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
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.47 more words