De Giorgi-Nash-Moser theory for local and nonlocal equations
| 구분 | 편미분방정식 |
|---|---|
| 일정 | 2025-07-14 16:00 ~ 18:00 |
| 강연자 | 옥지훈 (서강대학교) |
| 기타 | |
| 담당교수 | 정인지 |
De Giorgi-Nash-Moser theory addresses the H\"older continuity and the Harnack inequality for solutions of partial differential equations, which are among the most fundamental regularity results for second-order elliptic and parabolic equations. This theory was first developed for simple elliptic equations of the form
\[
\mathrm{div} (A(x)Du) = 0
\quad \text{in } \Omega \subset \mathbb{R}^n
\]
where the coefficient matrix $A(x)$ satisfies the following uniform ellipticity condition:
\[
\Lambda^{-1}|\xi|^2 \le A(x)\xi \cdot \xi \le \Lambda |\xi|^2,
\quad \text{for all }\ \xi \in \mathbb{R}^n \ \text{and } x \in \Omega,
\]
for some $\Lmabda\ge1$.
In this seminar, I will first introduce the De Giorgi-Nash-Moser theory for elliptic equations where the coefficient matrices may not satisfy the uniform ellipticity condition. Then, I will discuss the De Giorgi-Nash-Moser theory for nonlocal integro-differential equations of the form
\[
\mathrm{P.V.} \int_{\mathbb{R}^n} (u(x)-u(y))K(x,y)\, dy = 0,
\quad x\in \Omega,
\]
with various kernels.
