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*} Δu &=fχ_{Ω(u) \cap\{ u< ψ\} }+ Δψχ_{Ω(u)\cap \{u=ψ\}}, \qquad u\le ψ\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 $\psi$.

21 pages