Any weak solution of

is automatically holder continuous.

This is a corollary of the celebrated thm of De Giorgi, John Nash and J, Moser. It is obtained from the Harnack inequality of weak solution. Actually in dimension 2, this theorem is easily known by mathematicians before the technique of Moser iteration. In the following, we will give a short expository of the proof

From the Poincare inequality

, where , when ,

So this means that

Combining with Caccippoli’s inequality, we can get

with . This is the reverse holder inequality

So , . Thus by Gehring, . That is

The by the Morrey’s embedding thm, in dimension 2, .