An asymptotic expansion for the fractional p -Laplacian and for gradient-dependent nonlocal operators

Marco Squassina, Claudia Dalia Bucur, Claudia Bucur

Risultato della ricerca: Contributo in rivistaArticolo in rivista

Abstract

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well-known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are nowadays available. The nonlinear case, corresponding to the p-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the nonlocal, nonlinear mean value kernel, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) p-Laplacian (for p ≥ 2) and to other gradient-dependent nonlocal operators.
Lingua originaleEnglish
pagine (da-a)1-34
Numero di pagine34
RivistaCommunications in Contemporary Mathematics
Volume24
DOI
Stato di pubblicazionePubblicato - 2022

Keywords

  • fractional p -Laplacian
  • gradient-dependent operators
  • nonlocal p -Laplacian
  • Mean value formulas
  • infinite fractional Laplacian

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