A note on the Trace Theorem for domains which are locally subgraph of a Holder 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 ${\mathrm{\Omega }}_{\eta }$ be a domain which is locally subgraph of a function $\eta$. We prove that mapping ${\gamma }_{\eta }:u\mapsto u(\mathbf{x},\eta (\mathbf{x}))$ can be extended by continuity to a linear, continuous mapping from $H^1({\mathrm{\Omega }}_{\eta })$ to $H^s(\omega )$, $s<\alpha \mathrm{/}2$. This study is motivated by analysis of fluid-structure interaction problems.