Concentration Compactness

The following sections are included:

• Preface

• Contents

The following sections are included:

• Definitions and examples of functional spaces

• Hölder inequality. Young inequality for convolutions

• Arzelá–Ascoli theorem

• Hilbert space

• Weak convergence

• Linear operators in Hilbert space

• Differentiable functionals

• Continuous and differentiable functionals in L^p-spaces

The following sections are included:

• Weak derivatives. Definition of Sobolev spaces

• Chain rule

• Coordinate transformations. Trace domains and extension domains

• Friedrichs inequality

• Compactness lemma

• Poincaré inequality

• Space . Sobolev, Hardy and Nash inequalities

• Sobolev imbeddings

• Trace on the boundary

• Differentiable functionals in Sobolev spaces

• Sobolev spaces of higher order

The following sections are included:

• D-weak convergence and dislocation spaces

• D-weak convergence in l² with shifts

• Weak convergence decomposition

• Uniqueness in the weak convergence decomposition

• D-flask subspaces. D-weak compactness

• D-weak convergence with shift operators in ℝ^N

• Constrained minimization

• Compactness in the presence of symmetries

• The concentration compactness argument

• Bibliographic remarks

The following sections are included:

• Flask sets

• Existence of Sobolev minimizers on flask domains

• Rellich sets and compactness of Sobolev imbeddings

• Concentration compactness with symmetry

• Concentration compactness and the Friedrichs inequality

• Solvability in non-flask domains

• Convergence by penalty at infinity

• Minimizers with finite symmetry

• Positive non-extremal solutions

• Bibliographic remarks

The following sections are included:

• Semilinear elliptic equations with the critical exponent

• Oscillatory critical nonlinearity and the minimizer in the Sobolev inequality

• The Brézis–Nirenberg problem

• Minimizer for the critical trace inequality

• A singular subcritical problem

• Minimizer for the Hardy–Sobolev–Maz'ya inequality

• Bibliographic remarks

The following sections are included:

• The mountain pass theorem

• Functionals for the semilinear elliptic problems

• Critical points of the mountain pass type

• Mountain pass problems with the critical exponent

• Critical problem with punitive asymptotic values

• Bibliographic remarks

The following sections are included:

• Differentiable manifolds

• Tangent vectors and vector fields

• Cotangent vectors and 1-forms

• Tensor fields of degree 2

• Differential forms

The following sections are included:

• Riemannian manifolds

• Lie groups

• The exponential map

• Lie group actions

• Integration

• Bibliographic remarks

The following sections are included:

• Sobolev inequality on periodic manifolds

• “Magnetic” Sobolev space

• Magnetic shifts and D-convergence

• Subelliptic mollifiers and Sobolev spaces on Carnot groups

• Compactness of subelliptic Sobolev imbeddings

• Subelliptic Friedrichs and Poincaré inequalities

• Subelliptic Sobolev inequality

• Concentration compactness on Carnot groups due to shifts

• Concentration compactness on Carnot groups due to dilations

• Bibliographic remarks

The following sections are included:

• Dilations on the sphere and Yamabe problem

• Global compactness in spaces H^m(ℝ^N) and

• Minimizer in the Nash inequality

• A minimization problem with nonlocal term

• Concentration compactness with topological charge

• Bibliographic remarks

The following sections are included:

• Appendix A — Covering lemma

• Appendix B — Rearrangement inequalities

• Appendix C — Maximum principle

• Bibliography

• Index