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 C that depend smoothly on a real parameter lambda is an element of [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 Holder norms of the space variables are bounded uniformly in lambda.(c) 2023 Elsevier Masson SAS. All rights reserved.