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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

  

Direct Methods in the Calculus of Variations

  1. Elements of convex analysis, polar and bipolar function and their properties, convex envelopes. Semiclassical theory, Euler-Lagrange equations and relation with elliptic PDE’s. Regularity of minimizers.
  2. Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers.
  3. Relaxation theorems, representation of relaxed functionals; convex, quasiconvex, polyconvex and rank-one convex envelopes.

Introduction to the calculus of variations

  1. Elements of convex analysis, polar and bipolar function and their properties, convex envelopes. Semiclassical theory, Euler-Lagrange equations and relation with elliptic PDE’s. Regularity of minimizers.
  2. Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers.
  3. Relaxation theorems, representation of relaxed functionals; convex, quasiconvex, polyconvex and rank-one convex envelopes.

Semilinear equations in singularly perturbed domains with applications to phase transitions and geometrically constrained walls

  • Phase transitions models and the Cahn-Hilliard functional.
  • The Modica-Mortola Gamma-convergence result.
  • Structure of local minimizers of the Cahn-Hilliard functional in convex domains.
  • Construction of nonconstant local minimizers in nonconvex domains.
  • Extreme geometry and geometrically constrained magnetic walls:
    • construction of nontrivial local minimizers in singularly perturbed domains,
    • asymptotic analysis of the behavior of local minimizers in the singular limit. 

Variational models depending on curvatures in image segmentation

  • Variational models for the reconstruction problem:
    • the Mumford-Shah model, the Nitzberg-Mumford model,
    • the model on labelled contour graphs.
  • Three-dimensional scenes.
  • Stable maps from a two-manifold to the plane.
  • Apparent contours of embedded surfaces.
  • Labelling an apparent contour.
  • Visible contours
  • A solution to the completion problem:
    • completing a visible contour graph into an apparent contour.
    • Examples of the algorithmic proof.

Gamma-convergence

Direct methods in the calculus of variations:

  • semicontinuity and convexity,
  • coerciveness and reflexivity,
  • relaxation and minimizing sequences,
  • properties of integral functionals.

Gamma-convergence:

  • definition and elementary properties,
  • convergence of minima and of minimizers,
  • sequential characterization of Gamma-limits,
  • Gamma-convergence in metric spaces and Yosida transform,
  • Gamma-convergence of quadratic functionals.

G-convergence:

Calculus of Variations and Multiscale Analysis

Research topics

  • 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

 

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