The (fractional) Malmheden Theorem

We present a classical result by Malmheden based on a simple algorithm to determine the harmonic function in a ball, and we discuss the fractional counterpart, from which one obtains a representation formula for s-harmonic functions as a linear superposition of weighted classical harmonic functions and a new simple proof of the fractional Harnack inequality. This is based on a joint work with Giovanni Giacomin and Enrico Valdinoci.