The Dirichlet-Dirac problem

Question

Let $(M,g)$ be a spin manifold with Dirac operator $\cal D$. The Dirichlet-Dirac problem is to find a spinor $\psi$ such that:

$${\cal D} \psi +\psi =\phi$$

on $M$, and

$$\psi=0$$

on the boundary of $M$, $\phi$ is a fixed spinor.

Have we solutions of the Dirichlet-Dirac problem?