A Simple Proof of Regularity for C^(1,α)Interface Transmission Problems

We give an alternative proof of a recent result in[1]by Caffarelli,Soria-Carro,and Stinga about the C^(1,α)regularity of weak solutions to transmission problems with C^(1,α)interfaces.Our proof does not use the mean value property or the maximum principle,and also works for more general elliptic systems with variable coefficients.This answers a question raised in[1].Some extensions to C^(1,Dini)interfaces and to domains with multiple sub-domains are also discussed.