TheoremLet , satisfiesIf 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. , there exists such that

Since , we have

Also on since is non-negative on boundary. Then maximum principle tells in , where . Then we can first take and then take to get in .

RemarkThe condition is essential, otherwise one can take plus a large constant as a counterexample.

Advertisements