The fractional Malmheden theorem

Serena Dipierro; Giovanni Giacomin; Enrico Valdinoci

doi:10.3934/mine.2023024

The fractional Malmheden theorem

Serena Dipierro, Giovanni Giacomin, Enrico Valdinoci

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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.

Original language	English
Journal	Mathematics In Engineering
Volume	5
Issue number	2
Early online date	14 Apr 2022
DOIs	https://doi.org/10.3934/mine.2023024
Publication status	Published - 2023

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10.3934/mine.2023024Licence: CC BY

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title = "The fractional Malmheden theorem",

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 Laplacian, geometric properties of harmonic functions, Harnack inequality, Malmheden theorem, Poisson kernel, Schwarz theorem",

author = "Serena Dipierro and Giovanni Giacomin and Enrico Valdinoci",

note = "Funding Information: SD and EV are members of AustMS. Supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations” and by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications”. Publisher Copyright: {\textcopyright} 2023 the Author(s), licensee AIMS Press.",

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AU - Dipierro, Serena

AU - Giacomin, Giovanni

AU - Valdinoci, Enrico

N1 - Funding Information: SD and EV are members of AustMS. Supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations” and by the Australian Research Council DECRA DE180100957 “PDEs, free boundaries and applications”. Publisher Copyright: © 2023 the Author(s), licensee AIMS Press.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - fractional Laplacian

KW - geometric properties of harmonic functions

KW - Harnack inequality

KW - Malmheden theorem

KW - Poisson kernel

KW - Schwarz theorem

UR - https://www.scopus.com/pages/publications/85131743170

U2 - 10.3934/mine.2023024

DO - 10.3934/mine.2023024

M3 - Article

AN - SCOPUS:85131743170

SN - 2640-3501

VL - 5

JO - Mathematics In Engineering

JF - Mathematics In Engineering

IS - 2

ER -

The fractional Malmheden theorem

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Nonlocal equations and applications

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