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

(more…)

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