A note on the Trace Theorem for domains which are locally subgraph of a Hölder continuous function
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1.
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb
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Received:
01 May 2013
Revised:
01 July 2013
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Primary: 74F10; Secondary: 46E35.
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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.
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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