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

we get

or

.

By using the Sobolev inequality

and the Holder inequality

we conclude that

.

For

.

Set

and write the above inequalities in the form

For some positive numbers and we find

.

Now we choose and in such a way that for some

.

Solving these equations gives and .

Thus we arrive at

where .

Proposition. For any and any we havefor some .

Consequently, if we choose we get

Theorem. If then.

In particular when we have

Theorem. If then

and

Theorem. If then

From these two estimates we infer

Theorem. Solution of PDEis bounded and

.

We now come to the main result of this entry

Theorem(De Giorgi). If and belong to then is -Holder continuous. Moreover, for we have.

The proof of the Holder continuity is based an estimate concerning the oscillation

.

Precisely, there exists some constant such that

.

Thus by scaling and iterating this estimate, one obtains the Holder continuity of since the oscillation of is easily seen to decay as a power of radius. To see this, let pick and arbitrary in . Denote , then

.

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