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

Abstract

Let 1 ≤ p ≤ ∞. We show that a function u ∈ C(R N ) is a viscosity solution to the normalized p-Laplace equation n p u(x) = 0 if and only if the asymptotic formulaholds as ε → 0 in the viscosity sense. Here,with respect to λ ∈ R. This kind of asymptotic mean value property (AMVP) extends to the case p = 1 previous (AMVP)'s obtained when μ p (ε, u)(x) is replaced by other kinds of mean values. The natural definition of μ p (ε, 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.

Convergence of the naturalp-means for thep-Laplacian

“…Ishiwata, Magnanini, and Wadade have introduced the notion of natural p-means in their paper [13]. Let X be a compact topological measure space endowed with a positive finite Radon measure ν.…”

“…Existence, uniqueness, and several properties described below are established in [13]. While for general p there is no explicit formula for µ X p (u), this formula exists for the cases p = 1, 2, and p = ∞:…”

“…Note that a minor variation of the proof of the monotonicity Theorem 2.5 in [13] gives u 0 (x) ≤ T u 0 (x) for x ∈ Ω. Define the sequence u n = T u n−1 for n ≥ 1.…”

“…Outline of the paper: In Section 2 we recall the dynamic programming principle (DPP) proved in [9] and its relation with asymptotic mean value properties. In Section 3 we recall the properties of natural p-means following [13]. These are needed in order to prove the DPP for the natural p-means (Thm.…”

A mean value formula for the variational p-Laplacian

“…See for instance [2,7,9,14,[16][17][18], and [22]. We also want to mention [8] and [10], where two other nonlinear mean value formulas are studied, with some similarities with ours.…”