Trace operator on H 1 ($Ω$) for general open bounded domains
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912975573483520 |
|---|---|
| author | Eymard, Robert Gallouët, Thierry Maltese, David Vincent, Yannick |
| author_facet | Eymard, Robert Gallouët, Thierry Maltese, David Vincent, Yannick |
| contents | In the case of any bounded open set $Ω$ $\subset$ R d with boundary $\partial$$Ω$, we first construct a directional trace in any direction $θ$ of the unit sphere, for any u $\in$ L 2 ($Ω$) whose the directional derivative $\partial$ $θ$ u in the direction $θ$ belongs to L 2 ($Ω$). This directional trace is shown to belong to L 2 ($\partial$$Ω$, $μ$ $θ$ ), where $μ$ $θ$ is a measure supported by the closure of all points of $\partial$$Ω$ which are the extremity of an open segment directed by $θ$, included in $Ω$. This trace enables an integration by parts formula. We then show that the set H 1 tr ($Ω$) containing the elements of H 1 ($Ω$) whose the directional trace does not depend on $θ$ is closed. It therefore contains the closure of H 1 ($Ω$) $\cap$ C 0 ($Ω$) in H 1 ($Ω$). Examples where H 1 tr ($Ω$) = H 1 ($Ω$) and H 1 tr ($Ω$) __ = H 1 ($Ω$) are provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02107 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trace operator on H 1 ($Ω$) for general open bounded domains Eymard, Robert Gallouët, Thierry Maltese, David Vincent, Yannick Functional Analysis In the case of any bounded open set $Ω$ $\subset$ R d with boundary $\partial$$Ω$, we first construct a directional trace in any direction $θ$ of the unit sphere, for any u $\in$ L 2 ($Ω$) whose the directional derivative $\partial$ $θ$ u in the direction $θ$ belongs to L 2 ($Ω$). This directional trace is shown to belong to L 2 ($\partial$$Ω$, $μ$ $θ$ ), where $μ$ $θ$ is a measure supported by the closure of all points of $\partial$$Ω$ which are the extremity of an open segment directed by $θ$, included in $Ω$. This trace enables an integration by parts formula. We then show that the set H 1 tr ($Ω$) containing the elements of H 1 ($Ω$) whose the directional trace does not depend on $θ$ is closed. It therefore contains the closure of H 1 ($Ω$) $\cap$ C 0 ($Ω$) in H 1 ($Ω$). Examples where H 1 tr ($Ω$) = H 1 ($Ω$) and H 1 tr ($Ω$) __ = H 1 ($Ω$) are provided. |
| title | Trace operator on H 1 ($Ω$) for general open bounded domains |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2502.02107 |