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

The purpose of this note is to prove a version of the Trace Theorem for domains which are locally subgraph of a Hölder continuous function. More precisely, let η ∈ C0,α(ω), 0 < α < 1 and let Ωη be a domain which is locally subgraph of a function η. We prove that mapping γη : u 7→ u(x, η(x)) can be extended by continuity to a linear, continuous mapping from H(Ωη) to Hs(ω), s < α/2. This study is motivated by analysis of fluid-structure interaction problems.