A new description of global duality for a nonlinear geometric mass-minimization problem, and applications

Mardi 12 mai 2015 14:00-15:00 - Mircea Petrache - Université Pierre et Marie Curie

Résumé : The classical Plateau problem consists of minimizing the area of a 2D surface in $\mathbbR^3$ under the constraint of fixed boundary. This question can be interpreted in several ways, whose rigorous formulations led to the introduction of integral currents and flat chains with coefficients in a normed group $G$, starting from the 60’s. General tools for the global control of minimizers are mostly limited to the case of rectifiable currents with $G = \mathbbZ$, where a duality structure is present and we have the notion of a calibration. In 1D we can interpret this as a Kantorovich duality.
For the 1D case I will describe a recent result obtained in collaboration with Roger Züst where a duality structure appears for chains with coefficients in $G=\mathbbZ/2\mathbbZ$. This is the first case of « global » characterization of the dual to the Plateau problem for torsion coefficient groups (i.e. groups essentially different than $\mathbbR$ or $\mathbbZ$, allowing no algebraic duality).
The second half of the talk will concern applications :
1) I will shortly indicate the link with a recent very important counterexample of Bethuel for weak density questions in Nonlinear Sobolev spaces. In particular we will see that a corollary of our duality result could play a role in the study of the weak density problem, in the case where it is still open.
2) I will give a link to a large class of crystallization problems including Kohn-Sham’s Density Functional Theory, and I will describe some work in progress on that. Here again, knowing that there exist well-behaved dual formulations different than potential-theoretic methods might give new insights.

Lieu : bât. 425 - 113-115

A new description of global duality for a nonlinear geometric mass-minimization problem, and applications  Version PDF