Trace operator on H 1 ($\hat{I}\text{c◯}$) for general open bounded domains

Abstract

In the case of any bounded open set $\hat{I}\text{c◯}$ $\subset$ R d with boundary $\partial$$\hat{I}\text{c◯}$, we first construct a directional trace in any direction $\hat{I}¸$ of the unit sphere, for any u $\in$ L 2 ($\hat{I}\text{c◯}$) whose the directional derivative $\partial$ $\hat{I}¸$ u in the direction $\hat{I}¸$ belongs to L 2 ($\hat{I}\text{c◯}$). This directional trace is shown to belong to L 2 ($\partial$$\hat{I}\text{c◯}$, $\hat{I}¼$ $\hat{I}¸$ ), where $\hat{I}¼$ $\hat{I}¸$ is a measure supported by the closure of all points of $\partial$$\hat{I}\text{c◯}$ which are the extremity of an open segment directed by $\hat{I}¸$, included in $\hat{I}\text{c◯}$. This trace enables an integration by parts formula. We then show that the set H 1 tr ($\hat{I}\text{c◯}$) containing the elements of H 1 ($\hat{I}\text{c◯}$) whose the directional trace does not depend on $\hat{I}¸$ is closed. It therefore contains the closure of H 1 ($\hat{I}\text{c◯}$) $\cap$ C 0 ($\hat{I}\text{c◯}$) in H 1 ($\hat{I}\text{c◯}$). Examples where H 1 tr ($\hat{I}\text{c◯}$) = H 1 ($\hat{I}\text{c◯}$) and H 1 tr ($\hat{I}\text{c◯}$) __ = H 1 ($\hat{I}\text{c◯}$) are provided.