Work

A counterexample to the pointwise validity of an asymptotic mean value property for $p$-harmonic functions

Year: 2026

Type: preprint

Abstract: We give a counterexample showing that an asymptotic mean value property for $p$-harmonic functions in $\mathbb R^d$ does not necessarily hold in the pointwise sense for $d\geq 3$ and $p>2$ near $2$. The proof uses a perturbation argument from $p=2$ and the Implicit Function Theorem to establish the counterexample. We also use a computer assisted proof to give an explicit range of values of $p$ for which the counterexample holds. Finally, we also establish a counterexample for the pointwise asymptotic variational mean value property.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.28069

Authors: Arroyo, Ángel, Calder, Jeff, Parviainen, Mikko

Institutions: University of Alicante, University of Minnesota, University of Jyväskylä

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Agencia Estatal de Investigación

Award: Math-DT: Advancing the Mathematical Foundations for Dynamic Digital Twinning of Next-Generation Mobile Wireless Networks

OpenAlex ID: W7214181550

Cites: 1

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Related to: 10