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