Periodic solutions to integro-differential equations: variational formulation, symmetry, and regularity

We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in $\mathbb{R}$. We prove that, after an appropriate translation, each of them is necessarily an even function which is decreasing in half its period. In particular, it has only two critical points in half its period, the absolute maximum and minimum. If these statements hold for all nonconstant periodic solutions, and not only for constrained minimizers, remains as an open problem. Our results apply to operators with kernels in two different classes: kernels $K$ which are convex and kernels for which $K(\tau^{1/2})$ is a completely monotonic function of $\tau$. This last new class arose in our previous work on nonlocal Delaunay surfaces in $\mathbb{R}^n$. Due to their symmetry of revolution, it gave rise to a 1d problem involving an operator with a nonconvex kernel. Our proofs are based on a not so well-known Riesz rearrangement inequality on the circle $\mathbb{S}^1$ established in 1976. We also put in evidence a new regularity fact which is a truly nonlocal-semilinear effect and also occurs in the nonperiodic setting. Namely, for nonlinearities in $C^\beta$ and when $2s+\beta <1$ ($2s$ being the order of the operator), the solution is not always $C^{2s+\beta-\epsilon}$ for all $\epsilon>0$.