Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure
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...
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| Format: | Article |
| Language: | English |
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Texas State University
2020-01-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/08/abstr.html |
| Summary: | 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. |
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| ISSN: | 1072-6691 |