文档介绍：Weak KAM Theorem in
Lagrangian Dynamics
Seventh Preliminary
Version
Albert FATHI
Pisa, Version 16 February, 2005
ii
Contents
Preface vii
Introduction ix
The Hamilton-Jacobi Method . . . . . . . . . . . . xi
1 Convex Functions: Legendre and Fenchel 1
Convex Functions: General Facts . . . . . . . . . . 1
Linear Supporting Form and Derivative . . . . . . 7
The Fenchel Transform . . . . . . . . . . . . . . . . 11
Differentiable Convex Functions and Legendre Trans-
form . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Quasi-convex functions . . . . . . . . . . . . . . . . 28
Exposed Points of a Convex Set . . . . . . . . . . . 31
2 Calculus of Variations 37
Lagrangian, Action, Minimizers, and Extremal Curves 37
Lagrangians on Open Subsets of Rn . . . . . . . . 40
Lagrangians on Manifolds . . . . . . . . . . . . . . 48
The Euler-Lagrange Equation and its Flow . . . . 51
Symplectic Aspects . . . . . . . . . . . . . . . . . . 54
Lagrangian and Hamiltonians . . . . . . . . . . . . 58
Existence of Local Extremal Curves . . . . . . . . 63
The Hamilton-Jacobi method . . . . . . . . . . . . 71
3 Calculus of Variations for a Lagrangian Convex in
the Fibers: Tonelli’s Theory 81
Absolutely Continuous Curves. . . . . . . . . . . . 81
Lagrangian Convex in the Fibers . . . . . . . . . . 89
Tonelli’s Theorem . . . . . . . . . . . . . . . . . . 95
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Tonelli Lagrangians . . . . . . . . . . . . . . . . . . 98
Hamilton-Jacobi and Minimizers . . . . . . . . . . 101
Small Extremal Curves Are Minimizers . . . . . . 103
Regularity of Minimizers . . . . . . . . . . . . . . . 106
4 The Weak KAM Theorem 109
The Hamilton-Jacobi Equation Revisited . . . . . . 109
More on Dominated Functions and Calibrated Curves117
Properties . . . . . . . . . . . . . . . 130
The Lax-Oleinik Semigroup. . . . . . . . . .