$\displaystyle \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0, u=-K \quad\mbox{at} \quad x=\pm b, y=\pm h$
How to solve this?
If you let $\displaystyle u = v - K$ then $\displaystyle v $ satisfies
$\displaystyle \dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} = 0$
where $\displaystyle v = 0$ on the boundary of the rectangle. The solution of this second problem is $\displaystyle v \equiv 0$. Thus, the solution of your problem us $\displaystyle u = -K$ on the entire region.