Below, a "disk" means a compact subspace $D \subset \mathbb R^2$ whose boundary is a smooth simple closed curve.

><b>Task:</b> Find a procedure which takes as input a pairs
of disks
$
D_0 \subseteq D_1
$
in the plane,
and produces as output a smooth 1-parameter family of disks that interpolates between them:
$$
\{D_t\}_{t \in [0,1]}.
$$
The family should be monotonic in the sense that
$
t_1 \le t_2 \Rightarrow D_{t_1} \subseteq D_{t_2}.
$

The procedure should furthermore be (continuous and) smooth, meaning that if we have a family of pairs $D_0(x) \subseteq D_1(x)$ depending smoothly on some parameter $x\in\mathbb R^n$, then the output of the procedure $\{D_t(x)\}_{t \in [0,1]}$ should depend smoothly on $(t,x)\in [0,1]\times\mathbb R^n$.

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<b>Remark:</b> If $\partial D_0 \cap \partial D_1 = \emptyset$, then the level curves of the solution of the Dirichlet problem on $D_1 {\setminus} \mathring D_0$ with boundary values $0$ on $D_0$ and $1$ on $D_1$ provide a family of simple closed curves interpolating between $\partial D_0$ and $\partial D_1$ (hence a family of disks interpolating between $D_0$ and $D_1$). This procedure has all the desired good properties, except that it doesn't work when $\partial D_0 \cap \partial D_1 \neq \emptyset$.