All functions are locally $s$-harmonic up to a small error

Source abstract

We show that we can approximate every function f\in C^{k}(\overline{B_1}) by an s -harmonic function in B_1 that vanishes outside a compact set. That is, s -harmonic functions are dense in C^{k}_{\mathrm {loc}} .This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.