The dimension of the solution space of the equation $\lambda\, \overline{\partial}f=f$

Question

Let $\Omega$ be a bounded domain in the complex plane with smooth boundary. Fix $\lambda\in\mathbb{C}$ and consider the d-bar equation $\lambda\frac{\partial f}{\partial\overline{z}}=f$ in $\Omega$. For practical purposes lets restrict ourselves to solutions in $C^2(\overline{\Omega}).$ What can be said about the dimension of solution to such equation?What if we pose the same question but this time for $\lambda\frac{\partial f}{\partial z}=f$ ?

Answer 1

With the substitution $$ f(z) = e^{\bar z/\lambda } \phi(z) \tag{1} $$ the equation becomes
$$\frac{\partial \phi}{ \partial\bar z}=0$$ which says simply that $\phi$ is holomorphic. So, (1) gives all solutions as $\phi$ runs over holomorphic functions. That's an infinite-dimensional space.

Similarly for the second equation: $$ f(z) = e^{ z/\lambda } \overline{\phi(z)} $$