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

Serena Dipierro; Ovidiu Savin; Enrico Valdinoci

doi:10.4171/jems/684

JournalsjemsVol. 19, No. 4pp. 957–966

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

Serena Dipierro
University of Melbourne, Parkville, Australia
Ovidiu Savin
Columbia University, New York, USA
Enrico Valdinoci
University of Melbourne, Australia, University of Milano, Italy, and WIAS, Berlin, Germany

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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 $C_{loc}^{k}$ .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.

Cite this article

Serena Dipierro, Ovidiu Savin, Enrico Valdinoci, All functions are locally $s$ -harmonic up to a small error. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 957–966

DOI 10.4171/JEMS/684