Quantitative flatness results for nonlocal minimal surfaces in low dimensions.

We consider minimizers of nonlocal functionals, like the fractional perimeter, or the fractional anisotropic perimeter, in low dimensions. It is known that a minimizer for the nonlocal perimeter in \R^2 is necessarily an halfplane. We give a quantitative version of this result, in the following sense: we prove that minimizers in a ball of radius $R$ are nearly flat in $B_1$, when $R$ is large enough. Moreover we establish BV estimates and energy estimates in every dimension for the more general notion of stable critical sets. This is a joint work with Joaquim Serra and Enrico Valdinoci.