Trace operator on H 1 ($\Omega$) for general open bounded domains

In the case of any bounded open set $\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, we first construct a directional trace in any direction $\theta$ of the unit sphere, for any u $\in$ L 2 ($\Omega$) whose the directional derivative $\partial$ $\theta$ u in the direction $\theta$ belongs to L 2 ($\Omega$). This directional trace is shown to belong to L 2 ($\partial$$\Omega$, $\mu$ $\theta$ ), where $\mu$ $\theta$ is a measure supported by the closure of all points of $\partial$$\Omega$ which are the extremity of an open segment directed by $\theta$, included in $\Omega$. This trace enables an integration by parts formula. We then show that the set H 1 tr ($\Omega$) containing the elements of H 1 ($\Omega$) whose the directional trace does not depend on $\theta$ is closed. It therefore contains the closure of H 1 ($\Omega$) $\cap$ C 0 ($\Omega$) in H 1 ($\Omega$). Examples where H 1 tr ($\Omega$) = H 1 ($\Omega$) and H 1 tr ($\Omega$) __ = H 1 ($\Omega$) are provided.