A mean value formula for the p-Laplacian
Abstract
In this paper we introduce an asymptotic mean value formula of the form u(x) − Bεp (u) (x) =ε2 Δp u(x) + o(ε2) for the p−Laplacian, plap u := ∇ · ( |∇ u |p−2 ∇ u ). Here Bεp (u) is a nonlinear expression that involve the mean value and the surmemum/infimum of u in suitable balls. This asymptotic mean value formula allows us to characterize viscosity solutions to equations with different homogeneities in different sets of variables, like plapx' u(x)+qlapx''u(x)=f(x) (for x=(x',x'') ∈ℝd, x'=(x1,...,xn) and x''=(xn+1,...,xd)), and the parabolic p-Laplace heat equation ∂t u (x,t)−Δp u(x,t)=0. Moreover, the resulting asymptotic mean value formulas are related to two-player zero sum games whose value functions, characterized as the unique solution to the corresponding dynamic programming principle, converge as the parameter that controls the size of the steps in the game goes to zero, to the unique viscosity solution to the corresponding boundary value problem for the operator under consideration. Finally, we also include a Holder regularity result for plapx' u(x)+qlapx''u(x)=0 proved using the game theoretical interpretation of the problem.
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BibTeX
@misc{fuertesmoran2026mean,
title = {{A mean value formula for the p-Laplacian}},
author = {Carlos Fuertes-Moran and Julio D. Rossi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28022},
eprint = {2610.28022},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28022},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A mean value formula for the p-Laplacian
AU - Carlos Fuertes-Moran
AU - Julio D. Rossi
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28022
UR - https://arxiv.org/abs/2610.28022
AB - In this paper we introduce an asymptotic mean value formula of the form u(x) − B_ε^p (u) (x) =ε^2 Δ_p u(x) + o(ε^2) for the p−Laplacian, plap u := ∇ · ( |∇ u |^p−2 ∇ u ). Here B_ε^p (u) is a nonlinear expression that involve the mean value and the surmemum/infimum of u in suitable balls. This asymptotic mean value formula allows us to characterize viscosity solutions to equations with different homogeneities in different sets of variables, like plap^x' u(x)+qlap^x''u(x)=f(x) (for x=(x',x'') ∈ℝ^d, x'=(x_1,...,x_n) and x''=(x_n+1,...,x_d)), and the parabolic p-Laplace heat equation ∂_t u (x,t)−Δ_p u(x,t)=0. Moreover, the resulting asymptotic mean value formulas are related to two-player zero sum games whose value functions, characterized as the unique solution to the corresponding dynamic programming principle, converge as the parameter that controls the size of the steps in the game goes to zero, to the unique viscosity solution to the corresponding boundary value problem for the operator under consideration. Finally, we also include a Holder regularity result for plap^x' u(x)+qlap^x''u(x)=0 proved using the game theoretical interpretation of the problem.
ER -