Department of Mathematics
Western Washington University
Bellingham, WA 98225, USA
email: david.hartenstine@wwu.edu
Department of Mathematics
The University of the South
Sewanee, TN 37383, USA
email: mbrudd@sewanee.edu
David Hartenstine, Matthew Rudd
Abstract:
We show that Perron's method produces continuous p-harmonious functions
for 1<p<2. Such functions approximate p-harmonic functions and
satisfy a functional equation involving a convex combination of the
mean and median, generalizing the classical mean-value property
of harmonic functions. Simple sufficient conditions for the existence of
barriers are given. The p=1 situation, in which solutions to the
Dirichlet problem may not be unique, is also considered.
Finally, the relationship between 1-harmonious functions and functions
satisfying a local median value property is discussed.
Submitted March 2, 2016. Published May 16, 2016.
Math Subject Classifications: 35J92, 39B22, 35A35, 35B05, 35D40.
Key Words: Mean-value property; median; p-harmonic functions;
p-Laplacian; p-harmonious functions; Perron method.
Show me the PDF file (246 KB), TEX file for this article.
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David Hartenstine Department of Mathematics Western Washington University Bellingham, WA 98225, USA email: david.hartenstine@wwu.edu |
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| Matthew Rudd Department of Mathematics The University of the South Sewanee, TN 37383, USA email: mbrudd@sewanee.edu |
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