A natural approach to the asymptotic mean value property for the p-Laplacian

2016/04/02 by Ishiwata, Michinori, Magnanini, Rolando, Wadade, Hidemitsu · 1 citation
#35K55 #35K92 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 35J60 #Secondary 35J92

paper · doi:10.48550/arxiv.1604.00520

Abstract

Let 1≤ p≤∞. We show that a function u∈ C(\mathbb R^N) is a viscosity solution to the normalized p-Laplace equation Δ_pⁿ u(x)=0 if and only if the asymptotic formula u(x)=μ_p(\ve,u)(x)+o(\ve²) holds as \ve→ 0 in the viscosity sense. Here, μ_p(\ve,u)(x) is the p-mean value of u on B_\ve(x) characterized as a unique minimizer of inf_\la∈\RR\nr u-\la\nr_{L^p(B_\ve(x))}. This kind of asymptotic mean value property (AMVP) extends to the case p=1 previous (AMVP)'s obtained when μ_p(\ve,u)(x) is replaced by other kinds of mean values. The natural definition of μ_p(\ve,u)(x) makes sure that this is a monotonic and continuous (in the appropriate topology) functional of u. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic p-Laplace equation.

Cited by

• Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results

Related

• Game theoretical asymptotic mean value properties for non-homogeneous p-Laplace problems
• A mean value formula for the variational p-Laplacian
• An Asymptotic Mean Value Characterization for the Regularized p-Laplacian
• Improved regularity for the parabolic normalized p-Laplace equation
• On the Mean Value Property for the p-Laplace equation in the plane