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We consider the Cauchy problem for non-autonomous forms inducing ellipticoperators in divergence form with Dirichlet, Neumann, or mixed boundaryconditions on an open subset $\Omega$ $\subseteq$ R n. We obtain maximalregularity in L 2 ($\Omega$) if the coefficients are bounded, uniformlyelliptic, and satisfy a scale invariant bound on their fractionaltime-derivative of order one-half. Previous results even for such formsrequired control on a time-derivative of order larger than one-half.

On non-autonomous maximal regularity for elliptic operators in divergence form