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

TL;DR

This paper develops a directional trace operator for functions in L2 on arbitrary bounded open sets, enabling integration by parts and analyzing the structure of H1 functions with well-defined traces.

Contribution

It introduces a general directional trace for L2 functions on arbitrary domains, extending trace theory beyond smooth boundaries.

Findings

Constructed directional trace in any direction for L2 functions with directional derivatives in L2.

Proved the trace belongs to a measure-supported L2 space on boundary segments.

Identified conditions under which H1 trace space coincides with H1 or its closure.

Abstract

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.