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# Non-autonomous $L^q(L^p)$ maximal regularity for complex systems under mixed regularity in space and time

Abstract : We show non-autonomous $L^q(L^p)$ maximal regularity for families of complex second-order systems in divergence form under a mixed Hölder regularity condition in space and time. To be more precise, we let $p,q \in (1,\infty)$ and we consider coefficient functions in $C^{\beta + \varepsilon}$ with values in $C^{\alpha + \varepsilon}$ subject to the parabolic relation $2\beta + \alpha = 1$. To this end, we provide a weak $(p,q)$-solution theory with uniform constants and establish a priori higher spatial regularity. Furthermore, we show $p$-bounds for semigroups and square roots generated by complex elliptic systems under a minimal regularity assumption for the coefficients.
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https://hal.archives-ouvertes.fr/hal-03745208
Contributor : Sebastian Bechtel Connect in order to contact the contributor
Submitted on : Wednesday, August 3, 2022 - 8:39:24 PM
Last modification on : Friday, August 5, 2022 - 3:21:16 AM

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### Identifiers

• HAL Id : hal-03745208, version 1
• ARXIV : 2208.02527

### Citation

Sebastian Bechtel, Fabian Gabel. Non-autonomous $L^q(L^p)$ maximal regularity for complex systems under mixed regularity in space and time. 2022. ⟨hal-03745208⟩

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