Usually there exists a unique solution if the number of degrees of freedom equals the number conditions.

Say we have a PDE in space, so our domain is an N-ball inside R^(N).

Say we have M unknown scalar variables, (y\_1,y\_2,...,y\_M). That is we have M degrees of freedom.

Say our system is a system of first order linear PDE. More specifically our system of PDEs has a total of P equations. That is we have P conditions.

Say our boundary conditions consist of a total of Q equations. That is we have Q conditions.

What should we expect M,P,Q to be if we want a unique solution to exist? We expect M=P, and M=Q correct?

For example a Dirichlet Boundary condition totally determines our variables so M=Q.

Is it the same for a Neumann Boundary condition?

PS: If our PDE had a time component, some initial value and a boundary condition, nothing would change right? It would be the exact same?