Non-autonomous Lq(Lp) maximal regularity for complex systems under mixed regularity in space and time

Abstract

We show non-autonomous Lq(Lp) 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∈(1,∞) and we consider coefficient functions in Cβ+ε with values in Cα+ε subject to the parabolic relation 2β+α=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.