Bilinear embedding for divergence-form operators with complex coefficients on irregular domains

Let Ω ⊆ Rd be open and A a complex uniformly strictly accretive d× d matrix-valued function on Ω with L∞ coefficients. Consider the divergence-form operator LA=-div(A∇) with mixed boundary conditions on Ω. We extend the bilinear inequality that we proved in Carbonaro and Dragičević (J Eur Math Soc, to appear) in the special case when Ω = Rd. As a consequence, we obtain that the solution to the parabolic problem u′(t) + LAu(t) = f(t) , u(0) = 0 , has maximal regularity in Lp(Ω) , for all p> 1 such that A satisfies the p-ellipticity condition that we introduced in Carbonaro and Dragičević (to appear). This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on Ω , in particular, we do not assume any regularity of ∂Ω , nor the existence of a Sobolev embedding. The methods of Carbonaro and Dragičević (to appear) do not apply directly to the present case and a new argument is needed.