Math Help - Laplace's equation on the unit square

1. Laplace's equation on the unit square

Hello,

I'm hoping for some help with the following question...

v satisfies vxx + vyy = 0 on the open domain Q bounded by the unit square, with v(o,y)= y(1-y) and v = 0 on the other three sides of the square.
Prove that, in the domain Q,

0 < v(x,y) < 1/4(1-x).

Any help would be much appreciated, my exam's in a couple of weeks and I'm stumped with this question!

2. Re: Laplace's equation on the unit square

I think the left hand inequality is straightforward from the minimum principle. For the right hand side of the inequality, I migh suggest try letting

$w = v - \dfrac{1}{4}(1-x)$

create new boundary conditions for $w$ and use the maximum principle for $w$ to show that $w < 0$ inside the square..

3. Re: Laplace's equation on the unit square

ahh, it seems so obvious now!! thank you so much