The fractional Malmheden theorem

Serena Dipierro; Giovanni Giacomin; Enrico Valdinoci; Serena Dipierro; Giovanni Giacomin; Enrico Valdinoci

doi:10.3934/mine.2023024

Mathematics in Engineering

2023, Volume 5, Issue 2: 1-28. doi: 10.3934/mine.2023024

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The fractional Malmheden theorem

Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA6009 Crawley, Australia

Received: 14 November 2021 Revised: 12 March 2022 Accepted: 12 March 2022 Published: 14 April 2022

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.
- fractional Laplacian,
- Malmheden theorem,
- Schwarz theorem,
- Harnack inequality,
- Poisson kernel,
- geometric properties of harmonic functions
Citation: Serena Dipierro, Giovanni Giacomin, Enrico Valdinoci. The fractional Malmheden theorem[J]. Mathematics in Engineering, 2023, 5(2): 1-28. doi: 10.3934/mine.2023024

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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.

References

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