Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure
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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 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 ∈ Wl,∞ we show that strongly harmonic functions are also in Wl,∞ 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 lp-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 the 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.
CitationKijowski, A. (2020). Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure. Electronic Journal of Differential Equations, 2020(08), pp. 1-26.
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