Spectral analysis for nonlocal elliptic operator

Question

Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to $$-\Delta u=f,\quad u_{|\partial\Omega}=0$$

Now consider the elliptic operator $\Delta + b\cdot\nabla(-\Delta)^{-1}$ associated with, say, homogeneous Dirichlet boundary conditions, where $b$ is a smooth vector field (you can even take $b$ to be a constant vector field if that somehow makes this question answerable).

Is spectral analysis for such an operator possible? (i.e. existence of eigenvalues, properties of eigenfunctions, etc) I ask because such an operator (or rather a slightly more complicated version of it) appears as the linearization of a nonlinear operator that I'm studying. Any help/references would be greatly appreciated.

Answer 1

So that's $$ -\Delta u -b\cdot\nabla v = \lambda u $$ $$ -\Delta v + u =0 $$ which you can write, with $U=(u,v)$ $$ -\Delta U + B \nabla U + C U = \lambda Q U $$ where $\Delta$ is applied line wise, and so is the gradient, whereas $$ B=\begin{pmatrix} 0& b\cdot \\ 0 &0\end{pmatrix}, C=\begin{pmatrix} 0& 0 \\ 1 &0\end{pmatrix}\text{ and }Q=\begin{pmatrix} 1& 0 \\ 0 &0\end{pmatrix}. $$ In the form you wrote it, you see immediately that there is an associated compact operator (hence properties of the spectrum), and that the problem is not self adjoint. Whether it is a cooperative or a non cooperative system will let you prove properties on the eigensolutions. In any case, it is relatively nice, because it is weakly coupled, that is, the second order operator applies to each line separately. Mitidieri and Sweers have looked at these type of problems a while ago. More recent work probably quotes them.