Weak solutions of the nonlocal (1,p)-Laplace equation in the superquadratic case p≥2 are shown to lie in W^{γ,q}_{loc} for γ< spp/(p−1), and to have a gradient in L^q_{loc} when sp>(p−1)/p.
Regularity for the fractional $p$-Laplace equation
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abstract
Higher Sobolev and H\"older regularity is studied for local weak solutions of the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$. Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or} \quad\tfrac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev regularity estimates for the local $p$-Laplace are recovered. The case $p=2$ reproduces the almost $W^{1+s,2}_{\rm loc}$-regularity for the fractional Laplace equation of any order $s\in(0,1)$.
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Sobolev regularity for the nonlocal $(1, p)$-Laplace equations in the superquadratic case
Weak solutions of the nonlocal (1,p)-Laplace equation in the superquadratic case p≥2 are shown to lie in W^{γ,q}_{loc} for γ< spp/(p−1), and to have a gradient in L^q_{loc} when sp>(p−1)/p.