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:
where and are continuous functions.
Assume that for on . Prove that
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?