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

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.