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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 - Luigi Ambrosio

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


Published: 3rd December 2007
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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.

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

ISBN: 9783540759133
ISBN-10: 3540759131
Series: C.I.M.E. Foundation Subseries
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 206
Published: 3rd December 2007
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.27
Weight (kg): 0.34