In this entry, we introduce that De Giorgi’s class and prove that functions in 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 -solutions of

where is assumed to be of class . Our aim is to show that is in fact Holder continuous. The key idea is to use the Cacciopoli inequality on level sets of .

Definition(De Giorgi’s class). We define the De Giorgi’s class () to consist of all which satisfy.

The following remark is useful.

- If is a sub-solution then .
- If is a super-solution then .
- If then .
- In the definition of the De Giorgi class, exponent can be replaced by any .

If we denote by the level set of , that is

then if by choosing a cut-off function

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