vector valued pde's good reference

Question

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".

Answer 1

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.

Answer 2

It depends very much on how your equations are coupled. In the case you describe, that is, the second order part is just the laplacian on every line, if additionally the first order part is diagonal, this is called a weakly coupled elliptic system.

The paper you would like to read is probably 'Weakly Coupled Elliptic Systems and Positivity' by E. Mitidieri and G. Sweers (1994). The systems where you have the usual maximum principle are the cooperative ones. A typical example is when the coupling matrix is such that the diagonal is dominant, for example. They investigate in detail sufficient conditions for the system to be cooperative, and also discuss the non-cooperative case.