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Calculus of Variations

with X. Li-Jost; Cambridge University Press, 1998

Additions and Corrections: [PDF-Version (32k) | PostScript-Version (25k)]

Part one: One-dimensional variational problems

1 The classical theory

1.1 The Euler-Lagrange equations. Examples
1.2 The idea of the direct methods and some regularity results
1.3 The second variation. Jacobi fields
1.4 Free boundary conditions
1.5 Symmetries and the theorem of E. Noether

2 A geometrie example: geodesic curves

2.1 The length and energy of curves
2.2 Fields of geodesic curves
2.3 The existence of geodesics

3 Saddle point constructions

3.1 A finite dimensional example
3.2 The construction of Lyusternik-Schnirelman

4 The theory of Hamilton and Jacobi

4.1 The canonical equations
4.2 The Hamilton-Jacobi equation
4.3 Geodesics
4.4 Fields of extremals
4.5 Hilbert's invariant integral and Jacobi's theorem
4.6 Canonical transformations

5 Dynarnic optimization

5.1 Discrete control problems
5.2 Contimious control problems
5.3 The Pontryagin maximum principle

Part two: Multiple integrals in the calculus of variations

1 Lebesgue measure and integration theory

1.1 The Lebesgue measure and the Lebesgue integral
1.2 Convergence theorems

2 Banach spaces

2.1 Definition and basic properties of Banach and Hubert spaces
2.2 Dual spaces and weak convergence
2.3 Linear operators between Banach spaces
2.4 Calculus in Banach spaces

3 Lp and Sobolev spaces

3.1 Lp spaces
3.2 Approximation of Lp functions by smooth functions (mollification)
3.3 Sobolev spaces
3.4 Rellich's theorem and the Poincaré and Sobolev inequalities

4 The direct methods in the calculus of variations

4.1 Description of the problem and its solution
4.2 Lower semicontinuity
4.3 The existence of minimizers for convex variational problems
4.4 Convex functionals on Hubert spaces and Moreau-Yosida approximation
4.5 The Euler-Lagrange equations and regularity questions

5 Nonconvex functionals. Relaxation

5.1 Nonlower semicontinuous functionals and relaxation
5.2 Representation of relaxed functionals via convex envelopes

6 Γ-convergence

6.1 The definition of Γ-convergence
6.2 Homogenization
6.3 Thin insulating layers

7 BV-functionals and Γ-convergence: the example of Modica and Mortola

7.1 The space BV(Ω) 7.2 The example of Modica-Mortola

Appendix A The coarea formula
Appendix B The distance function from smooth hypersurfaces

8 Bifurcation theory

8.1 Bifurcation problems in the calculus of variations
8.2 The functional analytic approach to bifurcation theory
8.3 The existence of catenoids as an example of a bifurcation process

9 The Palais-Smale condition and unstable critical points of variational problems

9.1 The Palais-Smale condition
9.2 The mountain pass theorem
9.3 Topological indices and critical points

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