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

Boris Muha

doi:10.3934/nhm.2014.9.191

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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: April 30, 2013

Revised: June 30, 2013

Published: February 2014

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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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Mathematics Subject Classification: Primary: 74F10; Secondary: 46E35.

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