Oblique Derivative Boundary Value Problems on Families of Planar Domains

We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in $\mathbb{C}$ that depend smoothly on a real parameter $\lambda \in [0,1]$. We derive sharp regularity properties of the solutions in all variables, including the parameter $\lambda$. More specifically we show that the solution and its derivatives are continuous in all variables, and the H\"older norms of the space variables are bounded uniformly in $\lambda$.