Let $\Omega$ be a strictly convex domain, $\partial\Omega\in C^{2,1}$
We define a foliation $\{\Omega_t\}_{0\leq t\leq 1}\subset\Omega$ as follows.
Let $\Omega_0=B_r(x_0)$, a small ball centered at $x_0$ in $\Omega$ and $\Omega_1=\Omega$. We can have a uniformly concave defining functions $h_1,h_0$ such that $h_1>0$ in $\Omega$ and $h_0>0$ in $\Omega_0$
Define $\Omega_t=\{x|h_t(x)>0\}$, where $h_t=th_1+(1-t)h_0$
Suppose $\Omega_{t_0}$ is a fixed domain, $t_0\in(0,1)$. The question is, how can we construct a $C^{4}$ diffeomorphism $\Phi_t:\Omega_{t_0}\rightarrow\Omega_t$ such that it map every point in $\Omega_t$ back to $\Omega_{t_0}$ and $\Phi_{t_0}$ is the identity map?