Let me see if I understand the problem:

If

is an open set in

with a smooth boundary

. If

is a continous

-function so that

for some

and all

(i.e.

is constant on the boundary). And if

for all

. Then

must be a constant function.

This is true by the

__maximum principle__ of harmonic functions. The maximum principle states that

attains its maximum and minimum on the boundary. But since the function on the boundary is konstant it means

for all

. And so

is constant.

Though, I am not sure if the maximum principle has any limitations. If

happens to be bounded then it applies. What happens if

is the right-half plane? I do not know in that case. If the maximum principle still applies then the argument still works.