Which boundary condition dominates in elliptic boundary value problem?

Question

I am working on a solution to a boundary value problem (which is too complicated for me to reproduce here) but have a question about the boundary. In many dimensions, my function is infinite along one boundary, while zero along another. At the meeting of these two boundaries, what value does the function take? Infinite? Zero? Undefined? Or is this question not particularly important and should I just pick one of the two since the system will never actually arrive at this point? In any case, thanks for your input!