How to keep a spot cool?


  • Alexander Yu. Solynin Texas Tech University, Department of Mathematics and Statistics


Heat distribution, harmonic measure, quadratic differential, symmetrization



Let \(D\) be a planar domain, let \(a\) be a reference point fixed in \(D\), and let \(b_k\), \(k=1,\ldots,n\), be \(n\) 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 \(l_k\), \(k=1,\ldots,n\), 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 \(\mathbf{D}=\{z\colon|z|<1\}\) 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.





How to Cite

Solynin, A. Y. (2021). How to keep a spot cool?. Annales Fennici Mathematici, 46(2), 739–769. Retrieved from