De Giorgi-Nash-Moser theory for local and nonlocal equations

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De Giorgi-Nash-Moser theory for local and nonlocal equations

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구분 편미분방정식
일정 2025-07-14(월) 16:00~18:00
세미나실 27동 325호
강연자 옥지훈 (서강대학교)
담당교수 정인지
기타

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.

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