A generalization of Jordan-Schoenflies theorem on simple plane curves

Question

The well known Jordan-Schoenflies theorem says: let $C\subset \mathbb{R}^2$ be a closed simple curve. Then there exists a homeomorphism $f\colon\mathbb{R}^2\to \mathbb{R}^2$ such that $f(C)$ is the unit circle. Moreover if the curve $C$ is smooth then $f$ can be chosen to be diffeomorphism.

I need a slight generalization of the above statement.Let $D$ be the closed unit disk in the 2-plane. Let $C\subset D$ be a closed simple curve in the interior of $D$. Does there exist a homeomorphism $f\colon D\to D$ such that $f(C)$ is the circle of radus $1/2$ centered at 0? (For my purposes $C$ might be assumed to be a smooth curve, i.e. smooth immersion of the standard circle.) A reference would be helpful.

Remark. The point is that $f$ is a homeomorphism of the closed disk. For open one the statement is obviously equivalent to the original Jordan-Schoenflies theorem.

Answer 1

The answer is positive, of course. You can see it as follows. In the smooth case, the Schonfliess theorem says that your given Jordan smooth curve can be sent to the unit circle by a diffeomorphism which is in fact an isotopy $\phi_1$, i.e. the time 1 of the flow $(\phi_t)_{t\in\Bbb R}$ of a time-dependent vector field $(V_t)_{t\in\Bbb R}$. Let $K$ be the compact union of the intermediate positions $\phi_t(C)$, for $0\le t\le 1$. Let $u$ be on $\Bbb R^2$ a compactly supported smooth plateau function whose value is $1$ on $K$. Then, changing $(V_t)_{t\in\Bbb R}$ for $(uV_t)_{t\in\Bbb R}$, you can arrange that $(\phi_t)_{t\in\Bbb R}$ is compactly supported. It follows that any two smooth Jordan curves in the plane are conjugate by a compactly supported isotopy. Since the open unit disk is diffeomorphic to the plane, you get your disk version.