The regularity theory for the double obstacle problem

Abstract

In this paper, we prove a local $C^{1}$-regularity of the free boundary for the (hybrid) double obstacle problem with an upper obstacle $\psi$,

$$\begin{aligned} \Delta u&=f\chi _{\Omega (u) \cap \{ u< \psi \} }+ \Delta \psi \chi _{\Omega (u)\cap \{u=\psi \}}, \qquad u\le \psi \quad \text{ in } B_1, \end{aligned}$$

where $\Omega (u)=B_1 {\setminus } \left( \{u=0\} \cap \{ \nabla u =0\}\right)$ under a thickness assumption for u and $\psi$. The novelty of the paper is the study of points where two obstacles meet, here it refers to free boundary points where $\psi =0$. Our result is new, with a non-straightforward approach, as the analysis seems to require several subtle manoeuvres in finding the right conditions and methodology. A key point of difficulty lies in the classification of global solutions. This is due to the complex structure of global solutions for the double obstacle problem, and even more complex for the hybrid problem in this paper.