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| A Simple Proof of Regularity for $C^{1,\alpha}$\\ Interface Transmission Problems |
| Hongjie Dong |
| (Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA) |
| DOI： |
| Abstract: |
| We give an alternative proof of a recent result in~\cite{CST20} 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 with variable coefficients. This answers a question raised in~\cite{CST20}. Some extensions to $C^{1,\text{Dini}}$ interfaces and to domains with multiple sub-domains are also discussed. |
| Key words: Transmission problems, elliptic systems with variable coefficients, $C^{1,\alpha}$ and $C^{1,\text{Dini}}$ interfacial boundaries, $C^\alpha$ and Dini coefficients. |