Por Benoît Merlet (Laboratoire Paul Painlevé, Université de Lille).
Abstract: We introduce a family of functionals defined on the space of measurable functions $u(x,y)$ on a rectangle. These energies vanish on the non convex set $S$ of functions $u(x,y)$ which only depend on $x$ or only depend on $y$. We show that under some conditions the converse implication is true (if the energy vanishes then $u$ belongs to $S$). We establish quantitative versions of this result showing that the energy controls the distance from $u$ to $S$. We also obtain a rather precise description of the functions with finite energy. We present some generalization of these results in higher dimensions. Eventually, we restrict the setting to Lipschitz continuous functions $u$ and show that our work is related to some difficult regularity issues about scalar conservation laws. (Collaboration with Michael Goldman, CNRS-University of Paris).
