Improved H\"older regularity of fractional $(p,q)$-Poisson equation with regular data

TL;DR

This paper establishes improved regularity results, including Hölder continuity and Lipschitz estimates, for solutions to a fractional $(p,q)$-Poisson equation with Hölder continuous coefficients, extending previous findings.

Contribution

It extends existing regularity results to cases with Hölder continuous modulating coefficients and provides new Lipschitz estimates for weak solutions under broader conditions.

Findings

Viscosity solutions are locally Hölder continuous under specified conditions.

Solutions are locally Lipschitz when certain parameter thresholds are met.

The results improve and extend previous regularity theorems for fractional $p$-Laplacian equations.

Abstract

We prove a quantitative H\"{o}lder continuity result for viscosity solutions to the equation $$ (-\Delta_p)^{s}u(x) + {\rm PV} \int_{\mathbb{R}^n} |u(x)-u(x+z)|^{q-2}(u(x)-u(x+z))\frac{\xi(x,z)}{|z|^{n+ tq}} dz=f \quad \text{in}\; B_2, $$ where $t, s\in (0, 1), 1<p\leq q, tq\leq sp$ and $\xi\geq 0$. Specifically, we show that if $\xi$ is $\alpha$-H\"{o}lder continuous and $f$ is $\beta$-H\"{o}lder continuous then any viscosity solution is locally $\gamma$-H\"{o}lder continuous for any $\gamma<\gamma_\circ $, where \[ \gamma_\circ=\left\{\begin{array}{lll} \min\{1, \frac{sp+\alpha\wedge\beta}{p-1}, \frac{sp}{p-2}\} & \text{for}\; p>2, \\ \min\{1, \frac{sp+\alpha\wedge\beta}{p-1}\} & \text{for}\; p\in (1, 2]. \end{array} \right. \] Moreover, if $\min\{\frac{sp+\alpha\wedge\beta}{p-1}, \frac{sp}{p-2}\}>1$ when $p>2$, or $\frac{sp+\alpha\wedge\beta}{p-1}>1$ when $p\in (1, 2]$, the solution is locally Lipschitz. This extends the result of [20] to the case of H\"{o}lder continuous modulating coefficients. Additionally, due to the equivalence between viscosity and weak solutions, our result provides a local Lipschitz estimate for weak solutions of $(-\Delta_p)^{s}u(x)=0$ provided either $p\in (1, 2]$ or $sp>p-2$ when $p>2$, thereby improving recent works [9, 10, 24].