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.