The research of the group was focused essentially on the following topics:
1. Interface evolution problems related to image denoising and crystal growth
In particular, attention was devoted to the total variation approach to image denoising, to the related total variation flow, and to its anisotropic versions. The total variation approach to image denoising (introduced by Rudin-Osher-Fatemi in 1992) proposes to minimize the total variation under various constraints modeling the noise which corrupts the image. Qualitative properties of the total variation flow, and computation of explicit solutions of the denoising problem have been analyzed by the members of the group. The total variation flow turns out also to be strictly related to the problem of understanding the mechanism leading to crystal growth, in particular in the framework of the so-called crystalline mean curvature flow, as introduced by J. Taylor. In this direction, the group has investigated necessary and sufficient conditions for a facet of a crystal not to break or bend during the subsequent evolution, since a better understanding of facet-breaking phenomena may be of help in the understanding of the whole crystal growth process, at least in absence of exterior forcings.
2. Geometric evolution problems
This subject is extremely wide, and the members of the group focused attention to problems related to mean curvature flow in euclidean spaces. In particular, they have investigated the convergence to mean curvature flow of certain perturbed fourth order geometric equations, a sort of “viscosity” approximation in the geometric context suggested by De Giorgi. The complete description of the limiting process seems to be open also in the case of plane immersed curves evolving by curvature.
3. Ill-posed evolution problems in one space dimension.
This research belongs to the general problem of understanding evolution equations coming from the gradient flow of a non convex functional. It is also strictly related to the formation of microstructures, hence to material science and crystallization models. One of the simplest example is maybe represented by the Perona-Malik equation, an anisotropic diffusion equation proposed for image reconstruction, which inhibits the diffusion at the edges of the image, possibly permitting to recover them. The members of the group studied this kind of equations in one space dimension. They investigated the limits of various regularizations, such as the semi-discrete scheme and the approximation via a fourth order equation.
4. Minimal lorentzian submanifolds.
This field of research is related to problems arising in classical strings. The group has devoted some attention to the relation between a lorentzian minimal submanifold, and its approximation via the hyperbolic Ginzburg-Landau equations. The members of the group have also devoted some attention to the problem of understanding the singularities of lorentzian minimal surface equation in arbitrary codimension.The future research program of the group is the following.
1. Ill-posed evolution equations.
The members of the group want to focuse their research in characterizing, as much as possible, a solution to the Perona-Malik equation, possibly giving a theoretical explanation of the results obtained in various numerical regularization schemes. Another similar equation that is under investigation is the one obtained as gradient flow of a double-well potential type function depending on the gradient. In particular, the interest is focused on explaining the phenomena observed at the various time scales that seem to appear in such a kind of problems. It would be very nice to extend some of the arguments valid for the above mentioned equations, to the geometric case, namely to anisotropic curvature flow of plane curves, in presence of a non convex anisotropy.
2. Geometric evolution problems.
The members of the group are interested in continuing the research on the existence and characterization of the limit of the fourth order “viscous” approximations to mean curvature flow, in the case of immersed plane curves. In this framework, probably also partial answers could help to better understand the singularity formation. They are also interested in the investigation of a nontrivial generalization of mean curvature flow, namely the mean curvature motion of partitions. This is the case when more than two phases interact in the evolutionary mean curvature process, with the presence of triple junctions, for instance considering plane curves. The group would like to start to investigate the mean curvature evolution of a partition of the space, in the presence of triple curves and multiple junctions.
3. Minimal lorentzian submanifolds.
The members of the group would are interested in continuing their research in this direction, making use of some geometric measure techniques. It would be nice to understand if the notion of varifold solution to the lorenztian minimal surface equation enjoys some sort of compactness, and to study such a weak solution in the one-dimensional case.