Lipschitz regularity of fractional $p$-Laplacian

Abstract

In this article, we investigate the Hölder regularity of the fractional $p$-Laplace equation of the form $(-{\Delta }_p{)}^{s}u=f$ where $p>1,s\in (0,1)$ and $f\in L_{\mathrm{loc}}^{\infty }(\Omega )$. Specifically, we prove that $u\in C_{\mathrm{loc}}^{0,{\gamma }_{\circ }}(\Omega )$ for ${\gamma }_{\circ }=\min \{1,\frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\ne 1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.

arXiv comment31 pages, 1 figure, Lemma 4.2 corrected