- Minimization of functionals defined on classes of surfaces, depending on the area and the curvature of the surface.
These functionals arise in the study of minimal surfaces, surfaces with prescribed mean curvature, Willmore's conjecture, non-linear shell theory, image segmentation and surface reconstruction. Methods of geometric measure theory, as Caccioppoli sets, BV functions, rectifiable currents and varifolds are used.
Geometric and analytic properties characterizing some classes of subsets of an Euclidean space are investigated with regard to the order of rectifiability.
- Asymptotic methods for non linear problems.
Tools from Gamma-convergence and Geometric Measure Theory provide a powerful way to study a wealth of nonlinear phenomena in applied sciences.
As an example, functionals of the Ginzburg-Landau type arise from models in theoretical physics (in the context of superfludity, superconductivity, Bose-Einstein condensates). The study of Gamma-convergence for these functionals, in a Geometric Measure Theoretical setting, is expecially appealing because in the asymptotic limit solutions are known to develop singularities, whose properties we attempt to characterize.
Using the tool of Gamma-convergence one can also investigate the asymptotic behaviour of functionals modelling cellular elastic materials with fine microstructure and possible fractures (not assigned a priori) are investigated.
Further, in the context of fracture mechanics, some problems of relaxation of elastic energies with free discontinuities are considered.
We attempt to study the asymptotic behaviour of solutions of nonlinear elliptic equations of monotone type on Function Spaces of Sobolev type (with respect to measures).
- Topics of geometric measure theory in Carnot-Caratheodory spaces.
In particular the study of the intrinsic rectifiability of sets, regular surfaces, surface measures (like perimeter, Hausdorff measures and Minkowski content) and minimal hypersurfaces in this setting.