Research Group in Calculus of Variations and Geometric Measure Theory
Techniques from measure theory, and particularly from geometric measure theory, have been applied to different problems in partial differential equations since (at least) the first regularity results for minimal hypersurfaces in dimension larger than two.
In more recent years, the development of Gamma-convergence, and in a different direction, the progress in the analysis of nonlinear geometric problems were based on a consistent and systematic use of geometric measure theory, epitomized by the theory of Cartesian currents, and by the theory of Young measures and H-measures (or semiclassical measures).
The research of this group will be focussed in the following directions: the first and most established one concerns the applications of geometric measure theory to the asymptotics of certain classes of geometric variational problems along the line of the Modica-Mortola result for semilinear equations of reaction-diffusion type (Allen-Cahn equation), and to similar approaches for complex-valued Ginzburg-Landau system.
On a somewhat different line, measure theoretic tools have been applied quite successfully in the last four years to classes of pde's which display intrinsically "bad" solutions - in the sense of non-regular - such as those arising in mass transport problems and first order transport equations associated to the flow of non-Lipschitz vectorfield a là Di Perna-Lions.
We point out in particular that the recent proof of L. Ambrosio of the uniqueness for BV vectorfields is based on very effective combinations of measure theoretic tools which had been developed in the past years for completely different problems. In these same areas, it has also become apparent that sharp counterexamples can sometimes be provided only through deep and non-trivial constructions from classical measure theory and real analysis.
Indeed, one of the general goals of this group is to try to bridge the gap between real analysis and "hard" geometric measure theory on the one side, and the pde's and calculus of variations on the other side, in the hope that the wealth of results and geometric constructions available in the former might shed some light on the problems of the latter.
A more specific goal of future research is understanding the optimal assumptions in the Ambrosio-Di Perna-Lions type of results: so far the regularity conditions on the vectorfields have always been expressed in terms of coefficients in certain functionals spaces, i.e., in terms of linear quantities. Yet there is no specific reason for this, and indeed there are plenty of examples which suggest that a completely different and more geometric perspective could be quite successful, specifically for a system of conservation laws. Notice that despite the remarkable progress of the last four years, the picture of existence results for optimal transport problems is also far from being complete, and we plan to address some of the very basic questions which are still unanswered.
Calculus of Variations, Geometric Measure Theory,
Optimal Transport Theory, Harmonic maps, Evolution problems,
Analysis in Metric Spaces and in Carnot groups.