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