Examples of optimal H\"older regularity in semilinear equations involving the fractional Laplacian

TL;DR

This paper investigates the regularity of solutions to semilinear equations involving the fractional Laplacian, revealing a new phenomenon where the combined effects of nonlocality and nonlinearity limit the solutions' Hölder regularity.

Contribution

It identifies a novel regularity phenomenon specific to semilinear fractional Laplacian equations, showing solutions are less regular than previously expected under certain conditions.

Findings

Solutions are at most $C^{2s/(1-eta)}$ when $2s+eta<1$.

Regularity does not always reach $C^{2s+eta- ext{epsilon}}$ for all epsilon.

The phenomenon results from the interplay of nonlocality and semilinearity.

Abstract

We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities $f$ in $C^\beta$ and when $2s+\beta <1$, the solution is not always $C^{2s+\beta-\epsilon}$ for all $\epsilon >0$. Instead, in general the solution $u$ is at most $C^{2s/(1-\beta)}.$