The C^{p′}-regularity conjecture

For every dimension $n\ge 2$ and every $p>2$ , let $\Omega\subset\mathbb{R}^n$ be open and let $u\in W^{1,p}_{\mathrm{loc}}(\Omega)$ be a weak solution of the inhomogeneous $p$ -Laplace equation $-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=f$ in $\Omega$ , where $f\in L^\infty_{\mathrm{loc}}(\Omega)$ . Then $u\in C^{1,p'-1}_{\mathrm{loc}}(\Omega)=C^{1,1/(p-1)}_{\mathrm{loc}}(\Omega)$ , where $p'=p/(p-1)$ ; equivalently, $u$ has locally Hölder-continuous gradient of exponent $1/(p-1)$ .

References

Primary source

arXiv

Additional references

The $C^{p'}$ -regularity conjecture near $p=2$ — arXiv

calculus-of-variations partial-differential-equations regularity-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A new result proves the expected regularity when the exponent is close to two, but the full conjecture remains open.

The conjecture concerns the expected regularity of solutions to the inhomogeneous $p$ -Laplace equation. No proposer or original date is identified in the retrieved material.

Known results

In the planar case, for $p>2$ , bounded forcing gives local $C^{p'}$ regularity; higher dimensions remain open.

September 2026 near- $2$ result

A September 2026 preprint reports improved $p$ -harmonic gradient estimates proving the conjectured regularity for $p$ sufficiently close to $2$ , with an explicit threshold; this is a partial result, not an all- $p$ proof. Separately, a February 6, 2026 article reports that Cristiana De Filippis and Giuseppe Mingione completed a related threshold theorem for nonuniformly elliptic equations, but it does not establish that this is identical to the full conjecture here.

Current status (as of September 2026): The planar case and a claimed near- $2$ range are recorded, but the conjecture for all $p$ remains open and the new range result is unverified.

Sources

No solutions have been posted yet.