English

Sharp gradient integrability for $(s,p)$-Poisson type equations

Analysis of PDEs 2026-02-10 v1

Abstract

We prove local W1,qW^{1,q}-regularity for weak solutions to fractional pp-Laplacian type equations with right-hand side fLlocr(Ω)f\in L^r_{\mathrm{loc}}(\Omega). Assuming p>1p>1, s(0,1)s\in(0,1), and sp>1sp'>1, solutions belong to Wloc1,q(Ω)W^{1,q}_{\mathrm{loc}}(\Omega) for the optimal exponent q=q(n,p,s,r)q=q(n,p,s,r). We obtain quantitative local gradient estimates involving nonlocal tail terms. The optimality of qq is confirmed by a counterexample.

Keywords

Cite

@article{arxiv.2602.08944,
  title  = {Sharp gradient integrability for $(s,p)$-Poisson type equations},
  author = {Verena Bögelein and Frank Duzaar and Naian Liao and Kristian Moring},
  journal= {arXiv preprint arXiv:2602.08944},
  year   = {2026}
}
R2 v1 2026-07-01T10:28:24.442Z