Is it true that all boundary problems (ODE's n-order, PDE's, etc.) in any dimension always have an unique solution (if a solution exists)? If not, what are some counterexamples to this? Furthermore, is there a general set of restrictions that will allow the uniqueness of solution (ODE, PDE, etc.). For example, forcing both the derivative and the position function to have certain values at the boundary, or required the solution to be infinitely smooth and the space compact. Maybe it has to do with weak derivative, etc.
My intuition is that the answer should be no. Suppose two non-identical solutions for the BVP exist. Then I can "follow" both equations "out" (away) from the Boundary to a point where they differ. This means the vector field points in two directions at this point, which is not allowed to happen as learned in elementary differential equation.