Open AccessThe fractional Malmheden theorem
Author(s) -
Serena Dipierro,
Giovanni Giacomin,
Enrico Valdinoci
Publication year - 2022
Publication title -
mathematics in engineering
Language(s) - English
Resource type - Journals
ISSN - 2640-3501
DOI - 10.3934/mine.2023024
Subject(s) - mathematics , harnack's inequality , harmonic function , harnack's principle , fractional calculus , ball (mathematics) , harmonic , pure mathematics , representation (politics) , superposition principle , mathematical analysis , physics , quantum mechanics , politics , political science , law
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.