Density of $H^{1/2}(\partial \Omega)$ in $L_2(\partial\Omega)$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:28:32Z http://mathoverflow.net/feeds/question/94688 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94688/density-of-h1-2-partial-omega-in-l-2-partial-omega Density of $H^{1/2}(\partial \Omega)$ in $L_2(\partial\Omega)$ Mike 2012-04-20T20:59:40Z 2012-04-20T23:06:31Z <p>Hi,</p> <p>i know that the following statement is used extensively, but i cannot find a proof anywhere:</p> <p>For $\Omega$ a Lipschitz domain with boundary $\Gamma$, the space $H^{1/2}(\Gamma)$ is dense in $L_2(\Gamma)$.</p> <p>Here, the space $H^{1/2}(\Gamma)$ is defined as the trace space, i.e. as $\gamma_0(H^1(\Omega))$, where $\gamma_0$ is the trace operator.</p> <p>I read that one has to use density of $C^\infty(\overline\Omega)$ in $H^1(\Omega)$, but i dont see how - i can't extend an $L_2(\Gamma)$ function to a function in $H^1(\Omega)$ in a bounded way, right?</p> <p>I tried Google and looked in all the books I have access to, but didn't find a proof. Does anyone have a hint?</p> <p>Thanks, Mike</p> http://mathoverflow.net/questions/94688/density-of-h1-2-partial-omega-in-l-2-partial-omega/94698#94698 Answer by Piero D'Ancona for Density of $H^{1/2}(\partial \Omega)$ in $L_2(\partial\Omega)$ Piero D'Ancona 2012-04-20T23:06:31Z 2012-04-20T23:06:31Z <p>Lipschitz continuous functions on $\Gamma$ are dense in $L^2(\Gamma)$ and are contained in $H^{1/2}(\Gamma)$.</p>