The Regularity Theory for the Double Obstacle Problem

TL;DR

This paper establishes local $C^{1}$ regularity of free boundaries in the double obstacle problem, advancing understanding of the geometric structure of solutions under certain thickness conditions.

Contribution

It proves the regularity of free boundaries for the double obstacle problem with an upper obstacle, under specific thickness assumptions, which was previously unresolved.

Findings

Proves local $C^{1}$ regularity of free boundaries.

Establishes regularity results under thickness assumptions.

Provides a framework for analyzing double obstacle problems with upper obstacles.

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 $\psi$.