On the closability of differential operators
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913827410411520 |
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| author | Alberti, Giovanni Bate, David Marchese, Andrea |
| author_facet | Alberti, Giovanni Bate, David Marchese, Andrea |
| contents | We discuss the closability of directional derivative operators with respect to a general Radon measure $μ$ on $\mathbb{R}^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions $\mathrm{Lip}(\mathbb{R}^d)$ to $L^p(μ)$, for $1\leq p\leq\infty$. We also discuss the closability of the same operators from $L^q(μ)$ to $L^p(μ)$, and give necessary and sufficient conditions for closability, but we do not have an exact characterization. As a corollary we obtain that classical differential operators such as gradient, divergence and Jacobian determinant are closable from $L^q(μ)$ to $L^p(μ)$ only if $μ$ is absolutely continuous with respect to the Lebesgue measure. We finally consider the closability of a certain class of multilinear differential operators; these results are then rephrased in terms of metric currents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_08058 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the closability of differential operators Alberti, Giovanni Bate, David Marchese, Andrea Classical Analysis and ODEs Analysis of PDEs Functional Analysis 26B05, 49Q15, 26A27 We discuss the closability of directional derivative operators with respect to a general Radon measure $μ$ on $\mathbb{R}^d$; our main theorem completely characterizes the vectorfields for which the corresponding operator is closable from the space of Lipschitz functions $\mathrm{Lip}(\mathbb{R}^d)$ to $L^p(μ)$, for $1\leq p\leq\infty$. We also discuss the closability of the same operators from $L^q(μ)$ to $L^p(μ)$, and give necessary and sufficient conditions for closability, but we do not have an exact characterization. As a corollary we obtain that classical differential operators such as gradient, divergence and Jacobian determinant are closable from $L^q(μ)$ to $L^p(μ)$ only if $μ$ is absolutely continuous with respect to the Lebesgue measure. We finally consider the closability of a certain class of multilinear differential operators; these results are then rephrased in terms of metric currents. |
| title | On the closability of differential operators |
| topic | Classical Analysis and ODEs Analysis of PDEs Functional Analysis 26B05, 49Q15, 26A27 |
| url | https://arxiv.org/abs/2311.08058 |