On some energies penalizing oblique oscillations

In this talk I will present some results obtained with B. Merlet in recent years on a family of energies penalizing oscillations in oblique directions. These functionals, which first appeared in the study of an isoperimetric problem with non-local interactions, can be seen as a natural extension of the Bourgain-Brezis-Mironescu energies. A central insight is that these energies actually control second order derivatives rather than first order ones. Indeed, functions of finite energy have mixed (or oblique) derivatives given by bounded measures. The main focus of the talk is the study of the rectifiability properties of these 'defect' measures. Time permitting we will draw connections with branched transportation, PDE constrained measures and Aviles-Giga type differential inclusions.