Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 08, pp. 1-26.

Title: Characterization of mean value harmonic functions on norm induced metric
       measure spaces with weighted Lebesgue measure

Author: Antoni Kijowski (Polish Academy of Sciences, Warsaw, Poland)

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.
 
Submitted  March 21, 2019. Published January 14, 2020.
Math Subject Classifications: 31C05, 35J99, 30L99.
Key Words: Harmonic function; mean value property; metric measure space;
           Minkowski functional; norm induced metric; Pizzetti formula;
           weighted Lebesgue measure.