We know that for an ODE of $n^{th}$ order we need $n$ different boundary conditions. In PDEs, for example, for Laplace equation $\nabla^2 U=0$ (which is a second order PDE) we need only one B.C. (e.g Dirichlet or Neumann boundary conditions on the boundary $\partial \Omega$.)
$\large {\frac{\partial U}{\partial \vec n}|_{\partial \Omega}=0\space}$ or $\large { \space U|_{\partial \Omega}=0}$
I would like to know what is a sufficient boundary condition to a more general equation like $\Large {\frac{\partial^m U}{\partial x^m}+\frac{\partial^n U}{\partial y^n}=0}$ on a boundary $\partial \Omega$. How do we know if a given boundary condition is sufficient to solve a general PDE? What is the relation between the order of a PDE and the number and type of boundary conditions needed on $\partial \Omega$.
Thanks.