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Seminars

Seminar
FIELD Math:Analysis
DATE January 10 (Mon), 2022
TIME 10:00-11:00
PLACE 1424
SPEAKER Park, Jinwan
HOST Lee, Taehun
INSTITUTE 서울대학교
TITLE The Regularity Theory for the Double Obstacle Problem
ABSTRACT 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.
FILE 71421641301553764_1.pdf
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