Non-autonomous maximal regularity for forms given by elliptic operators of bounded variation

We show maximal Lp-regularity for non-autonomous Cauchy problems provided the trace spaces are stable in some parameterized sense and the time dependence is of bounded variation. In particular on L2(Ω), for Lipschitz domains Ω and under mixed boundary conditions, we obtain maximal Lp-regularity for all p∈(1,2] for elliptic operators with coefficients aij:Ω→C satisfying aij(⋅,x)∈BV uniformly in x∈Ω.