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 $(\phi,\lambda)$?
For each $\lambda$,the solution is of the form $\phi(z)=h(z)\exp(\lambda \overline{z}+\overline{\lambda} z)$ where h is holomorphic and purely imaginary along the boundary.
For $\psi(z):=\exp(-\lambda\overline{z})\phi(z)$ is holomorphic by the PDE and subject to boundary condition $$ \exp(\lambda\overline{z})\psi(z)=-\exp(\overline{\lambda}z)\overline{\psi(z)} $$ thus $\exp(-\overline{\lambda} z)\psi(z)$ is holomorphic with boundary value being pure imaginary.
