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

Nevertheless, the conjecture has remained open for general solutions in higher dimensions.

— The $C^{p'}$-regularity conjecture near $p=2$  (2609.09966 - Lee et al., 9 Sep 2026) in Section 1, Introduction

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

The precise optimal exponent is unknown in general dimensions.

— Qualitative-to-quantitative regularity for degenerate equations  (2610.13137 - Sobral et al., 9 Oct 2026) in Introduction, paragraph beginning “Interior $C^{1,\alpha}$ regularity for $p$-harmonic functions is classical.”