Improvement of flatness for nonlocal phase transitions

TL;DR

This paper proves an improvement of flatness for nonlocal phase transitions, showing that solutions with flat level sets are one-dimensional, extending classical results to fractional and more general operators.

Contribution

It extends flatness improvement results to nonlocal equations including fractional Laplacians, and shows solutions with flat level sets are one-dimensional.

Findings

Entire solutions with flat level sets are 1D for s in (0,1).

Results apply to a broad class of nonlocal operators.

New insights for fractional Laplacian and anisotropic operators.

Abstract

We prove an improvement of flatness result for nonlocal phase transitions. For a class of nonlocal equations that includes $(-\Delta)^{s/2} u = u-u^3$, with~$s\in(0,1)$, we obtain a result in the same spirit of a celebrated theorem of Savin for the equation $-\Delta u = u-u^3$. As a consequence, we deduce that entire solutions to~$(-\Delta)^{s/2} u = u-u^3$ with asymptotically flat level sets are $1$D when~$s\in(0,1)$. The results presented are completely new even for the case of the fractional Laplacian, but the robustness of the proofs allows us to treat also more general, possibly anisotropic, integro-differential operators.