A symmetry result in R2 for global minimizers of a general type of nonlocal energy

In this paper, we are interested in a general type of nonlocal energy, defined on a ball BR⊂ Rn for some R> 0 as E(u,BR)=∬R2n\(CBR)2F(u(x)-u(y),x-y)dxdy+∫BRW(u)dx.We prove that in R2, under suitable assumptions on the functions F and W, bounded continuous global energy minimizers are one-dimensional. This proves a De Giorgi conjecture for minimizers in dimension two, for a general type of nonlocal energy.