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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2311.01628 |
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| _version_ | 1866916330619273216 |
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| author | Shao, Arick Vergara, Bruno |
| author_facet | Shao, Arick Vergara, Bruno |
| contents | We consider heat operators on a bounded domain $Ω\subseteq \mathbb{R}^n$, with a critically singular potential diverging as the inverse square of the distance to $\partial Ω$. While null boundary controllability for such operators was recently proved in all dimensions in arXiv:2112.04457, it crucially assumed (i) $Ω$ was convex, (ii) the control must be prescribed along all of $\partial Ω$, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of $Ω$, (ii) allow for the control to be localized near any $x_0 \in \partial Ω$, and (iii) treat the full range of strength parameters for the singular potential. Morever, we lower the regularity required for $\partial Ω$ and the lower-order coefficients. The key novelty is a local Carleman estimate near $x_0$, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of $\partial Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01628 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Approximate boundary controllability for parabolic equations with inverse square infinite potential wells Shao, Arick Vergara, Bruno Analysis of PDEs We consider heat operators on a bounded domain $Ω\subseteq \mathbb{R}^n$, with a critically singular potential diverging as the inverse square of the distance to $\partial Ω$. While null boundary controllability for such operators was recently proved in all dimensions in arXiv:2112.04457, it crucially assumed (i) $Ω$ was convex, (ii) the control must be prescribed along all of $\partial Ω$, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of $Ω$, (ii) allow for the control to be localized near any $x_0 \in \partial Ω$, and (iii) treat the full range of strength parameters for the singular potential. Morever, we lower the regularity required for $\partial Ω$ and the lower-order coefficients. The key novelty is a local Carleman estimate near $x_0$, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of $\partial Ω$. |
| title | Approximate boundary controllability for parabolic equations with inverse square infinite potential wells |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2311.01628 |