Pointwise characterization of p-harmonicity without viscosity solutions

Determine whether the pointwise asymptotic mean value formula \(\lim_{\varepsilon\to0}\varepsilon^{-p}\fint_{B_\varepsilon(x)}|u(y)-u(x)|^{p-2}(u(y)-u(x))\,dy=0\) characterizes \(p\)-harmonic functions in dimensions \(d\geq3\) without interpreting the limit in the viscosity sense.

Background

The paper studies pointwise asymptotic mean value properties at critical points of pp-harmonic functions, where ordinary Taylor expansion arguments are unavailable. For the asymptotic formula proposed by del Teso and collaborators, the authors show that the homogeneous counterexamples constructed in the paper nevertheless satisfy the formula pointwise for p>2p>2 sufficiently close to $2$, while they fail it for p<2p<2 sufficiently close to $2$.

The established results therefore do not settle whether the formula characterizes pp-harmonicity pointwise in dimensions d≥3d\geq3, as opposed to only characterizing it in the viscosity sense. The unresolved issue is whether the viscosity interpretation can be removed from the characterization.

References

Surprisingly, for $d\geq3$ the $p$-harmonic functions constructed in Section~\ref{sec:main} satisfy this asymptotic expansion pointwise in the range $p>2$, so the question of whether this limit characterizes $p$-harmonicity without the viscosity interpretation for $d\geq 3$ is still an open problem.

— A counterexample to the pointwise validity of an asymptotic mean value property for $p$-harmonic functions  (2609.28069 - Arroyo et al., 23 Sep 2026) in Section 5, “Other mean value properties,” final paragraph preceding Theorem dTL-counterexample