hope you all are doing well. I am working on minimal surfaces (Chemical engineering background), and I am stuck at a particular problem. I need to solve Laplace equation with the following boundary conditions
$\frac{\partial^2z}{\partial x^2} + \frac{\partial^2z}{\partial y^2} = 0$
boundary conditions: (1) $\frac{\partial z(x, 0)}{\partial y}=C_1$; (2) $\frac{\partial z(x, 1)}{\partial y}=C_2$; (3) $\frac{\partial z(0, y)}{\partial x}=0$; (4) $\frac{\partial z(1, y)}{\partial x}=0$;
I am interested in knowing $\textbf{if a unique solution to this problem exists?}$
Any help would be highly appreciated.
Thanks in advance.