Interpolating between disks in the plane

Question

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

Task: 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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Remark: If $D_0$ is contained in the interior of $D_1$, then the level curves of the solution of the Dirichlet problem on $D_1 {\setminus} \mathring D_0$ with boundary values $0$ on $\partial D_0$ and $1$ on $\partial D_1$ 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, but it doesn't obviously work when $\partial D_0 \cap \partial D_1 \neq \emptyset$.

Answer 1

Here is an incomplete approach that was too long for a comment.

1. In some smooth way, choose $\epsilon>0$ such that

   (i) the region $C_\epsilon\subset \overline{D_1}$ within distance $\epsilon$ of $\partial D_1$ is a collar neighbourhood of $\partial D_1$ (i.e. there is a smooth embedding $\phi:\partial D_1 \times [0,\epsilon) \to C$ such that $\phi(x,t)$ is the point in $C$ at distance $t$ along the normal $\nu_x$ to $\partial D_1$ at $x$),

   (ii) $C_\epsilon \cap \partial D_0$ is the set $\{\phi(x,u(x)):x \in V_\epsilon\}$ for some set $V_\epsilon \subset \partial D_1$ and smooth function $u:V_\epsilon \to [0,\epsilon)$, and

   (iii) $|\nabla u|<\tilde{\epsilon}$ on $V_\epsilon$ for some $\tilde{\epsilon}>0$.

2. Let $\eta\in C^\infty(\mathbb{R})$ be a smooth function with $\eta=0$ on $(-\infty,0)$ and $\eta=1$ on $[1,\infty)$. For some $0<\delta<\epsilon$, let $\tilde{D}_0 \subset D_1$ be the set such that $$ \partial \tilde{D}_0 \cap C_{\frac{\epsilon}{2}} =\left\{\phi\left(x,\frac{\epsilon}{2}+\left(u(x)-\frac{\epsilon}{2}\right)\eta\left(\frac{2u(x)}{\epsilon}\right)\right):x \in V_\epsilon, u(x)<\frac{\epsilon}{2}\right\}, $$ and $\tilde{D}_0 \backslash C_{\frac{\epsilon}{2}}=D_0 \backslash C_{\frac{\epsilon}{2}}$. Then $\tilde{D}_0$ has smooth boundary and $\partial \tilde{D}_0 \cap \partial D_1=\emptyset$.

3. Let $w \in C^\infty(D_1 \backslash \tilde{D}_0$ denote the solution to the Dirichlet problem on $D_1 \backslash \tilde{D}_0$ with boundary conditions $w=0$ on $\partial \tilde{D}_0$ and $w=1$ on $\partial D_1$. I believe, but have no proof, that by choosing $\epsilon,\tilde{\epsilon}$ small enough, we can ensure that for each $x \in V_\epsilon$, $s \mapsto w(\phi(x,s))$ is increasing for $s \in (0,u(x))$. For $t \in (0,1)$ and $x \in V_x$, let $u_t(x)$ denote the distance from $x$ to $w^{-1}(s)$ along $\nu_x$.

4. Define $D_t$ such that $D_t \backslash C_\epsilon$ is the $t$-superlevel set of $w$ in $D_1 \backslash C_\epsilon$, and $$ \partial D_t \cap C_\epsilon=\left\{\left(x, u_t(x) + \left(t u(x) - u_t(x)\right) \eta\left(\frac{2u(x)}{\epsilon}-1\right)\right):x \in V_x \right\}. $$ That is, $D_t$ moves linearly from $D_0$ to $D_1$ within $C_{\frac{\epsilon}{2}}$, $D_t$ is a level set of $w$ on $D_1 \backslash C_\epsilon$. Also, $D_t$ is smooth, and on $C_\epsilon$ (as well as on $D_1 \backslash C_\epsilon$), the $D_t$ are correctly nested since $u_t(x) + \left(t u(x) - u_t(x)\right) \eta\left(\frac{2u(x)}{\epsilon}-1\right)$ is increasing in $t$.

Hopefully, someone more familiar with the Dirichlet problem can show that the solution to the DIrichlet problem on a thin enough strip has gradients which are never parallel to the strip.