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

Boris Muha; Boris Muha

doi:10.3934/nhm.2014.9.191

Networks and Heterogeneous Media

2014, Volume 9, Issue 1: 191-196. doi: 10.3934/nhm.2014.9.191

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

Boris Muha ^{1
,}

1.
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb

Received: 01 May 2013 Revised: 01 July 2013
Primary: 74F10; Secondary: 46E35.

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 $\eta\in C^{0,\alpha}(\omega)$, $0<\alpha<1$ 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
Citation: Boris Muha. A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function[J]. Networks and Heterogeneous Media, 2014, 9(1): 191-196. doi: 10.3934/nhm.2014.9.191

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Abstract

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

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