Entire solutions to semilinear nonlocal equations in $\RR^2$

TL;DR

This paper proves that stable solutions to certain nonlocal equations in two dimensions are one-dimensional, using a novel Liouville theorem approach to establish flatness of level sets.

Contribution

It introduces a new method employing Liouville theorems to prove flatness and one-dimensionality of solutions to nonlocal equations, extending De Giorgi type results.

Findings

Stable solutions are one-dimensional under natural assumptions.

The method applies to monotone solutions in one direction.

First successful use of Liouville theorem approach for flatness in nonlocal equations.

Abstract

We consider entire solutions to $L u= f(u)$ in $\RR^2$, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our result applies to any solution $u$ which is monotone in one direction. Compared to other proofs of the De Giorgi type results on nonlocal equations, our method is the first successfull attempt to use the Liouville theorem approach to get flatness of the level sets.