Recent zbMATH articles in MSC 31Bhttps://zbmath.org/atom/cc/31B2021-04-16T16:22:00+00:00WerkzeugArsove-Huber theorem in higher dimensions.https://zbmath.org/1456.310052021-04-16T16:22:00+00:00"Ma, Shiguang"https://zbmath.org/authors/?q=ai:ma.shiguang"Qing, Jie"https://zbmath.org/authors/?q=ai:qing.jieThe aim is to extend the Arsove-Huber theory of surfaces to higher dimensions. A basic tool is the \(n\)-Laplace equation \[\text{div}(|\nabla u|^{n-2}\nabla u)= 0\] in exactly \(n\) dimensions (the so-called borderline case). The Brezis-Merle inequality (a refined version of Trudinger's inequality) is applied in \(n\) dimensions. An ingredient is the Wolff potential.
The theory is applied for hypersurfaces in a hyperbolic space, having nonnegative Ricci curvature. Even some unpublished results are announced.
For the entire collection see [Zbl 1446.58001].
Reviewer: Peter Lindqvist (Trondheim)On \(C^m\)-reflection of harmonic functions and \(C^m\)-approximation by harmonic polynomials.https://zbmath.org/1456.310062021-04-16T16:22:00+00:00"Paramonov, Petr V."https://zbmath.org/authors/?q=ai:paramonov.peter-v"Fedorovskiy, Konstantin Yu."https://zbmath.org/authors/?q=ai:fedorovskiy.konstantin-yuThe paper deals with sharp \(C^m\) continuity conditions for harmonic reflection operators of functions over the boundary of simple Carathéodory domains in \(\mathbb{R}^N\). A simple Carathéodory domain \(D \subset \mathbb{R}^N\), \(N\geq 2\), is a nonempty bounded domain sucht that the set \(\Omega \equiv \mathbb{R}^N \setminus \overline{D}\) is a domain, \(\partial D=\partial \Omega\) and, if \(N\geq 3\), both \(D\) and \(\Omega\) are regular with respect to the Dirichlet problem for harmonic functions. The harmonic reflection operator \(R_D\) is the map which takes a sufficiently regular harmonic function \(f\) in \(D\) to the solution \(g\) of the exterior Dirichlet problem for the Laplace operator in the complement of the closure of \(D\) and with Dirichlet data the trace of \(f\) on \(\partial D\). The authors establish sharp conditions on a Carathéodory domain ensuring that the operator \(R_D\) preserves the \(C^m\) continuity, \(m\in(0,1)\), of functions. This is done by means of a new criterion for the \(C^m\) continuity of the Poisson operator, i.e., the operator that maps a smooth function \(\phi\) on \(\partial D\) to the solution \(f\) of the Dirichlet problem with boundary data \(\phi\) on \(\partial D\). Finally, the authors deduce sufficient conditions for the \(C^m\) approximation of functions by harmonic polynomials on \(\partial D\).
Reviewer: Paolo Musolino (Padova)Balayage of measures with respect to (sub-)harmonic functions.https://zbmath.org/1456.310042021-04-16T16:22:00+00:00"Khabibullin, B. N."https://zbmath.org/authors/?q=ai:khabibullin.b-nSummary: We investigate some properties of balayage, or, sweeping (out), of measures with respect to subclasses of subharmonic functions. The following issues are considered: relationships between balayage of measures with respect to classes of harmonic or subharmonic functions and balayage of measures with respect to significantly smaller classes of specific classes of functions; integration of measures and balayage of measures; sensitivity of balayage of measures to polar sets, etc.