The Regularity Theory for the Double Obstacle Problem

Abstract: 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 $\Omega(u)=B_1 \setminus \left( {u=0} \cap { \nabla u =0}\right)$ under a thickness assumption for $u$ and $\psi$.