The fractional Malmheden theorem | Zendy

Serena Dipierro | Zendy; Giovanni Giacomin | Zendy; Enrico Valdinoci | Zendy

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The 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.

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