Cover image for 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
Title:
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
Series:
Lecture notes in mathematics, 1927
Publication Information:
Berlin : Springer, 2008
Physical Description:
xi, 196 p. : ill. ; 24 cm.
ISBN:
9783540759133

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30000010178998 QA315 C344 2008 Open Access Book Book
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Summary

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

Luigi AmbrosioLuis Caffarelli and Luis SilvestreMichael G. CrandallLawrence C. EvansNicola FuscoElvira Mascolo
Transport Equation and Cauchy Problem for Non-Smooth Vector Fieldsp. 1
1 Introductionp. 1
2 Transport Equation and Continuity Equation within the Cauchy-Lipschitz Frameworkp. 4
3 ODE Uniqueness versus PDE Uniquenessp. 8
4 Vector Fields with a Sobolev Spatial Regularityp. 19
5 Vector Fields with a BV Spatial Regularityp. 27
6 Applicationsp. 31
7 Open Problems, Bibliographical Notes, and Referencesp. 34
Referencesp. 37
Issues in Homogenization for Problems with Non Divergence Structurep. 43
1 Introductionp. 43
2 Homogenization of a Free Boundary Problem: Capillary Dropsp. 44
2.1 Existence of a Minimizerp. 46
2.2 Positive Density Lemmasp. 47
2.3 Measure of the Free Boundaryp. 51
2.4 Limit as ¿ → 0p. 53
2.5 Hysteresisp. 54
2.6 Referencesp. 57
3 The Construction of Plane Like Solutions to Periodic Minimal Surface Equationsp. 57
3.1 Referencesp. 64
4 Existence of Homogenization Limits for Fully Nonlinear Equationsp. 65
4.1 Main Ideas of the Proofp. 67
4.2 Referencesp. 73
Referencesp. 74
A Visit with the ∞-Laplace Equationp. 75
1 Notationp. 78
2 The Lipschitz Extension/Variational Problemp. 79
2.1 Absolutely Minimizing Lipschitz iff Comparison With Conesp. 83
2.2 Comparison With Cones Implies ∞-Harmonicp. 84
2.3 ∞-Harmonic Implies Comparison with Conesp. 86
2.4 Exercises and Examplesp. 86
3 From ∞-Subharmonic to ∞-Superharmonicp. 88
4 More Calculus of ∞-Subharmonic Functionsp. 89
5 Existence and Uniquenessp. 97
6 The Gradient Flow and the Variational Problem for \parallel|Du|\parallel_{{L^\infty}}p. 102
7 Linear on All Scalesp. 105
7.1 Blow Ups and Blow Downs are Tight on a Linep. 105
7.2 Implications of Tight on a Line Segmentp. 107
8 An Impressionistic History Lessonp. 109
8.1 The Beginning and Gunnar Aronossonp. 109
8.2 Enter Viscosity Solutions and R. Jensenp. 111
8.3 Regularityp. 113
Modulus of Continuityp. 113
Harnack and Liouvillep. 113
Comparison with Cones, Full Bornp. 114
Blowups are Linearp. 115
Savin's Theoremp. 115
9 Generalizations, Variations, Recent Developments and Gamesp. 116
9.1 What is ¿ ∞ for H(x, u, Du)?p. 116
9.2 Generalizing Comparison with Conesp. 118
9.3 The Metric Casep. 118
9.4 Playing Gamesp. 119
9.5 Miscellanyp. 119
Referencesp. 120
Weak KAM Theory and Partial Differential Equationsp. 123
1 Overview, KAM theoryp. 123
1.1 Classical Theoryp. 123
The Lagrangian Viewpointp. 124
The Hamiltonian Viewpointp. 125
Canonical Changes of Variables, Generating Functionsp. 126
Hamilton-Jacobi PDEp. 127
1.2 KAM Theoryp. 127
Generating Functions, Linearizationp. 128
Fourier seriesp. 128
Small divisorsp. 129
Statement of KAM Theoremp. 129
2 Weak KAM Theory: Lagrangian Methodsp. 131
2.1 Minimizing Trajectoriesp. 131
2.2 Lax-Oleinik Semigroupp. 131
2.3 The Weak KAM Theoremp. 132
2.4 Dominationp. 133
2.5 Flow invariance, characterization of the constant cp. 135
2.6 Time-reversal, Mather setp. 137
3 Weak KAM Theory: Hamiltonian and PDE Methodsp. 137
3.1 Hamilton-Jacobi PDEp. 137
3.2 Adding P Dependencep. 138
3.3 Lions-Papanicolaou-Varadhan Theoryp. 139
A PDE construction of \bar {{H}}p. 139
Effective Lagrangianp. 140
Application: Homogenization of Nonlinear PDEp. 141
3.4 More PDE Methodsp. 141
3.5 Estimatesp. 144
4 An Alternative Variational/PDE Constructionp. 145
4.1 A new Variational Formulationp. 145
A Minimax Formulap. 146
A New Variational Settingp. 146
Passing to Limitsp. 147
4.2 Application: Nonresonance and Averagingp. 148
Derivatives of {{\overline {{\bf H}}}}^kp. 148
Nonresonancep. 148
5 Some Other Viewpoints and Open Questionsp. 150
Referencesp. 152
Geometrical Aspects of Symmetrizationp. 155
1 Sets of finite perimeterp. 155
2 Steiner Symmetrization of Sets of Finite Perimeterp. 164
3 The Pòlya-Szegö Inequalityp. 171
Referencesp. 180
CIME Courses on Partial Differential Equations and Calculus of Variationsp. 183