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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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