of HB(RN) ........................................... 81 17. Proving that a Point is too Small .......................... 85 18. Compact Embeddings ......................................... 89 19. Lax-Milgram Lemma .......................................... 93 20. The Space H(div; Ω) ........................................ 99 21. Background on Interpolation; the Complex Method ........... 103 22. Real Interpolation; K-Method .............................. 109 23. Interpolation of L2 Spaces with Weights ................... 115 24. Real Interpolation; J-Method .............................. 119 25. Interpolation Inequalities, the Spaces (E0,E1)θ,1 .......... 123 26. The Lions-Peetre Reiteration Theorem ...................... 127 27. Maximal Functions ......................................... 131 28. Bilinear and Nonlinear Interpolation ...................... 137 29. Obtaining Lp by Interpolation, with the Exact Norm ........ 141 30. My Approach to Sobolev's Embedding Theorem ................ 145 31. My Generalization of Sobolev's Embedding Theorem .......... 149 32. Sobolev's Embedding Theorem for Besov Spaces .............. 155 33. The Lions-Magenes Space H001/2(Ω) .......................... 159 34. Denning Sobolev Spaces and Besov Spaces for Ω ............. 163 35. Characterization of Ws,p(RN) ............................... 165 36. Characterization of Ws,p(Ω)................................. 169 37. Variants with BV Spaces ................................... 173 38. Replacing BV by Interpolation Spaces ...................... 177 39. Shocks for Quasi-Linear Hyperbolic Systems ................ 183 40. Interpolation Spaces as Trace Spaces ...................... 191 41. Duality and Compactness for Interpolation Spaces .......... 195 42. Miscellaneous Questions ................................... 199 43. Biographical Information .................................. 205 44. Abbreviations and Mathematical Notation ................... 209 References .................................................... 213 Index ......................................................... 215