Weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems

Abstract

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 $C^{{\hat{I}}^2+\epsilon }$ with values in $C^{\hat{I}\pm +\epsilon }$ subject to the parabolic relation $2{\hat{I}}^2+\hat{I}\pm =1$. If $p<\frac{d}{\hat{I}}\pm$, we can likewise deal with spatial $H^{\hat{I}\pm +\epsilon ,\frac{d}{\hat{I}}\pm }$ 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.

31 pages. Final version, published in JDE