A note on the Trace Theorem for domains which are locally subgraph of a Holder continuous function

Abstract

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 $η\in C^{0,α}(ω)$, $0<α<1$ and let $Ω_η$ be a domain which is locally subgraph of a function $η$. We prove that mapping $γ_η:u\mapsto u({\bf x},η({\bf x}))$ can be extended by continuity to a linear, continuous mapping from $H^1(Ω_η)$ to $H^s(ω)$, $s<α/2$. This study is motivated by analysis of fluid-structure interaction problems.