Abstract
We study the mean-value harmonic functions on open subsets of \(\mathbb{R}^n\) equipped with weighted Lebesgue measures and norm induced metrics. Our main result is a necessary condition stating that all such functions solve a certain homogeneous system of elliptic PDEs. Moreover, a converse result is established in case of analytic weights. Assuming the Sobolev regularity of the weight \(w \in W^{l,\infty}\) we show that strongly harmonic functions are also in \(W^{l,\infty}\) and that they are analytic, whenever the weight is analytic. The analysis is illustrated by finding all mean-value harmonic functions in \(\mathbb{R}^2\) for the \(l^p\)-distance \({1 \leq p \leq \infty}\). The essential outcome is a certain discontinuity with respect to \(p\), i.e. that for all \(p \ne 2\) there are only finitely many linearly independent mean-value harmonic functions, while for p=2 there are infinitely many of them. We conclude with the remarkable observation that strongly harmonic functions in \(\mathbb{R}^n\) possess the mean value property with respect to infinitely many weight functions obtained from a given weight.
For more information see https://ejde.math.txstate.edu/Volumes/2020/08/abstr.html