In order to solve a 2D Laplace Equation in x-y coordinates, we need to prescribe 4 boundary conditions (2 in x and 2 in y). For example, if the domain of interest is a 1x1 rectangle, we can specify BCs at x=0, x=1, y=0, and y=1.
But, what if the domain is arbitrary, say a hexagon and there are values on all 6 sides that I'm interested to prescribe ? It seems like I will end up with more than 4 BCs in such case ? But, BCs have to be specified all along the boundary for the PDE to be well-posed.
Perhaps, I'm missing something very basic here. Could one of you help me understand this ? Your help will be much appreciated.
Thanks in advance,