The Regularity Theory for the Double Obstacle Problem

In this paper, we prove local $C^1$ regularity of free boundaries for the double obstacle problem with an upper obstacle $\psi$, \begin{align*} \Delta u &=f\chi_{\Omega(u) \cap{ u< \psi} }+ \Delta \psi \chi_{\Omega(u)\cap {u=\psi}}, \qquad u\le \psi \quad \text { in } B_1, \end{align*} where $\mathrm{\Omega }(u)=B_1\setminus (\{u=0\}\cap \{\mathrm{\nabla }u=0\})$ under a thickness assumption for $u$ and $\psi$.