Let n≥ 2, 1 0 be an admissible interior Hölder exponent for gradients of local p-harmonic functions. Is it true that, for every 0<alpha<alpha_rm loc(n,p) and every s∈((p-1)/p,1), every bounded weak solution of (-Delta_p)^s u=0 in B_2⊂R^n satisfies u∈ C^1,alpha(B_1/2)? Here (-Delta_p)^s denotes the fractional p-Laplacian, formally given by (-Delta_p)^s u(x)=2,P.V.!int_R^nfrac|u(x)-u(y)|^p-2(u(x)-u(y))|x-y|^n+sp,dy.

Open C¹,α regularity problem for singular fractional p-Laplacian equations

Let $n\ge 2$ , $1<p<2$ , and let $\alpha_{\rm loc}(n,p)>0$ be an admissible interior Hölder exponent for gradients of local $p$ -harmonic functions. Is it true that, for every $0<\alpha<\alpha_{\rm loc}(n,p)$ and every $s\in((p-1)/p,1)$ , every bounded weak solution of $(-\Delta_p)^s u=0$ in $B_2\subset\mathbb{R}^n$ satisfies $u\in C^{1,\alpha}(B_{1/2})$ ? Here $(-\Delta_p)^s$ denotes the fractional $p$ -Laplacian, formally given by $(-\Delta_p)^s u(x)=2\,\mathrm{P.V.}\!\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Near-local-order formulation
For every $n\ge 2$ , $1<p<2$ , and $0<\alpha<\alpha_{\rm loc}(n,p)$ , there exists $s_*=s_*(n,p,\alpha)<1$ such that every bounded weak solution of $(-\Delta_p)^s u=0$ in $B_2$ belongs to $C^{1,\alpha}(B_{1/2})$ whenever $s\in(s_*,1)$ .
source: Zhang, Towards gradient Hölder regularity for singular fractional $p$-Laplace equations

References

Primary source

arXiv

Additional references

Towards gradient Hölder regularity for singular fractional p-Laplace equations — arXiv — Chao Zhang

nonlocal-pde regularity-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A 2026 preprint proves the desired regularity only when the fractional order is sufficiently close to one, so the full problem remains open.

The problem asks whether singular fractional $p$ -Laplace solutions have interior $C^{1,\alpha}$ regularity for $1<p<2$ , with $\alpha$ below the admissible local $p$ -harmonic exponent. The complete range of fractional orders remains unsettled.

Known results

For $p<2$ and $s<(p-1)/p$ , earlier work gives $u\in W^{\sigma,p}$ for every $\sigma\in(0,ps/(p-1))$ .
For $p<2$ and $s>(p-1)/p$ , earlier work gives differentiability and $\nabla u\in W^{\sigma,p}$ for $\sigma\in(0,ps-p+1)$ .
Global boundary results establish only $C^\beta$ regularity for some $\beta\in(0,s]$ , not $C^{1,\alpha}$ regularity.

2026 partial result near $s=1$

Chao Zhang’s preprint reports that for $n\ge2$ , $1<p<2$ , and $0<\alpha<\alpha_{\rm loc}(n,p)$ , there is $s_*<1$ such that bounded weak solutions belong to $C^{1,\alpha}(B_{1/2})$ whenever $s\in(s_*,1)$ . It explicitly presents this as partial progress; orders away from $1$ remain untreated.

Current status (as of August 2026): $C^{1,\alpha}$ regularity is established for $s$ sufficiently close to $1$ in the singular range $1<p<2$ , while the full range $s\in(0,1)$ remains open.

Sources

No solutions have been posted yet.

Near-local-order formulation

Primary source

Additional references

Progress summary