You are here

Calculus of Variations and Multiscale Analysis

  • Homogenization of variational problems
  • Gamma-Convergence and relaxation
  • Variational methods in continuum mechanics
  • Variational methods in rate independent evolution prooblems
  • Variational methods in phase transitions
  • Variational methods in micromagnetics
  • Applications of geometric measure theory
  • Existence problems in the calculus of variations
  • Hamilton-Jacobi equations


The Classical Isoperimetric Problem and its Nonlocal Variants

  • Preliminary results on Geometric Measure Theory:
    • Hausdorff measures,
    • tangent measures, 
    • rectifiable sets.
  • The theory of sets of finite perimeter. 
  • The De Giorgi's solution to the Isoperimetric Problem. 
  • Partial regularity theory of Lambda-minimizers of the perimeter functional. 
  • A nonlocal variant of the perimeter: the sharp interface Ohta-Kawasaki energy. 
  • Regularity of  local minimizers. 
  • The second variation of the perimeter and of the Ohta-Kawasaki energy. 

Advanced Topics in Calculus of Variations

Program of the course:

  • Preliminaries on Gamma-convergence.
  • Homogenization: an example in dimension one.
  • Dimension reduction in elasticity: preliminaries on the mathematical theory of elasticity;
    Korn’s inequality; rigorous derivation of a plate theory in linearized elasticity
    rigidity theorem; rigorous derivation of Kirchhoff plate theory.
  • Gradient theory of phase transitions


Sign in