Maximal $L_p$-regularity for fractional problem driven by non-autonomous forms
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| Format: | Preprint |
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2025
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| _version_ | 1866916654978433024 |
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| author | He, Jia Wei Li, Shi Long Zhou, Yong |
| author_facet | He, Jia Wei Li, Shi Long Zhou, Yong |
| contents | We investigate the maximal $L_p$-regularity in J.L. Lions' problem involving a time-fractional derivative and a non-autonomous form $a(t;\cdot,\cdot)$ on a Hilbert space $H$. This problem says whether the maximal $L_p$-regularity in $H$ hold when $t \mapsto a(t ; u, v)$ is merely continuous or even merely measurable. We prove the maximal $L_p$-regularity results when the coefficients satisfy general Dini-type continuity conditions. In particular, we construct a counterexample to negatively answer this problem, indicating the minimal Hölder-scale regularity required for positive results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10010 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal $L_p$-regularity for fractional problem driven by non-autonomous forms He, Jia Wei Li, Shi Long Zhou, Yong Classical Analysis and ODEs 26A33, 35B65, 45D05 We investigate the maximal $L_p$-regularity in J.L. Lions' problem involving a time-fractional derivative and a non-autonomous form $a(t;\cdot,\cdot)$ on a Hilbert space $H$. This problem says whether the maximal $L_p$-regularity in $H$ hold when $t \mapsto a(t ; u, v)$ is merely continuous or even merely measurable. We prove the maximal $L_p$-regularity results when the coefficients satisfy general Dini-type continuity conditions. In particular, we construct a counterexample to negatively answer this problem, indicating the minimal Hölder-scale regularity required for positive results. |
| title | Maximal $L_p$-regularity for fractional problem driven by non-autonomous forms |
| topic | Classical Analysis and ODEs 26A33, 35B65, 45D05 |
| url | https://arxiv.org/abs/2503.10010 |