Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem: $$ \overline{\partial} \phi = \lambda\, \phi \quad \text{on $D$},$$ subject to the boundary condition $\textrm{Re}(\phi) =0$ on $\partial D$.

Does there exist a nontrivial pair $(\phi,\lambda)$?