Research

 

My research is focused on calculus of variations and geometric measure theory.


I am particularly interested in variational models in material science, where the competition of bulk and concentrated terms in the energy is the mechanisms of fascinating phenomena as formation of singularities (cracks) and topological singularities (vortices, dislocations).

Dislocations are line defects in the periodic structure of crystals, and are responsible of plasticity in metals. I am interested in deriving macroscopic material dependent models in plasticity by fundamental atomistic models. Part of my research consists in developing a Ginzburg Landau approach to dislocations, exploiting the analogies between vortices in superconductivity and dislocations in crystals.

Collaborators: Roberto Alicandro, Marco Cicalese, Lucia De Luca, Adriana Garroni, Giovanni Leoni.

Griffith criterium says that the stability of a crack is determined by a competition between the energy needed to break atomic bonds (to enlarge the crack) and the corresponding release of elastic energy in the body. This principle has been revisited by Francfort and Marigo to determine the path of the crack. The mathematical formulation involves SBV functions and the geometric measure theory. Many interesting questions are still open, concerning for instance crack initiation, kinking, derivation of linear and continuous theories from discrete fundamental systems. Collaborators: Fausto Acanfora, Antonin Chambolle, Gianni Dal Maso, Francois Ebobisse, Matteo Focardi, Maria Stella Gelli, Alessandro Giacomini.

I am interested in some variational methods in image processing. The model problem consists in denoising a given image (the fingerprint on the left) to obtain a better image (on the right). The celebrated Mumford Shah approach is based on the minimization of a functional given by the sum of a fidelity term, and a surface term penalizing the presence of too many boundaries in the picture (associated to noise). I am particularly interested in models where this surface term is non-local.

Collaborators: Marco Barchiesi, Antonin Chambolle, Massimiliano Morini, Sung Ha Kang, Triet Lee.

Selected Publications (All the papers can be consulted at http://cvgmt.sns.it/people/ponsiglio/)


Giacomini A., Ponsiglione M.: A Γ-convergence approach to stability of unilateral minimality properties in fracture mechanics and applications.

Arch. Ration. Mech. Anal. 180 (2006).

Chambolle A., Giacomini A., Ponsiglione M.: Piecewise Rigidity. J. Funct. Anal. 244 (2007).

Chambolle A., Giacomini A., Ponsiglione M.: Crack initiation in brittle materials. Arch. Ration. Mech. Anal. 188 (2008).

Garroni A., Leoni G., Ponsiglione M.: Gradient theory for plasticity via homogenization of discrete dislocations. JEMS 5 (2010).

De Luca L., Garroni A., Ponsiglione M.: Γ-convergence analysis of systems of edge dislocations: the self energy regime.

Arch. Ration. Mech. Anal. 206 (2012).