Oblique Derivative Boundary Value Problems on Families of Planar Domains

TL;DR

This paper studies elliptic equations with oblique boundary conditions on smoothly varying planar domains, establishing sharp regularity results for solutions and their derivatives uniformly across the domain family.

Contribution

It provides new regularity estimates for solutions of elliptic boundary value problems on parameter-dependent domains, including uniform bounds in the parameter.

Findings

Solutions are continuous in all variables, including the parameter.

Hölder norms of solutions are uniformly bounded across the domain family.

Sharp regularity properties are established for solutions and their derivatives.

Abstract

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 $\lambda$.