FIELD Math:Analysis January 10 (Mon), 2022 10:00-11:00 1424 Park, Jinwan Lee, Taehun ¼­¿ï´ëÇÐ±³ The Regularity Theory for the Double Obstacle Problem In this talk, I will introduce the regularity of the free boundary for the double obstacle problems. First, I am going to introduce the proof of local $C^{1}$ regularity of free boundaries for the elliptic 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 $\psi$. The function $\psi$ satisfies $$\psi \in C^{1,1}(B_1) \cap C^{2,1}(\overline{\Omega(\psi)}),\quad \Omega(\psi)=B_1 \setminus \left( \{\psi=0\} \cap \{ \nabla \psi =0\}\right).$$This is a joint work with Ki-ahm Lee and Henrik Shahgholian.Next, I will talk about the parabolic problem. This is a joint work with Ki-ahm Lee. 71421641301553764_1.pdf