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Gradient regularity for $(s,p)$-harmonic functions
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abstract
We study the local regularity properties of $(s,p)$-harmonic functions, i.e. local weak solutions to the fractional $p$-Laplace equation of order $s\in (0,1)$ in the case $p\in (1,2]$. It is shown that $(s,p)$-harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power $q\geq 1$. As a result, $(s,p)$-harmonic functions are H\"older continuous to arbitrary H\"older exponent in $(0,1)$. In addition, the weak gradient of $(s,p)$-harmonic functions has certain fractional differentiability. All estimates are stable when $s$ reaches $1$, and the known regularity properties of $p$-harmonic functions are formally recovered, in particular the local $W^{2,2}$-estimate.
Forward citations
Cited by 2 Pith papers
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Improved H\"older regularity of fractional $(p,q)$-Poisson equation with regular data
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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.
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