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arXiv · 2602.08944

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

Verena Bögelein·Frank Duzaar·Naian Liao·Kristian Moring
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Abstract

We prove local $W^{1,q}$-regularity for weak solutions to fractional $p$-Laplacian type equations with right-hand side $f\in L^r_{\mathrm{loc}}(\Omega)$. Assuming $p>1$, $s\in(0,1)$, and $sp'>1$, solutions belong to $W^{1,q}_{\mathrm{loc}}(\Omega)$ 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.

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Verena Bögelein, Frank Duzaar, Naian Liao, Kristian Moring. 2026-02-09. Sharp gradient integrability for $(s,p)$-Poisson type equations. https://arxiv.org/abs/2602.08944

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