# How to keep a spot cool?

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## Abstract

Let $D$ be a planar domain, let $a$ be a<br>\emph{reference} point fixed in $D$, and let $b_k$,<br>$k=1,\ldots,n$, be $n$ \emph{controlling} points fixed in<br>$D\setminus\{a\}$. Suppose further that each $b_k$ is connected to<br>the boundary $\partial D$ by an arc $l_k$. In this paper, we<br>propose a problem to find a shape of arcs $l_k$, $k=1,\ldots,n$,<br>which provides the minimum to the harmonic measure<br>$\omega(a,\cup_{k=1}^n l_k,D\setminus \cup_{k=1}^n l_k)$. This<br>problem can also be interpreted as a problem on the minimal<br>temperature at $a$, in the steady-state regime, when the arcs<br>$l_k$ are kept at constant temperature $T_1$ while the boundary<br>$\partial D$ is kept at constant temperature $T_0<T_1$.<br><br>In this paper, we mainly discuss the first non-trivial case of<br>this problem when $D$ is the unit disk $\mathbb{D}=\{z:\,|z|<1\}$<br>with the reference point $a=0$ and two controlling points<br>$b_1=ir$, $b_2=-ir$, $0<r<1$. It appears, that even in
Original language English 23pp. Ann. Acad. Sci. Fenn. Math. Published - Sep 10 2021