TY - JOUR
T1 - An asymptotic expansion for the fractional p -Laplacian and for gradient-dependent nonlocal operators
AU - Bucur, Claudia
AU - Bucur, Claudia Dalia
AU - Squassina, Marco
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 - Mean value formulas
KW - fractional p -Laplacian
KW - gradient-dependent operators
KW - infinite fractional Laplacian
KW - nonlocal p -Laplacian
KW - Mean value formulas
KW - fractional p -Laplacian
KW - gradient-dependent operators
KW - infinite fractional Laplacian
KW - nonlocal p -Laplacian
UR - http://hdl.handle.net/10807/203970
U2 - 10.1142/S0219199721500218
DO - 10.1142/S0219199721500218
M3 - Article
SN - 0219-1997
VL - 24
SP - 1
EP - 34
JO - Communications in Contemporary Mathematics
JF - Communications in Contemporary Mathematics
ER -