How to keep a spot cool?

Abstract

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.