I need help finding th series solution to the laplace equation $u_{xx} +u_{yy}=0\\$ in the infinite rectangle $\Pi =(-\infty,0]$x$[0,\pi]$ in $R^2(x,y)$ provided that sup$_{(x,y)\in\Pi}|u(x,y)|<\infty$ and the function u=u(x,y) satisfy the boundary values $u_y(x,0)=u_y(x,\pi)=0, u(0,y)=h(y).$
I have seperated the variables to be $\frac{X''(x)}{-X(x)}=\frac{Y''(y)}{Y(y)}=\lambda$ and split them up to two ODE, but i cant figure out what to do next to meet the boundry conditions when the conditions are with respect to derivative of y. I also dont quite understand the meaning of the notation with sup.
All help is apperciated!
Given that the function is a sum of terms of the form $X(x)Y(y)$, to satisfy the boundary conditions on $y$ we need $X(x)Y'(0)=X(x)Y'(\pi)=0$. For this to have a nontrivial solution we must have $Y'(0)=Y'(\pi)=0$. So we need $$ Y''=\lambda Y, \quad Y'(0)=Y'(\pi)=0. $$ There are two sorts of solution to this equation: if $\lambda=0$, $1$ is a solution. If $\lambda<0$, $\cos{\sqrt{-\lambda}y}$ is a solution provided that $\lambda = -n^2$ for some integer $n$.
For $X$, boundedness means we must take $e^{nx}$ as the solution to $X''=n^2X$ to keep the solution bounded. Thus we have $$ u(x,y) = A_0 + \sum_{n=1}^{\infty} A_n e^{nx} \cos{ny} $$ as the putative solution, and you then find the $A_0$ in the usual way using the Fourier series of $h(y)$.