Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces. (English) Zbl 1333.30071
In [A. Björn et al., J. Reine Angew. Math. 556, 173–203 (2003; Zbl 1018.31004)] harmonic extensions
\[
H_f(x)=\int_{\partial \Omega} f(y)P_{x_0}(x,y)\, d\nu_{x_0}(y)
\]
for \(f\in L^1(\partial \Omega, \nu_{x_0})\) were constructed, where \(x_0\in \Omega\) is a fixed point, \(\nu_{x_0}\) is a harmonic measure on \(\partial \Omega\) evaluated at \(x_0\) and \(P_{x_0}\) is a real-valued function defined on \(\Omega\times \partial\Omega\).
In the paper under review, the authors determine the Poisson kernel \(P_{x_0}\). To do so, they identify the harmonic measure. It turns out that it is the outward normal derivative of the Green function.
The authors define a metric Laplacian and investigate its properties. An important result is the verification of a Gauss-Green formula. A further result is that harmonic (the definition of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure) functions have the mean value property and the converse holds as well.
In the paper under review, the authors determine the Poisson kernel \(P_{x_0}\). To do so, they identify the harmonic measure. It turns out that it is the outward normal derivative of the Green function.
The authors define a metric Laplacian and investigate its properties. An important result is the verification of a Gauss-Green formula. A further result is that harmonic (the definition of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure) functions have the mean value property and the converse holds as well.
Reviewer: Thomas Zürcher (Coventry)
MSC:
| 30L99 | Analysis on metric spaces |
| 31B05 | Harmonic, subharmonic, superharmonic functions in higher dimensions |
Citations:
Zbl 1018.31004References:
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