Weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911051734319104 |
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| author | Bechtel, Sebastian |
| author_facet | Bechtel, Sebastian |
| contents | We show weighted non-autonomous $L^q(L^p)$ maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let $p,q \in (1,\infty)$ and we consider coefficient functions in $C^{β+ \varepsilon}$ with values in $C^{α+ \varepsilon}$ subject to the parabolic relation $2β+ α= 1$. If $p < \frac{d}α$, we can likewise deal with spatial $H^{α+ \varepsilon, \frac{d}α}$ regularity. The starting point for this result is a weak $(p,q)$-solution theory with uniform constants. Further key ingredients are a commutator argument that allows us to establish higher a priori spatial regularity, operator-valued pseudo differential operators in weighted spaces, and a representation formula due to Acquistapace and Terreni. Furthermore, we show $p$-bounds for semigroups and square roots generated by complex elliptic systems under a minimal regularity assumption for the coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02527 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems Bechtel, Sebastian Analysis of PDEs Classical Analysis and ODEs We show weighted non-autonomous $L^q(L^p)$ maximal regularity for families of complex second-order systems in divergence form under a mixed regularity condition in space and time. To be more precise, we let $p,q \in (1,\infty)$ and we consider coefficient functions in $C^{β+ \varepsilon}$ with values in $C^{α+ \varepsilon}$ subject to the parabolic relation $2β+ α= 1$. If $p < \frac{d}α$, we can likewise deal with spatial $H^{α+ \varepsilon, \frac{d}α}$ regularity. The starting point for this result is a weak $(p,q)$-solution theory with uniform constants. Further key ingredients are a commutator argument that allows us to establish higher a priori spatial regularity, operator-valued pseudo differential operators in weighted spaces, and a representation formula due to Acquistapace and Terreni. Furthermore, we show $p$-bounds for semigroups and square roots generated by complex elliptic systems under a minimal regularity assumption for the coefficients. |
| title | Weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2208.02527 |