- 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

Research Group:

## Direct Methods in the Calculus of Variations

- 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.
- Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers.
- Relaxation theorems, representation of relaxed functionals; convex, quasiconvex, polyconvex and rank-one convex envelopes.

## Introduction to the calculus of variations

- 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.
- Direct method, quasiconvexity, polyconvexity, rank-one convexity and their relations. Semicontinuity theorems for scalar and vectorial functionals; existence of minimizers.
- 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 problems
- Variational methods in phase transitions
- Variational methods in micromagnetics
- Applications of geometric measure theory
- Existence problems in the calculus of variations
- Hamilton-Jacobi equations

## Grants

- National Research Project (PRIN 2015) “Calculus of Variations”, funded by the Italian Ministry of Education, University, and Research, February 5, 2017 – February 5, 2020, National Coordinator: Luigi Ambrosio.