We will show a basic argument why Harnack inequality implies Holder continuity. Assume is a weak solution to a uniform elliptic equation

in . Assume we have Harnack inequality (Indeed we do have due to Moser), i.e. 83 more words

We will show a basic argument why Harnack inequality implies Holder continuity. Assume is a weak solution to a uniform elliptic equation

in . Assume we have Harnack inequality (Indeed we do have due to Moser), i.e. 83 more words

— 1. Definition of Newtonian Potential —

882 more wordsDefinition 1 Let be a bounded open set, we define the Newtonian potential of is the function on by…

Theorem Let , satisfies

If there exists such that

If as , then in .

*Proof:* Without loss of generality assume on , or we can add a large constant on to make it positive on boundary.