Two smooth bounded domains in $\mathbb{R}^2$ with the same boundary are equal Given two bounded smooth open subsets of the plane, can I say that if their boundary are equal then they are equal ?
This question received a positive answer under a connectedness assumption on both sets, but I would like to know if this assumption can be removed.
Edit: by smooth set I mean a set whose interior and boundary are locally the epigraph and the graph of a smooth (say $C^{\infty}$) function.
Edit: I guess the result reduces to proving that for every connected component $E$ of the first set, there is a unique connected component $F$ of the second set such that $\partial E=\partial F$. One could then conclude using the answer to the linked question.
 A: I am assuming that for smooth domain you mean a subset of $\mathbb R^2$ whose boundary is a smooth curve. Then, the answer is no.
Let $D_r$ be the open disk of radius $r$ centred in the origin and $S_r$ be its boundary.
Let $U = \mathrm{int}(D_2\setminus D_1)$, where int denotes the interior. In other words you have $$U=\{(x, y): x^2+y^2\in(1, 4)\}\,.$$ Clearly $U$ is a open smooth region and its boundary is $S_1\cup S_2$.
Now, consider $V = D_1\cup U$, that is $$V=\{(x, y): x^2+y^2\in[0,1)\cup(1, 4)\}\,.$$ Again, this is an open smooth region and its boundary is $S_1\cup S_2$.
$U$ and $V$ are bounded, smooth, and with the same boundary. Yet, $U\neq V$.
A: Another example (at least for the topological boundary).
Begin with a countable set of numbers $r_k \in (1,2)$ that cluster only at $1$ and $2$.  The boundary set consists of circles centered at the origin with these radii.  The two sets $E, F$ consist of alternating annuli.  The annuli get narrower and narrower as we approach radius $1$ or radius $2$.  The topological boundary consists of these circles, together with the circles of radii $1$ and $2$.  Does the geometric boundary consist only of these circles, without the two extreme circles?

There is a $C^\infty$ function that vanishes on these circles, and is alternately positive and negative on the annuli.
