Calculus of variations and non-linear partial differential equations : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005 /

Calculus of variations and non-linear partial differential equations : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005 /

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Bibliographic Details
Meeting name:CIME Session "Enumerative Invariants in Algebraic Geometry and String Theory" (2005 : Cetraro, Italy)
Imprint:Berlin ; New York : Springer, 2008.
Description:xi, 196 p. : ill. ; 24 cm.
Language:English
Series:Lecture notes in mathematics, 0075-8434 ; 1927
Lecture notes in mathematics (Springer-Verlag) ; 1927.
Subject:
Calculus of variations.
Calculus of variations -- Congresses.
Differential equations, Partial -- Congresses.
Calculus of variations.
Differential equations, Nonlinear.
Differential equations, Partial.
Conference papers and proceedings.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/6655806
Hidden Bibliographic Details
Other authors / contributors:Mascolo, Elvira.
Dacorogna, Bernard, 1953-
Marcellini, Paolo.
ISBN:3540759131
9783540759133
Notes:Includes bibliographical references.
Table of Contents:
  • Transport Equation and Cauchy Problem for Non-Smooth Vector Fields
  • 1. Introduction
  • 2. Transport Equation and Continuity Equation within the Cauchy-Lipschitz Framework
  • 3. ODE Uniqueness versus PDE Uniqueness
  • 4. Vector Fields with a Sobolev Spatial Regularity
  • 5. Vector Fields with a BV Spatial Regularity
  • 6. Applications
  • 7. Open Problems, Bibliographical Notes, and References
  • References
  • Issues in Homogenization for Problems with Non Divergence Structure
  • 1. Introduction
  • 2. Homogenization of a Free Boundary Problem: Capillary Drops
  • 2.1. Existence of a Minimizer
  • 2.2. Positive Density Lemmas
  • 2.3. Measure of the Free Boundary
  • 2.4. Limit as ¿ → 0
  • 2.5. Hysteresis
  • 2.6. References
  • 3. The Construction of Plane Like Solutions to Periodic Minimal Surface Equations
  • 3.1. References
  • 4. Existence of Homogenization Limits for Fully Nonlinear Equations
  • 4.1. Main Ideas of the Proof
  • 4.2. References
  • References
  • A Visit with the ∞-Laplace Equation
  • 1. Notation
  • 2. The Lipschitz Extension/Variational Problem
  • 2.1. Absolutely Minimizing Lipschitz iff Comparison With Cones
  • 2.2. Comparison With Cones Implies ∞-Harmonic
  • 2.3. ∞-Harmonic Implies Comparison with Cones
  • 2.4. Exercises and Examples
  • 3. From ∞-Subharmonic to ∞-Superharmonic
  • 4. More Calculus of ∞-Subharmonic Functions
  • 5. Existence and Uniqueness
  • 6. The Gradient Flow and the Variational Problem for \parallel|Du|\parallel_{{L^\infty}}
  • 7. Linear on All Scales
  • 7.1. Blow Ups and Blow Downs are Tight on a Line
  • 7.2. Implications of Tight on a Line Segment
  • 8. An Impressionistic History Lesson
  • 8.1. The Beginning and Gunnar Aronosson
  • 8.2. Enter Viscosity Solutions and R. Jensen
  • 8.3. Regularity
  • Modulus of Continuity
  • Harnack and Liouville
  • Comparison with Cones, Full Born
  • Blowups are Linear
  • Savin's Theorem
  • 9. Generalizations, Variations, Recent Developments and Games
  • 9.1. What is ¿ ∞ for H(x, u, Du)?
  • 9.2. Generalizing Comparison with Cones
  • 9.3. The Metric Case
  • 9.4. Playing Games
  • 9.5. Miscellany
  • References
  • Weak KAM Theory and Partial Differential Equations
  • 1. Overview, KAM theory
  • 1.1. Classical Theory
  • The Lagrangian Viewpoint
  • The Hamiltonian Viewpoint
  • Canonical Changes of Variables, Generating Functions
  • Hamilton-Jacobi PDE
  • 1.2. KAM Theory
  • Generating Functions, Linearization
  • Fourier series
  • Small divisors
  • Statement of KAM Theorem
  • 2. Weak KAM Theory: Lagrangian Methods
  • 2.1. Minimizing Trajectories
  • 2.2. Lax-Oleinik Semigroup
  • 2.3. The Weak KAM Theorem
  • 2.4. Domination
  • 2.5. Flow invariance, characterization of the constant c
  • 2.6. Time-reversal, Mather set
  • 3. Weak KAM Theory: Hamiltonian and PDE Methods
  • 3.1. Hamilton-Jacobi PDE
  • 3.2. Adding P Dependence
  • 3.3. Lions-Papanicolaou-Varadhan Theory
  • A PDE construction of \bar {{H}}
  • Effective Lagrangian
  • Application: Homogenization of Nonlinear PDE
  • 3.4. More PDE Methods
  • 3.5. Estimates
  • 4. An Alternative Variational/PDE Construction
  • 4.1. A new Variational Formulation
  • A Minimax Formula
  • A New Variational Setting
  • Passing to Limits
  • 4.2. Application: Nonresonance and Averaging
  • Derivatives of {{\overline {{\bf H}}}}^k
  • Nonresonance
  • 5. Some Other Viewpoints and Open Questions
  • References
  • Geometrical Aspects of Symmetrization
  • 1. Sets of finite perimeter
  • 2. Steiner Symmetrization of Sets of Finite Perimeter
  • 3. The Pòlya-Szegö Inequality
  • References
  • CIME Courses on Partial Differential Equations and Calculus of Variations

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