By the maximum principle I know that attains it's maximum on the boundary. Hence I know that the maximum of and are and respectively.Let be a bounded set with smooth boundary. Assume that and are and that they satisfy:

in

on

and

in

on

where and are continuous functions.

Assume that for on . Prove that

for

State any theorems used in the proof.

Since I know that for .

I need to justify that where .

Since the laplacian is negative I know that and .

This is where I get my problem. , and do not necessarily imply that for .

Can anyone help?