I recently came across a Dirichlet problem for a vector valued functions. In broad terms the problem is as follows. 


Suppose $\Omega \subset \Bbb R^n$ is a smooth bounded domain, $P:C^\infty(X)^n \to C^\infty(X)^n$ is a second order vector valued differential operator with elliptic symbol (in my case the second order part is just the Laplacian on each component) and $f \in C^\infty(\bar \Omega)^n$. Under what conditions does the boundary value problem 


$P(u)=f$


$u|\_{\partial \Omega}=0$


have a unique solution for $u \in C^\infty(X)$? 


Clearly, to provide a weak solution one can still apply the Fredholm alternative to see that uniqueness guarantees existence. So then the obvious question arises: what guarantees uniqueness? In the scalar case, when the zero order term is negative this is automatic by a weak version of the maximum principle. But I don't know what is the analogous condition in the vector valued case.


It would be nice to have a reference for these type of "PDE systems".



vector valued pde's good reference
There is another famous criterion which is used a lot in geometry, which is the Bochner technique. If the system is symmetric (and self-adjoint with Dirichlet conditions) and
there is a good integration by parts formula


$\int Pu \cdot u = \int |\nabla u|^2 + A |u|^2 $ 


(which would require u = 0 on the boundary) and if $A$ is nonnegative, then there is
no nullspace. This is used, for example, to prove that there are no harmonic 1-forms on manifolds (with or without boundary) with positive Ricci curvature - though for the Hodge Laplacian there are slightly different boundary conditions which are more geometric.


