Counterexamples to maximal regularity for operators in divergence form
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| Format: | Preprint |
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2024
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| _version_ | 1866908803405971456 |
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| author | Bechtel, Sebastian Mooney, Connor Veraar, Mark |
| author_facet | Bechtel, Sebastian Mooney, Connor Veraar, Mark |
| contents | In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such operators admit maximal $L^2$-regularity on $H^{-1}$ under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal $L^p$-regularity on $H^{-1}(\mathbb{R}^d)$ or $L^2$-regularity on $L^2(\mathbb{R}^d)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05550 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counterexamples to maximal regularity for operators in divergence form Bechtel, Sebastian Mooney, Connor Veraar, Mark Analysis of PDEs Classical Analysis and ODEs Primary: 35K90, Secondary: 35B65 In this paper, we present counterexamples to maximal $L^p$-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions' theory that such operators admit maximal $L^2$-regularity on $H^{-1}$ under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal $L^p$-regularity on $H^{-1}(\mathbb{R}^d)$ or $L^2$-regularity on $L^2(\mathbb{R}^d)$. |
| title | Counterexamples to maximal regularity for operators in divergence form |
| topic | Analysis of PDEs Classical Analysis and ODEs Primary: 35K90, Secondary: 35B65 |
| url | https://arxiv.org/abs/2401.05550 |