On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients

TL;DR

This paper establishes boundary regularity results for weak and strong solutions of elliptic equations with Dini mean oscillation coefficients, extending previous work to include conormal and oblique derivative problems.

Contribution

It proves boundary differentiability of solutions under Dini continuity conditions for coefficients and boundary geometry, extending Safonov's earlier results.

Findings

Weak solutions are continuously differentiable up to the boundary.

Strong solutions are twice continuously differentiable up to the boundary.

Results extend regularity theory to more general boundary conditions.

Abstract

We show that weak solutions to conormal derivative problem for elliptic equations in divergence form are continuously differentiable up to the boundary provided that the mean oscillations of the leading coefficients satisfy the Dini condition, the lower order coefficients satisfy certain suitable conditions, and the boundary is locally represented by a $C^1$ function whose derivatives are Dini continuous. We also prove that strong solutions to oblique derivative problem for elliptic equations in nondivergence form are twice continuously differentiable up to the boundary if the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a $C^1$ function whose derivatives are double Dini continuous. This in particular extends a result of M. V. Safonov (Comm. Partial Differential Equations 20:1349--1367, 1995)