Nonlocal Meyers' Example

TL;DR

This work investigates sharp regularity limits for nonlocal equations of order $2s$ by constructing explicit nonlocal Meyers-type examples that bound higher integrability and differentiability in terms of coefficient oscillation. The authors develop a Taylor-approximation-based nonlocal energy and a renormalized energy to mimic the local Meyers framework, and they employ Fourier analysis to compute the action of the nonlocal operator on carefully chosen test functions $u_{\delta}(x)=|x|^{1-\delta}\widehat{x}_1$. They prove a precise coupling between the nonlocal coefficient parameter $\varepsilon$ and the regularity index $\delta$, yielding $(-\Delta_{\mathbb{A}_{s,\varepsilon}})^s u_{\delta}=0$ for suitably related pairs, and they show robustness of these results as $s\uparrow 1$, recovering the classical local Meyers example. The paper also extends the nonlocal Meyers construction to models based on Riesz fractional derivatives, establishing analogous regularity limitations and the $s\to1$ consistency with local theory. These results provide sharp, quantifiable barriers to self-improving properties in nonlocal elliptic problems and illuminate how nonlocality interacts with coefficient oscillations to constrain regularity.

Abstract

We present nonlocal variants of the famous Meyers' example of limited higher integrability and differentiability. In the limit $s \nearrow 1$ we recover the standard Meyers' example. We consider the fractional Laplacian based on differences as well as the one based on fractional derivatives defined by Riesz potentials.

Theorems & Definitions (106)

• remark 1
• definition 1: Meyers-type energy
• lemma 1
• proof
• remark 2
• lemma 2
• proposition 1
• proof
• lemma 3
• proof