Periodic solutions to integrodifferential equations: variational formulation, symmetry, and regularity
Abstract
We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integrodifferential 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 wellknown 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 nonlocalsemilinear 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 $\epsilon>0$.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.06462
 arXiv:
 arXiv:2404.06462
 Bibcode:
 2024arXiv240406462C
 Keywords:

 Mathematics  Analysis of PDEs;
 35J61;
 35B10;
 35A15;
 35S05
 EPrint:
 47 pages