Electron. J. Diff. Equ., Vol. 2016 (2016), No. 123, pp. 1-12.

Perron's method for p-harmonious functions

David Hartenstine, Matthew Rudd

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.

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David Hartenstine
Department of Mathematics
Western Washington University
Bellingham, WA 98225, USA
email: david.hartenstine@wwu.edu
  Matthew Rudd
Department of Mathematics
The University of the South
Sewanee, TN 37383, USA
email: mbrudd@sewanee.edu

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