Show thatIf f attains its maximum at some point in L, show that f is a constant function.

Question

Let L be the integer lattice, that is, the set of points (i,j) in the plane where i and j are integers. Let f be a real valued function defined on L. Suppose that for all $ (i,j) \in L $

$$ f(i,j) = {f(i-1,j) +f(i+1,j) +f(i,j+1) + f(i,j-1) \over 4 }$$

If f attains its maximum at some point in L, show that f is a constant function.

Answer 1

If $f$ achieve maximum value $M$ at $f(i,j)$ then we have: $$4M = f(i-1,j) +f(i+1,j) +f(i,j+1) + f(i,j-1) \leq 4M $$ so $$ f(i-1,j) =f(i+1,j) =f(i,j+1) = f(i,j-1)=M $$

So each neighbor of point with maximum value has also maximu value. Clearly we can extend this whole plane. So $f$ is a constant function.