Calculus of Variations and Nonlinear Partial Differential Equations : Lectures Given at the C. I. M. E. Summer School Held in Cetraro, Italy, June 27-July 2 2005

by Ambrosio, Luigi; Caffarelli, Luis; Crandall, Michael G.; Evans, Lawrence C.; Fusco, Nicola

• ISBN13: 9783540759133
• ISBN10: 3540759131
• Edition: 1st
• Format: Paperback
• Copyright: 2008-01-15
• Publisher: Springer Verlag

Summary

This volume provides the texts of lectures given by L. Ambrosio, L. Caffarelli, M. Crandall, L.C. Evans, N. Fusco at the Summer course held in Cetraro, Italy in 2005. These are introductory reports on current research by world leaders in the fields of calculus of variations and partial differential equations. Coverage includes transport equations for nonsmooth vector fields, viscosity methods for the infinite Laplacian, and geometrical aspects of symmetrization.

Table of Contents

Transport Equation and Cauchy Problem for Non-Smooth Vector Fields	p. 1
Introduction	p. 1
Transport Equation and Continuity Equation within the Cauchy-Lipschitz Framework	p. 4
ODE Uniqueness versus PDE Uniqueness	p. 8
Vector Fields with a Sobolev Spatial Regularity	p. 19
Vector Fields with a BV Spatial Regularity	p. 27
Applications	p. 31
Open Problems, Bibliographical Notes, and References	p. 34
References	p. 37
Issues in Homogenization for Problems with Non Divergence Structure	p. 43
Introduction	p. 43
Homogenization of a Free Boundary Problem: Capillary Drops	p. 44
Existence of a Minimizer	p. 46
Positive Density Lemmas	p. 47
Measure of the Free Boundary	p. 51
Limit as ¿ → 0	p. 53
Hysteresis	p. 54
References	p. 57
The Construction of Plane Like Solutions to Periodic Minimal Surface Equations	p. 57
References	p. 64
Existence of Homogenization Limits for Fully Nonlinear Equations	p. 65
Main Ideas of the Proof	p. 67
References	p. 73
References	p. 74
A Visit with the ∞-Laplace Equation	p. 75
Notation	p. 78
The Lipschitz Extension/Variational Problem	p. 79
Absolutely Minimizing Lipschitz iff Comparison With Cones	p. 83
Comparison With Cones Implies ∞-Harmonic	p. 84
∞-Harmonic Implies Comparison with Cones	p. 86
Exercises and Examples	p. 86
From ∞-Subharmonic to ∞-Superharmonic	p. 88
More Calculus of ∞-Subharmonic Functions	p. 89
Existence and Uniqueness	p. 97
The Gradient Flow and the Variational Problem for <$>\parallel Du \parallel_{L^\infty}<$>	p. 102
Linear on All Scales	p. 105
Blow Ups and Blow Downs are Tight on a Line	p. 105
Implications of Tight on a Line Segment	p. 107
An Impressionistic History Lesson	p. 109
The Beginning and Gunnar Aronosson	p. 109
Enter Viscosity Solutions and R. Jensen	p. 111
Regularity	p. 113
Modulus of Continuity	p. 113
Harnack and Liouville	p. 113
Comparison with Cones, Full Born	p. 114
Blowups are Linear	p. 115
Savin's Theorem	p. 115
Generalizations, Variations, Recent Developments and Games	p. 116
What is ¿∞ for H(x, u, Du)?	p. 116
Generalizing Comparison with Cones	p. 118
The Metric Case	p. 118
Playing Games	p. 119
Miscellany	p. 119
References	p. 120
Weak KAM Theory and Partial Differential Equations	p. 123
Overview, KAM theory	p. 123
Classical Theory	p. 123
The Lagrangian Viewpoint	p. 124
The Hamiltonian Viewpoint	p. 125
Canonical Changes of Variables, Generating Functions	p. 126
Hamilton-Jacobi PDE	p. 127
KAM Theory	p. 127
Generating Functions, Linearization	p. 128
Fourier series	p. 128
Small divisors	p. 129
Statement of KAM Theorem	p. 129
Weak KAM Theory: Lagrangian Methods	p. 131
Minimizing Trajectories	p. 131
Lax-Oleinik Semigroup	p. 131
The Weak KAM Theorem	p. 132
Domination	p. 133
Flow invariance, characterization of the constant c	p. 135
Time-reversal, Mather set	p. 137
Weak KAM Theory: Hamiltonian and PDE Methods	p. 137
Hamilton-Jacobi PDE	p. 137
Adding P Dependence	p. 138
Lions-Papanicolaou-Varadhan Theory	p. 139
A PDE construction of <$>\bar {H}<$>	p. 139
Effective Lagrangian	p. 140
Application: Homogenization of Nonlinear PDE	p. 141
More PDE Methods	p. 141
Estimates	p. 144
An Alternative Variational/PDE Construction	p. 145
A new Variational Formulation	p. 145
A Minimax Formula	p. 146
A New Variational Setting	p. 146
Passing to Limits	p. 147
Application: Nonresonance and Averaging	p. 148
Derivatives of <$>{\overline {\bf H}}^k<$>	p. 148
Nonresonance	p. 148
Some Other Viewpoints and Open Questions	p. 150
References	p. 152
Geometrical Aspects of Symmetrization	p. 155
Sets of finite perimeter	p. 155
Steiner Symmetrization of Sets of Finite Perimeter	p. 164
The Pòlya-Szegö Inequality	p. 171
References	p. 180
CIME Courses on Partial Differential Equations and Calculus of Variations	p. 183
Table of Contents provided by Publisher. All Rights Reserved.