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An asymptotic mean value characterization for $$p$$-harmonic functions. (English) Zbl 1187.35115
Summary: We characterize $$p$$-harmonic functions in terms of an asymptotic mean value property. A $$p$$-harmonic function $$u$$ is a viscosity solution to $$\Delta_pu= \text{div}(|\nabla u|^{p-2}\nabla u)=0$$ with $$1<p\leq\infty$$ in a domain $$\Omega$$ if and only if the expansion
$u(x)= \frac{\alpha}{2} \bigg\{\max_{\overline{B_\varepsilon(x)}}u+ \min_{\overline{B_\varepsilon(x)}}u\bigg\}+ \frac{\beta}{|B_\varepsilon(x)|} \int_{B_\varepsilon(x)} u\,dy+o(\varepsilon^2)$
holds as $$\varepsilon\to 0$$ for $$x\in\Omega$$ in a weak sense, which we call the viscosity sense. Here the coefficients $$\alpha,\beta$$ are determined by $$\alpha+\beta=1$$ and $$\alpha/\beta= (p-2)/(N+2)$$.

##### MSC:
 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35J60 Nonlinear elliptic equations 35J70 Degenerate elliptic equations
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