Fractional De Giorgi conjecture in dimension 2 via complex-plane methods

We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in $R^2$ for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the s-harmonic extension of the fractional Laplacian in the half-space $R_{+}^3$ introduced by L. Caffarelli and L. Silvestre in 2007.

We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space $R_{+}^{n+1}$ under Neumann boundary conditions. This generalizes the s-harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace.