TY - JOUR

T1 - An asymptotic expansion for the fractional p -Laplacian and for gradient-dependent nonlocal operators

AU - Squassina, Marco

AU - Bucur, Claudia Dalia

AU - Bucur, Claudia

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - fractional p -Laplacian

KW - gradient-dependent operators

KW - nonlocal p -Laplacian

KW - Mean value formulas

KW - infinite fractional Laplacian

KW - fractional p -Laplacian

KW - gradient-dependent operators

KW - nonlocal p -Laplacian

KW - Mean value formulas

KW - infinite fractional Laplacian

UR - http://hdl.handle.net/10807/203970

U2 - 10.1142/S0219199721500218

DO - 10.1142/S0219199721500218

M3 - Article

VL - 24

SP - 1

EP - 34

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

ER -