Sharp gradient integrability for $(s,p)$-Poisson type equations
Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring
TL;DR
This work establishes a sharp Calderón–Zygmund-type theory for the fractional p-Poisson equation with general nonlocal coefficients. By combining a discrete second-order difference framework with a two-stage comparison (inhomogeneity and coefficient freezing) and a meticulous stopping-time/level-set strategy, the authors prove local gradient regularity u ∈ W^{1,q}_{loc}(Ω) for q = rn(p−1)/(n − r(p−1)(sp'−1)) under sp'>1 and f ∈ L^r_{loc}(Ω). The results include quantitative gradient estimates featuring nonlocal tail terms and are shown to be optimal via explicit counterexamples; they extend the regularity theory to variable coefficients and provide higher differentiability in Besov-type scales. The methods reconcile nonlocal nonlinear structure with classical Calderón–Zygmund ideas, enabling sharp control of ∇u by f, tail, and the nonlocal operator data, with potential impact on nonlinear nonlocal PDE analysis and applications where fractional diffusion with irregular media arises.
Abstract
We prove local $W^{1,q}$-regularity for weak solutions to fractional $p$-Laplacian type equations with right-hand side $f\in L^r_{\mathrm{loc}}(Ω)$. Assuming $p>1$, $s\in(0,1)$, and $sp'>1$, solutions belong to $W^{1,q}_{\mathrm{loc}}(Ω)$ for the optimal exponent $q=q(n,p,s,r)$. We obtain quantitative local gradient estimates involving nonlocal tail terms. The optimality of $q$ is confirmed by a counterexample.