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

2018/04/26 by Kijowski, Antoni
#30L99 (Secondary) #31C05 (Primary) #35J99 #Analysis of PDEs (math.AP) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1804.10005

Abstract

We study the mean-value harmonic functions on open subsets of ℝⁿ equipped with weighted Lebesgue measures and norm induced metrics. Our main result is a necessary condition saying that all such functions solve a certain homogeneous system of elliptic PDEs. Moreover, a converse result is established in case of analytic weights. Assuming Sobolev regularity of weight w ∈ W^l,∞ we show that strongly harmonic functions are as well in W^l,∞ and that they are analytic, whenever the weight is analytic. The analysis is illustrated by finding all mean-value harmonic functions in ℝ² for the l^p-distance 1 ≤ p ≤ ∞. The essential outcome is a certain discontinuity with respect to p, i.e. that for all p ≠ 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 a remarkable observation that strongly harmonic functions in ℝⁿ possess the mean value property with respect to infinitely many weight functions obtained from a given weight.

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