PhD research topic - Marcello Ponsiglione | Department of Mathematics

PhD research topic - Marcello Ponsiglione

Research area

Calculus of Variations

Keywords

Gamma-convergence, Elasticity, Dislocations, Materials Science, Geometric flows

Research topics

The first research topic deals with multi-scale variational models for topological singularities in materials science. Vortices in superconductors, XY spin systems, liquid crystals and dislocations in crystals represent prototypical examples. I aim at exploiting the universality character of topological singularities, providing variational models able to describe and predict the common features of several apparently unrelated physical systems. Topological singularities are characterized by concentration of the energy and of the relevant fields, and formation of fascinating microstructures, which strongly influence the physical properties of materials. Their modeling is a central problem in materials science, and a challenging mathematical task. The main goal consists in building up and analyzing macroscopic models, which are rigorously derived by fundamental microscopic theories, rich enough to be predictive, but overcoming the untreatable complexity of discrete and microscopic systems. Calculus of variations is a fruitful framework to this task, providing natural tools for the asymptotic analysis across different length scales, and describing in an efficient way complex phenomena such as the formation and evolution of microstructures, driven by energy minimization [1].

A second topic concerns local and non-local mean curvature flows. Very recently, existence and uniqueness for a large class of non-local [3] and crystalline [4] geometric flows have been established. Still, the theory should be generalized to include more and more examples of relevant geometric flows. In particular, nonlocal evolutions that are not invariant by translations and higher codimension geometric evolutions will be investigated both within the level-set formulation and the minimizing movement approach [2]. Nonlinear-evolution laws and geometric flows for partitions will be also investigated.

Bibliography

[1] Alicandro R., De Luca L., Garroni A., Ponsiglione M.: Metastability and dynamics of discrete topological singularities in two dimensions: a Γ-convergence approach. Arch. Ration. Mech. Anal. 214 (2014) 1-62.

[2] Almgren F., Taylor J. E., Wang L.-H.: Curvature-driven fows: a variational approach. SIAM J. Control Optim. 31 (2) (1993), 387-438.

[3] Chambolle A., Morini M., Ponsiglione M.: Nonlocal curvature flows. Arch. Ration. Mech. Anal. 218 (2015), no. 3, 1263–1329.

[4] Chambolle A., Morini M., Ponsiglione M.: Existence and uniqueness for a crystalline mean curvature flow. Comm. Pure Appl. Math. 7 (2017)