How to keep a spot cool?

Abstract

Let $D$ be a planar domain, let $a$ be a reference point fixed in $D$, and let $b_k$, $k=1,\dots ,n$, 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 $l_k$, $k=1,\dots ,n$, 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.