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 H{\"o}lder 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