Median values, 1-harmonic functions, and functions of least gradient

Matthew B. Rudd; Heather A. Van Dyke

doi:10.3934/cpaa.2013.12.711

Article Contents

2013, Volume 12, Issue 2: 711-719. Doi: 10.3934/cpaa.2013.12.711

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Median values, 1-harmonic functions, and functions of least gradient

Matthew B. Rudd^1, and
Heather A. Van Dyke^2,

1.
Department of Mathematics, Sewanee: The University of the South, Sewanee, TN 37383
2.
Department of Mathematics, Washington State University, Pullman, WA 99164

Received: September 30, 2011

Revised: May 31, 2012

Published: February 2013

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Abstract

Motivated by the mean value property of harmonic functions, we introduce the local and global median value properties for continuous functions of two variables. We show that the Dirichlet problem associated with the local median value property is either easy or impossible to solve, and we prove that continuous functions with this property are $1$-harmonic in the viscosity sense. We then close with the following conjecture: a continuous function having the global median value property and prescribed boundary values coincides with the function of least gradient having those same boundary values.

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Mathematics Subject Classification: Primary: 35J92, 39B22, 35A35; Secondary: 35B05, 35D40, 35A16.

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