Optimal regularity of p-harmonic functions in dimensions three and higher

Determine the optimal regularity of p-harmonic functions in dimensions N3, particularly whether regularity stronger than the currently known estimates holds in general.

Background

The paper notes that p-harmonic functions generally cannot be expected to have local C{1,1} regularity. In dimension two, an optimal C{1,1] theory is available, and examples demonstrate the failure of C{1,1}_{loc} regularity for p>2. The corresponding optimal regularity theory in dimensions N3 is unresolved.

This regularity question is relevant to comparison and moving-plane methods because existing approaches may require differentiability or higher regularity that weak solutions do not generally possess.

References

To the best of our knowledge, the optimal regularity issue for $p$-harmonic functions in $N\ge 3$ remains widely open.

— Comparison principles and symmetry for subquadratic fractional $p$-Laplacian equations  (2609.02626 - Ye et al., 2 Sep 2026) in Section 1, Introduction