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

A natural approach to the asymptotic mean value property for the p-Laplacian Let $$1\le p\le \infty$$ 1 ≤ p ≤ ∞ . We show that a function $$u\in C(\mathbb R^N)$$ u ∈ C ( R N ) is a viscosity solution to the normalized p-Laplace equation $$\Delta _p^n u(x)=0$$ Δ p n u ( x ) = 0 if and only if the asymptotic formula \begin{aligned} u(x)=\mu _p(\varepsilon ,u)(x)+o(\varepsilon ^2) \end{aligned} u ( x ) = μ p ( ε , u ) ( x ) + o ( ε 2 ) holds as $$\varepsilon \rightarrow 0$$ ε → 0 in the viscosity sense. Here, $$\mu _p(\varepsilon ,u)(x)$$ μ p ( ε , u ) ( x ) is the p-mean value of u on $$B_\varepsilon (x)$$ B ε ( x ) characterized as a unique minimizer of \begin{aligned} \Vert u-\lambda \Vert _{L^p(B_\varepsilon (x))} \end{aligned} ‖ u - λ ‖ L p ( B ε ( x ) ) with respect to $$\lambda \in {\mathbb {R}}$$ λ ∈ R . This kind of asymptotic mean value property (AMVP) extends to the case $$p=1$$ p = 1 previous (AMVP)’s obtained when $$\mu _p(\varepsilon ,u)(x)$$ μ p ( ε , u ) ( x ) is replaced by other kinds of mean values. The natural definition of $$\mu _p(\varepsilon ,u)(x)$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

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

, Volume 56 (4) – Jun 10, 2017
22 pages

/lp/springer_journal/a-natural-approach-to-the-asymptotic-mean-value-property-for-the-p-Yu6htSXDpf
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1188-7
Publisher site
See Article on Publisher Site

### Abstract

Let $$1\le p\le \infty$$ 1 ≤ p ≤ ∞ . We show that a function $$u\in C(\mathbb R^N)$$ u ∈ C ( R N ) is a viscosity solution to the normalized p-Laplace equation $$\Delta _p^n u(x)=0$$ Δ p n u ( x ) = 0 if and only if the asymptotic formula \begin{aligned} u(x)=\mu _p(\varepsilon ,u)(x)+o(\varepsilon ^2) \end{aligned} u ( x ) = μ p ( ε , u ) ( x ) + o ( ε 2 ) holds as $$\varepsilon \rightarrow 0$$ ε → 0 in the viscosity sense. Here, $$\mu _p(\varepsilon ,u)(x)$$ μ p ( ε , u ) ( x ) is the p-mean value of u on $$B_\varepsilon (x)$$ B ε ( x ) characterized as a unique minimizer of \begin{aligned} \Vert u-\lambda \Vert _{L^p(B_\varepsilon (x))} \end{aligned} ‖ u - λ ‖ L p ( B ε ( x ) ) with respect to $$\lambda \in {\mathbb {R}}$$ λ ∈ R . This kind of asymptotic mean value property (AMVP) extends to the case $$p=1$$ p = 1 previous (AMVP)’s obtained when $$\mu _p(\varepsilon ,u)(x)$$ μ p ( ε , u ) ( x ) is replaced by other kinds of mean values. The natural definition of $$\mu _p(\varepsilon ,u)(x)$$ 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.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jun 10, 2017

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