@article{Dipierro2023, 
author = {Serena Dipierro and Giovanni Giacomin and Enrico Valdinoci},
title = {The fractional Malmheden theorem},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {2},
pages = {1-28},
keywords = {fractional Laplacian, Malmheden theorem, Schwarz theorem, Harnack inequality, Poisson kernel, geometric properties of harmonic functions},
url = {https://www.sciopen.com/article/10.3934/mine.2023024},
doi = {10.3934/mine.2023024},
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.}
}