Recent zbMATH articles in MSC 30C85https://www.zbmath.org/atom/cc/30C852022-05-16T20:40:13.078697ZWerkzeugHarmonic measure of the outer boundary of colander setshttps://www.zbmath.org/1483.310032022-05-16T20:40:13.078697Z"Glücksam, Adi"https://www.zbmath.org/authors/?q=ai:glucksam.adiSummary: We present two companion results: Phragmén-Lindelöf type tight bounds on the minimal possible growth of subharmonic functions with a recurrent zero set, and tight bounds on the maximal possible decay of the harmonic measure of the outer boundary of colander sets.How to keep a spot cool?https://www.zbmath.org/1483.310112022-05-16T20:40:13.078697Z"Solynin, Alexander Yu."https://www.zbmath.org/authors/?q=ai:solynin.alexander-yuSummary: Let \(D\) be a planar domain, let \(a\) be a \textit{reference} point fixed in \(D\), and let \(b_k, k=1,\dots,n\), be \(n\) \textit{controlling} points fixed in \(D\setminus\{a\}\). Suppose further that each \(b_k\) is connected to the boundary \(\partial D\) by an arc \(l_k\). In this paper, we propose the problem of finding a shape of arcs \(l_k\), \(k=1,\dots,n\), which provides the minimum to the harmonic measure \(\omega(a,\bigcup_{k=1}^n l_k,D\setminus \bigcup_{k=1}^n l_k)\). This problem can also be interpreted as a problem on the minimal temperature at \(a\), in the steady-state regime, when the arcs \(l_k\) are kept at constant temperature \(T_1\) while the boundary \(\partial D\) is kept at constant temperature \(T_0< T_1\).
In this paper, we mainly discuss the first non-trivial case of this problem when \(D\) is the unit disk \(\mathbf{D}=\{z:|z|< 1\}\) with the reference point \(a=0\) and two controlling points \(b_1=ir\), \(b_2=-ir\), \(0< r< 1\). It appears, that even in this case our minimization problem is highly nontrivial and the arcs \(l_1\) and \(l_2\) providing minimum for the harmonic measure are not the straight line segments as it could be expected from symmetry properties of the configuration of points under consideration.