Electron. J. Differential Equations,
Vol. 2020 (2020), No. 08, pp. 126.
Characterization of mean value harmonic functions on norm induced metric
measure spaces with weighted Lebesgue measure
Antoni Kijowski
Abstract:
We study the meanvalue harmonic functions on open subsets of~
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
we show that strongly harmonic functions are also in
and that they are analytic, whenever the weight is analytic.
The analysis is illustrated by finding all meanvalue harmonic functions in
for the
distance
.
The essential outcome
is a certain discontinuity with respect to
, i.e. that for all
there
are only finitely many linearly independent meanvalue harmonic functions,
while for p=2 there are infinitely many of them. We conclude with the remarkable
observation that strongly harmonic functions in
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.
Show me the PDF file (427 KB),
TEX file for this article.

Antoni Kijowski
Institute of Mathematics
Polish Academy of Sciences, ul. Sniadeckich 8
00656 Warsaw, Poland
email: akijowski@impan.pl

Return to the EJDE web page