We establish quantitative properties of minimizers and stable sets for nonlocal interaction functionals, including the s-fractional perimeter as a particular case.On the one hand, we establish universal BV -estimates in every dimension n 2 for stable sets. Namely, we prove that any stable set in B 1 has finite classical perimeter in B 1/2 , with a universal bound. This nonlocal result is new even in the case of s-perimeters and its local counterpart (for classical stable minimal surfaces) was known only for simply connected two-dimensional surfaces immersed in R 3 .On the other hand, we prove quantitative flatness estimates for minimizers and stable sets in low dimensions n = 2, 3. More precisely, we show that a stable set in B R , with R large, is very close in measure to being a half space in B 1 -with a quantitative estimate on the measure of the symmetric difference. As a byproduct, we obtain new classification results for stable sets in the whole plane.

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation 1/2 in R n. Our energy estimates hold for every nonlinearity and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable.\ud As a consequence, in dimension , we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation in R n.Peer ReviewedPostprint (published version

Abstract. We study the nonlinear fractional equation (−∆) s u = f (u) in R n , for all fractions 0 < s < 1 and all nonlinearities f . For every fractional power s ∈ (0, 1), we obtain sharp energy estimates for bounded global minimizers and for bounded monotone solutions. They are sharp since they are optimal for solutions depending only on one Euclidian variable. As a consequence, we deduce the one-dimensional symmetry of bounded global minimizers and of bounded monotone solutions in dimension n = 3 whenever 1/2 ≤ s < 1. This result is the analogue of a conjecture of De Giorgi on one-dimensional symmetry for the classical equationIt remains open for n = 3 and s < 1/2, and also for n ≥ 4 and all s.

AbstractWe prove that half spaces are the only stable nonlocal s-minimal cones in {\mathbb{R}^{3}}, for {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from {s=1}. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation ( 1) 1/2 u = f (u) in the whole space R 2m , where f is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate.More precisely, we prove the existence of a saddle-shaped solution in every even dimension 2m, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions 2m = 4 and 2m = 6.These results are relevant in connection with the analog for fractional equations of a conjecture of De Giorgi on the 1-D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1-D solutions, to be global minimizers in high dimensions, a property not yet established.