An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions

We study nonlinear elliptic equations for operators corresponding to non-stable Lévy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) Lévy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen–Cahn type equations with a non-stable Lévy diffusion. These operators may still be realized as local operators using a system of PDEs — in the spirit of the extension problem of Caffarelli and Silvestre.