Contents -- Preface -- 1. Functional spaces and convergence -- 1.1 Definitions and examples of functional spaces -- 1.2 Holder inequality. Young inequality for convolution -- 1.3 Arzela-Ascoli theorem -- 1.4 Hilbert space -- 1.5 Weak convergence -- 1.6 Linear operators in Hilbert space -- 1.7 Differentiable functionals -- 1.8 Continuous and differentiable functionals in LP-spaces -- 2. Sobolev spaces -- 2.1 Weak derivatives . Definition of Sobolev spaces -- 2.2 Chain rule -- 2.3 Coordinate transformations Trace domains and extension domain -- 2.4 Friedrichs inequality -- 2.5 Compactness lemma -- 2.6 Poincar inequality -- 2.7 Space D1,2(RN). SobIove, Hardy and Nash inequalities -- 2.8 Sobolev imbeddings -- 2.9 Trace on the boundary -- 2.10 Differentiable functionals in Sobolev spaces -- 2.11 Sobolev spaces of higher order -- 3. Weak convergence decomposition -- 3.1 D-weak convergence and dislocation spaces -- 3.2 D-weak convergence in l2 with shifts -- 3.3 Weak convergence decomposition -- 3.4 Uniqueness in the weak convergence decomposition -- 3.5 D-flask subspaces. D-weak compactness -- 3.6 D-weak convergence with shift operators in RN -- 3.7 Constrained minimization -- 3.8 Compactness in the presence of symmetries -- 3.9 The concentration compactness argument -- 3.10 Bibliographic remarks -- 4. Concentration compactness with Euclidean shifts -- 4.1 Flask sets -- 4.2 Existence of Sobolev minimizers on flask domains -- 4.3 Rellich sets and compactness of Sobolev imbeddings -- 4.4 Concentration compactness with symmetry -- 4.5 Concentration compactness and the Friedrichs inequality -- 4.6 Solvability in non-flask domains -- 4.7 Convergence by penalty at infinity -- 4.8 Minimizers with finite symmetry -- 4.9 Positive non-extremal solutions -- 4.10 Bibliographic remarks -- 5. Concentration compactness with dilations.

5.1 Semilinear elliptic equations with the critical exponent -- 5.2 Oscillatory critical nonlinearity and the minimizer in the Sobolev inequality -- 5.3 The Brezis-Nirenberg problem -- 5.4 Minimizer for the critical trace inequality -- 5.5 A singular subcritical problem -- 5.6 Minimizer for the Hardy-Sobolev-Maz'ya inequality -- 5.7 Bibliographic remarks -- 6. Minimax problems -- 6.1 The mountain pass theorem -- 6.2 Functionals for the semilinear elliptic problems -- 6.3 Critical points of the mountain pass type -- 6.4 Mountain pass problems with the critical exponent -- 6.5 Critical problem with punitive asymptotic values -- 6.6 Bibliographic remarks -- 7 . Differentiable manifolds -- 7.1 Differentiable manifolds -- 7.2 Tangent vectors and vector fields -- 7.3 Cotangent vectors and 1-forms -- 7.4 Tensor fields of degree 2 -- 7.5 Differential forms -- 8 . Riemannian manifolds and Lie groups -- 8.1 Riemannian manifolds -- 8.2 Lie groups -- 8.3 The exponential map -- 8.4 Lie group actions -- 8.5 Integration -- 8.6 Bibliographic remarks -- 9 . Sobolev spaces on manifolds and subelliptic problems -- 9.1 Sobolev inequality on periodic manifolds -- 9.2 "Magnetic" Sobolev space -- 9.3 Magnetic shifts and D-convergence -- 9.4 Subelliptic mollifiers and Sobolev spaces on Carnot groups -- 9.5 Compactness of subelliptic Sobolev imbeddings -- 9.6 Subelliptic Friedrichs and Poincark inequalities -- 9.7 Subelliptic Sobolev inequality -- 9.8 Concentration compactness on Carnot groups due to shifts -- 9.9 Concentration compactness on Carnot groups due to dilations -- 9.10 Bibliographic remarks -- 10 . Further applications -- 10.1 Dilations on the sphere and Yamabe problem -- 10.2 Global compactness in spaces Hm(RN) and Dm. 2(RN) -- 10.3 Minimizer in the Nash inequality -- 10.4 A minimization problem with nonlocal term.

10.5 Concentration compactness with topological charge -- 10.6 Bibliographic remarks -- Appendix A Covering lemma -- Appendix B Rearrangement inequalities -- Appendix C Maximum principle -- Bibliography -- Index.