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
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010178998 | QA315 C344 2008 | Open Access Book | Book | Searching... |
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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
Transport Equation and Cauchy Problem for Non-Smooth Vector Fields | p. 1 |
1 Introduction | p. 1 |
2 Transport Equation and Continuity Equation within the Cauchy-Lipschitz Framework | p. 4 |
3 ODE Uniqueness versus PDE Uniqueness | p. 8 |
4 Vector Fields with a Sobolev Spatial Regularity | p. 19 |
5 Vector Fields with a BV Spatial Regularity | p. 27 |
6 Applications | p. 31 |
7 Open Problems, Bibliographical Notes, and References | p. 34 |
References | p. 37 |
Issues in Homogenization for Problems with Non Divergence Structure | p. 43 |
1 Introduction | p. 43 |
2 Homogenization of a Free Boundary Problem: Capillary Drops | p. 44 |
2.1 Existence of a Minimizer | p. 46 |
2.2 Positive Density Lemmas | p. 47 |
2.3 Measure of the Free Boundary | p. 51 |
2.4 Limit as ¿ → 0 | p. 53 |
2.5 Hysteresis | p. 54 |
2.6 References | p. 57 |
3 The Construction of Plane Like Solutions to Periodic Minimal Surface Equations | p. 57 |
3.1 References | p. 64 |
4 Existence of Homogenization Limits for Fully Nonlinear Equations | p. 65 |
4.1 Main Ideas of the Proof | p. 67 |
4.2 References | p. 73 |
References | p. 74 |
A Visit with the ∞-Laplace Equation | p. 75 |
1 Notation | p. 78 |
2 The Lipschitz Extension/Variational Problem | p. 79 |
2.1 Absolutely Minimizing Lipschitz iff Comparison With Cones | p. 83 |
2.2 Comparison With Cones Implies ∞-Harmonic | p. 84 |
2.3 ∞-Harmonic Implies Comparison with Cones | p. 86 |
2.4 Exercises and Examples | p. 86 |
3 From ∞-Subharmonic to ∞-Superharmonic | p. 88 |
4 More Calculus of ∞-Subharmonic Functions | p. 89 |
5 Existence and Uniqueness | p. 97 |
6 The Gradient Flow and the Variational Problem for \parallel|Du|\parallel_{{L^\infty}} | p. 102 |
7 Linear on All Scales | p. 105 |
7.1 Blow Ups and Blow Downs are Tight on a Line | p. 105 |
7.2 Implications of Tight on a Line Segment | p. 107 |
8 An Impressionistic History Lesson | p. 109 |
8.1 The Beginning and Gunnar Aronosson | p. 109 |
8.2 Enter Viscosity Solutions and R. Jensen | p. 111 |
8.3 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 |
9 Generalizations, Variations, Recent Developments and Games | p. 116 |
9.1 What is ¿ ∞ for H(x, u, Du)? | p. 116 |
9.2 Generalizing Comparison with Cones | p. 118 |
9.3 The Metric Case | p. 118 |
9.4 Playing Games | p. 119 |
9.5 Miscellany | p. 119 |
References | p. 120 |
Weak KAM Theory and Partial Differential Equations | p. 123 |
1 Overview, KAM theory | p. 123 |
1.1 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 |
1.2 KAM Theory | p. 127 |
Generating Functions, Linearization | p. 128 |
Fourier series | p. 128 |
Small divisors | p. 129 |
Statement of KAM Theorem | p. 129 |
2 Weak KAM Theory: Lagrangian Methods | p. 131 |
2.1 Minimizing Trajectories | p. 131 |
2.2 Lax-Oleinik Semigroup | p. 131 |
2.3 The Weak KAM Theorem | p. 132 |
2.4 Domination | p. 133 |
2.5 Flow invariance, characterization of the constant c | p. 135 |
2.6 Time-reversal, Mather set | p. 137 |
3 Weak KAM Theory: Hamiltonian and PDE Methods | p. 137 |
3.1 Hamilton-Jacobi PDE | p. 137 |
3.2 Adding P Dependence | p. 138 |
3.3 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 |
3.4 More PDE Methods | p. 141 |
3.5 Estimates | p. 144 |
4 An Alternative Variational/PDE Construction | p. 145 |
4.1 A new Variational Formulation | p. 145 |
A Minimax Formula | p. 146 |
A New Variational Setting | p. 146 |
Passing to Limits | p. 147 |
4.2 Application: Nonresonance and Averaging | p. 148 |
Derivatives of {{\overline {{\bf H}}}}^k | p. 148 |
Nonresonance | p. 148 |
5 Some Other Viewpoints and Open Questions | p. 150 |
References | p. 152 |
Geometrical Aspects of Symmetrization | p. 155 |
1 Sets of finite perimeter | p. 155 |
2 Steiner Symmetrization of Sets of Finite Perimeter | p. 164 |
3 The Pòlya-Szegö Inequality | p. 171 |
References | p. 180 |
CIME Courses on Partial Differential Equations and Calculus of Variations | p. 183 |