Let D be a differential operator on a manifold with boundary. We consider the differential equation $Df = 0$ with Dirichlet-boundary-conditions $f|_{\partial M}= g$.
Are there cases, where not every $g$ leads to a unique solution, but every $g$ which satisfies additional constraints, e.g. satisfying a differential equation on the boundary, leads to a unique solution?
Thank you for your help!