JournalsjemsVol. 19, No. 4pp. 957–966

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

We show that we can approximate every function fCk(B1)f\in C^{k}(\overline{B_1}) by an ss-harmonic function in B1B_1 that vanishes outside a compact set.

That is, ss-harmonic functions are dense in ClockC^{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.

Cite this article

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

DOI 10.4171/JEMS/684