A simple proof of regularity for $C^{1,\alpha}$ interface transmission problems

TL;DR

This paper presents a straightforward proof demonstrating the $C^{1,eta}$ regularity of weak solutions to transmission problems with $C^{1,eta}$ interfaces, extending previous results with a more general approach.

Contribution

It provides a simplified proof of regularity for transmission problems that avoids traditional methods and applies to more general elliptic systems and interfaces.

Findings

Proof does not rely on mean value property or maximum principle.

Extends regularity results to $C^{1, ext{Dini}}$ interfaces.

Applicable to domains with multiple sub-domains.

Abstract

We give a simple proof of a recent result in [1] by Caffarelli, Soria-Carro, and Stinga about the $C^{1,\alpha}$ regularity of weak solutions to transmission problems with $C^{1,\alpha}$ interfaces. Our proof does not use the mean value property or the maximum principle, and also works for more general elliptic systems. Some extensions to $C^{1,\text{Dini}}$ interfaces and to domains with multiple sub-domains are also discussed.