Suppose we know the following theorem

Let , satisfy the Possion equation in . Then and if , we have

- ,
- ,

Use the above theorem and cut-off function

,

when and when

to prove the interior holder estimate of possion equation

Let be a domain in , in . If and , then and for any two concentric balls , we have

WLOG assume , let , is the cut off fuction.

then .

Since has compact support, we have representation

Let

Then from the representation. We will bound them respectively.

For , note that , and theorem 4.5 implies

So satisfies (1)

For , since when , we get that in fact

Since we have when and , this means

So satisfies (1)

For , by the same reason,

Apply the Green identity on the domin

So we can estimate as , which means also satisfies (1).

Gilbarg Trudinger’s book. chapter 4, exercise 4.3.