On the asymptotic mean value property for planar \(p\)-harmonic functions (Q2814406)

Let \(u\in C(\Omega )\cap W_{\mathrm{loc}}^{1,p}(\Omega )\), where \(\Omega \subset \mathbb{R}^{2}\) is a domain and \(1<p<\infty \), and let \(A(u;x,r)\) denote the mean value (when defined) of \(u\) over the open disc \(D(x,r)\) of centre \(x\) and radius \(r\). The authors show that \(u\) is \(p\)-harmonic on \(\Omega \) if and only if NEWLINE\[NEWLINE u(x)=\frac{p-2}{p+2}\cdot \frac{1}{2}\left( \sup_{D(x,r)}u+\inf_{D(x,r)}u\right) +\frac{4}{p+2}A(u;x,r)+o(r^{2}), \quad \text{for } r\rightarrow 0^+ NEWLINE\]NEWLINE for each \(x\in \Omega \). This equivalence had previously been established for a restricted range of \(p\) by \textit{P. Lindqvist} and \textit{J. Manfredi} [Proc. Am. Math. Soc. 144, No. 1, 143--149 (2016; Zbl 1327.35124)].