These might just be errors in your describing what you have done, but
D[u[2, 0]] == 0
doesn't include which variable you are differentiating with respect to when all your other derivatives do include which variable
D[u[0, 0] == 0]
looks very questionable, I suspect you don't mean you are differentiating solutions to u and perhaps this is again a derivative that needs a variable and the result is then equal to zero.