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Characterization of domains through families of measures. (English) Zbl 1041.31001

Let \(\Omega\) be a plane domain bounded by a regular curve \(\gamma\). For each point \(x\in\Omega\), let \(\lambda_x\) be a measure defined on the Lebesgue measurable subsets of \(\gamma\). For example, for each measurable set \(A\subset\gamma\), we could have \(\lambda_x(A)= \omega(x,\Omega, A)\), the harmonic measure of \(A\) at the point \(x\). Assume that \(\Omega\) is starlike with respect to the point \(x\), and let \(\rho= \rho(x,\theta)\), \(0\leq\theta\leq 2\pi\), denote the polar equation for \(\gamma\) relative to the point \(x\). If \(E\) is an arc of \(\gamma\), then the total internal visual angle for \(E\) relative to the point \(x\) is given by \[ \Theta(x, E)= \{\theta: (\rho(x,\theta), \theta)\in E\}.\tag{1} \] The author considers optimization problems for which the goal is to find \[ \max\{\Theta(x, E): E\subset\gamma,\,\lambda_x(E)= C\},\tag{2} \] where \(C\) is a constant. If, for each choice of \(C< \lambda_x(C)\), the problem is solved by a single arc, then the visual angle is said to be coalescent. The author proves two basic results: (1) The circle is the unique Dini-smooth domain \(\Omega\) for which the harmonic measure \(\omega(x,\Omega,-)\) is coalescent for each point \(x\in\Omega\); and (2) the circle is the unique convex \(C^1\) domain such that the internal visual angle \(\Theta(x,-)\) is coalescent with respect to arc length for each point \(x\in\Omega\). The proof of (1) uses a previous result of the author [Rev. Mat. Complut. 12, 477–509 (1999; Zbl 0960.49029)]. The proof of (2) involves a lengthy argument using facts about the function \(f(\theta)= \sqrt{\rho^2(x,\theta)+ [\rho'(x,\theta)]^2}\), the Radon-Nikodym derivative of the arc length measure on \(\gamma\).

MSC:

31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
28A12 Contents, measures, outer measures, capacities

Citations:

Zbl 0960.49029
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