Given a smooth function $u$ defined on $\mathbb{R}^2$, restrict $u$ to a subset $\Omega \subset \mathbb{R}^2$ (possibly not simply connected) foliated by level sets of a smooth function $\psi: \Omega \to \mathbb{R}$. i.e. $\psi^{-1}(t_0)=\partial \Omega$, and$$\Omega=\cup_{t\in [t_0, t_M]} \psi^{-1}(t).$$
Can $u$ and $\psi$ be symmetrize to some function $u^*$ and $\psi^*$ defined on a simply connected region $\Omega^*$ such that $\int_{\{\psi >t\}}e^u=\int_{\{\Omega^*>t\}}e^{u^*}$, and$$\int_{\{\psi^*=t\}}|\nabla u^*| \leq \int_{\{\psi^=t\}}|\nabla u|. $$
Any insights or comments on the formal definition and properties of $u^*$ would be greatly appreciated.