On the normal trace space of extended divergence-measure fields
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917545285517312 |
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| author | Irving, Christopher |
| author_facet | Irving, Christopher |
| contents | We characterise the normal trace space associated to extended (measure-valued) divergence-measure fields on the boundary of a set $E \subset \mathbb R^n$, as the Arens-Eells space $\mathrm{AE}(\partial E)$. Such a trace operator is constructed for any Borel set $E$, and under a mild regularity condition, which includes Lipschitz domains, this trace operator is shown to moreover be surjective. This relies in part on a new pointwise description of the Anzellotti pairing $\overline{\nabla ϕ\cdot {\boldsymbol F}}$ between a $\mathrm{W}^{1,\infty}$ function $ϕ$ and extended divergence-measure field ${\boldsymbol F}$. As an application, we prove extension theorems for divergence-measure fields and divergence-free measures. Results for $\mathrm{L}^1$-fields are also obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09536 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the normal trace space of extended divergence-measure fields Irving, Christopher Functional Analysis Analysis of PDEs 35R06, 49Q15 We characterise the normal trace space associated to extended (measure-valued) divergence-measure fields on the boundary of a set $E \subset \mathbb R^n$, as the Arens-Eells space $\mathrm{AE}(\partial E)$. Such a trace operator is constructed for any Borel set $E$, and under a mild regularity condition, which includes Lipschitz domains, this trace operator is shown to moreover be surjective. This relies in part on a new pointwise description of the Anzellotti pairing $\overline{\nabla ϕ\cdot {\boldsymbol F}}$ between a $\mathrm{W}^{1,\infty}$ function $ϕ$ and extended divergence-measure field ${\boldsymbol F}$. As an application, we prove extension theorems for divergence-measure fields and divergence-free measures. Results for $\mathrm{L}^1$-fields are also obtained. |
| title | On the normal trace space of extended divergence-measure fields |
| topic | Functional Analysis Analysis of PDEs 35R06, 49Q15 |
| url | https://arxiv.org/abs/2503.09536 |