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Research Article | Open Access

The fractional Malmheden theorem

Serena Dipierro( ), Giovanni Giacomin, Enrico Valdinoci
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA6009 Crawley, Australia
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Abstract

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for s-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.

Keywords

fractional LaplacianMalmheden theoremSchwarz theoremHarnack inequalityPoisson kernelgeometric properties of harmonic functions

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Mathematics in Engineering
Volume 5 Issue 2,
April 2023
Pages 1-28
DOI: 10.3934/mine.2023024

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Cite this article:
Dipierro S, Giacomin G, Valdinoci E. The fractional Malmheden theorem. Mathematics in Engineering, 2023, 5(2): 1-28. https://doi.org/10.3934/mine.2023024

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Received: 14 November 2021
Revised: 12 March 2022
Accepted: 12 March 2022
Published: 15 April 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)