Observability inequality, log-type Hausdorff content and heat equations
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| Format: | Preprint |
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2024
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| _version_ | 1866917852759457792 |
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| author | Huang, Shanlin Wang, Gengsheng Wang, Ming |
| author_facet | Huang, Shanlin Wang, Gengsheng Wang, Ming |
| contents | This paper studies observability inequalities for heat equations on both bounded domains and the whole space $\mathbb{R}^d$. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension $s$ for any $s\in (d-1,d]$, but also for certain sets of Hausdorff dimension $d-1$. On the whole space $\mathbb{R}^d$, we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality.
To obtain these observability inequalities, we use the adapted Lebeau-Robiano strategy from \cite{Duyckaerts2012resolvent}. For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: We establish a spectral inequality/a Logvinenko-Sereda uncertainty principle; we set up a quantitative propagation of smallness of analytic functions; we build up a Remez' inequality; and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate in Lubinsky \cite{Lubinsky1997small}, which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11573 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Observability inequality, log-type Hausdorff content and heat equations Huang, Shanlin Wang, Gengsheng Wang, Ming Analysis of PDEs This paper studies observability inequalities for heat equations on both bounded domains and the whole space $\mathbb{R}^d$. The observation sets are measured by log-type Hausdorff contents, which are induced by certain log-type gauge functions closely related to the heat kernel. On a bounded domain, we derive the observability inequality for observation sets of positive log-type Hausdorff content. Notably, the aforementioned inequality holds not only for all sets with Hausdorff dimension $s$ for any $s\in (d-1,d]$, but also for certain sets of Hausdorff dimension $d-1$. On the whole space $\mathbb{R}^d$, we establish the observability inequality for observation sets that are thick at the scale of the log-type Hausdorff content. Furthermore, we prove that for the 1-dimensional heat equation on an interval, the Hausdorff content we have chosen is an optimal scale for the observability inequality. To obtain these observability inequalities, we use the adapted Lebeau-Robiano strategy from \cite{Duyckaerts2012resolvent}. For this purpose, we prove the following results at scale of the log-type Hausdorff content, the former being derived from the latter: We establish a spectral inequality/a Logvinenko-Sereda uncertainty principle; we set up a quantitative propagation of smallness of analytic functions; we build up a Remez' inequality; and more fundamentally, we provide an upper bound for the log-type Hausdorff content of a set where a monic polynomial is small, based on an estimate in Lubinsky \cite{Lubinsky1997small}, which is ultimately traced back to the classical Cartan Lemma. In addition, we set up a capacity-based slicing lemma (related to the log-type gauge functions) and establish a quantitative relationship between Hausdorff contents and capacities. These tools are crucial in the studies of the aforementioned propagation of smallness in high-dimensional situations. |
| title | Observability inequality, log-type Hausdorff content and heat equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.11573 |