Concentration compactness: functional-analytic grounds and applications

Contents......Page 10
Preface......Page 6
1.1 Definitions and examples of functional spaces......Page 14
1.2 Holder inequality. Young inequality for convolution......Page 17
1.3 Arzela—Ascoli theorem......Page 20
1.4 Hilbert space......Page 21
1.5 Weak convergence......Page 26
1.6 Linear operators in Hilbert space......Page 29
1.7 Differentiable functionals......Page 33
1.8 Continuous and differentiable functionals in LP-spaces......Page 36
2.1 Weak derivatives . Definition of Sobolev spaces......Page 42
2.2 Chain rule......Page 45
2.3 Coordinate transformations Trace domains and extension domain......Page 47
2.4 Friedrichs inequality......Page 50
2.5 Compactness lemma......Page 52
2.6 Poincar inequality......Page 54
2.7 Space D1,2(RN). SobIove, Hardy and Nash inequalities......Page 56
2.8 Sobolev imbeddings......Page 60
2.9 Trace on the boundary......Page 64
2.10 Differentiable functionals in Sobolev spaces......Page 69
2.11 Sobolev spaces of higher order......Page 70
3. Weak convergence decomposition......Page 72
3.1 D-weak convergence and dislocation spaces......Page 73
3.2 D-weak convergence in l2 with shifts......Page 74
3.3 Weak convergence decomposition......Page 75
3.4 Uniqueness in the weak convergence decomposition......Page 81
3.5 D-flask subspaces. D-weak compactness......Page 82
3.6 D-weak convergence with shift operators in RN......Page 83
3.7 Constrained minimization......Page 88
3.8 Compactness in the presence of symmetries......Page 90
3.9 The concentration compactness argument......Page 92
3.10 Bibliographic remarks......Page 93
4.1 Flask sets......Page 96
4.2 Existence of Sobolev minimizers on flask domains......Page 102
4.3 Rellich sets and compactness of Sobolev imbeddings......Page 103
4.4 Concentration compactness with symmetry......Page 104
4.5 Concentration compactness and the Friedrichs inequality......Page 105
4.6 Solvability in non-flask domains......Page 108
4.7 Convergence by penalty at infinity......Page 111
4.8 Minimizers with finite symmetry......Page 113
4.9 Positive non-extremal solutions......Page 115
4.10 Bibliographic remarks......Page 120
5.1 Semilinear elliptic equations with the critical exponent......Page 122
5.2 Oscillatory critical nonlinearity and the minimizer in the Sobolev inequality......Page 129
5.3 The Brezis-Nirenberg problem......Page 134
5.4 Minimizer for the critical trace inequality......Page 139
5.5 A singular subcritical problem......Page 144
5.6 Minimizer for the Hardy-Sobolev-Maz'ya inequality......Page 149
5.7 Bibliographic remarks......Page 150
6. Minimax problems......Page 154
6.1 The mountain pass theorem......Page 155
6.2 Functionals for the semilinear elliptic problems......Page 158
6.3 Critical points of the mountain pass type......Page 162
6.4 Mountain pass problems with the critical exponent......Page 168
6.5 Critical problem with punitive asymptotic values......Page 170
6.6 Bibliographic remarks......Page 172
7.1 Differentiable manifolds......Page 174
7.2 Tangent vectors and vector fields......Page 177
7.3 Cotangent vectors and 1-forms......Page 183
7.4 Tensor fields of degree 2......Page 185
7.5 Differential forms......Page 188
8.1 Riemannian manifolds......Page 194
8.2 Lie groups......Page 202
8.3 The exponential map......Page 206
8.4 Lie group actions......Page 210
8.5 Integration......Page 212
8.6 Bibliographic remarks......Page 214
9.1 Sobolev inequality on periodic manifolds......Page 216
9.2 “Magnetic” Sobolev space......Page 218
9.3 Magnetic shifts and D-convergence......Page 219
9.4 Subelliptic mollifiers and Sobolev spaces on Carnot groups......Page 223
9.5 Compactness of subelliptic Sobolev imbeddings......Page 230
9.6 Subelliptic Friedrichs and Poincark inequalities......Page 231
9.7 Subelliptic Sobolev inequality......Page 234
9.8 Concentration compactness on Carnot groups due to shifts......Page 235
9.9 Concentration compactness on Carnot groups due to dilations......Page 237
9.10 Bibliographic remarks......Page 240
10.1 Dilations on the sphere and Yamabe problem......Page 244
10.2 Global compactness in spaces Hm(RN) and Dm. 2(RN)......Page 245
10.3 Minimizer in the Nash inequality......Page 249
10.4 A minimization problem with nonlocal term......Page 250
10.5 Concentration compactness with topological charge......Page 253
10.6 Bibliographic remarks......Page 257
Appendix A Covering lemma......Page 260
Appendix B Rearrangement inequalities......Page 262
Appendix C Maximum principle......Page 264
Bibliography......Page 266
Index......Page 274