Saddle-shaped solutions for the fractional Allen-Cahn equation

We establish existence and qualitative properties of solutions to the fractional Allen-Cahn equation, which vanish on the Simons cone and are even with respect to the coordinate axes. These solutions are called saddle-shaped solutions. More precisely, we prove monotonicity properties, asymptotic behaviour, and instability in dimensions 2m = 4, 6. We extend to any fractional power s of the Laplacian, some results obtained for the case s = 1/2 in [19]. The interest in the study of saddle-shaped solutions comes in connection with a celebrated De Giorgi conjecture on the one-dimensional symmetry of monotone solutions and of minimizers for the Allen-Cahn equation. Saddle-shaped solutions are candidates to be (not one-dimensional) minimizers in high dimension, a property which is not known to hold yet.