Serena Dipierro, Giovanni Giacomin and Enrico Valdinoci
The Fractional Malmheden Theorem
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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 simple proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.