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

TL;DR

This paper proves a version of the Trace Theorem for domains that are locally subgraphs of Hölder continuous functions, extending the trace operator to Sobolev spaces with a regularity loss depending on the Hölder exponent.

Contribution

It establishes a trace theorem for domains defined by Hölder continuous functions, which was previously not well-understood, especially for applications in fluid-structure interaction.

Findings

The trace operator extends continuously from H^1(Ω_η) to H^s(ω) for s<α/2.

The result applies to domains that are locally subgraphs of Hölder continuous functions.

This work is motivated by fluid-structure interaction analysis.

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 $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ 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.