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A note on the Trace Theorem for domains which are locally subgraph of a H\" older continuous function

The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a H\" older continuous function. More precisely, let $\eta\in C^{; ; ; ; 0, \alpha}; ; ; ; (\omega)$, $0<\alpha<1$ and let $\Omega_{; ; ; ; \eta}; ; ; ; $ be a domain which is locally subgraph of a function $\eta$. We prove that mapping $\gamma_{; ; ; ; \eta}; ; ; ; :u\mapsto u({; ; ; ; \bf x}; ; ; ; , \eta({; ; ; ; \bf x}; ; ; ; ))$ can be extended by continuity to a linear, continuous mapping from $H^1(\Omega_{; ; ; ; \eta}; ; ; ; )$ to $H^s(\omega)$, $s<\alpha/2$. This study is motivated by analysis of fluid-structure interaction problems.

Trace Theorem; Fluid-structure interaction; Sobolev spaces; non-Lipschitz domain.

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