# Ngô Quốc Anh

## June 16, 2010

### De Giorgi’s class and De Giorgi’s theorem

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 18:55

In this entry, we introduce that De Giorgi’s class $DG(\Omega)$ and prove that functions in $DG(\Omega)$ are Holder continuous. This has as a consequence the celebrated De Giorgi’s theorem saying that solutions of second-order elliptic equations with bounded measurable coefficients are Holder continuous.

Let us study $H^1$-solutions of $\displaystyle\int_\Omega {{A^{\alpha \beta }}(x){\nabla _\alpha }u{\nabla _\beta }\varphi dx} = 0, \quad \varphi \in H_0^1(\Omega )$

where $A$ is assumed to be of class $L^\infty(\Omega)$. Our aim is to show that $u$ is in fact Holder continuous. The key idea is to use the Cacciopoli inequality on level sets of $u$.

Definition (De Giorgi’s class). We define the De Giorgi’s class ( $DG(\Omega)$) to consist of all $u \in H^1(\Omega)$ which satisfy $\displaystyle \int_{{B_\rho }} {{{\left| {\nabla {{(u - k)}^ + }} \right|}^2}dx} \leqslant \frac{c}{{{{(R - \rho )}^2}}}\int_{{B_R}} {{{\left| {{{(u - k)}^ + }} \right|}^2}dx} , \quad \forall k \in \mathbb{R}$.

The following remark is useful.

1. If $u$ is a sub-solution then $u \in DG(\Omega)$.
2. If $u$ is a super-solution then $-u \in DG(\Omega)$.
3. If $u \in DG(\Omega)$ then $u+{\rm const} \in DG(\Omega)$.
4. In the definition of the De Giorgi class, exponent $p=2$ can be replaced by any $p>1$.

If we denote by $A$ the level set of $u$, that is $A(k,r)=\{x \in B_r : u(x)>k\}$

then if $u \in DG(\Omega)$ by choosing a cut-off function $\displaystyle \eta \in C_0^\infty\left(B_\frac{R+\rho}{2}\right), \quad \eta \equiv 1, \quad {\rm on } B_\rho$