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

Let Ωᵈ be open, A a complex uniformly strictly accretive d× d matrix-valued function on Ω with L∞ coefficients, b and c two d-dimensional vector-valued functions on Ω with L∞ coefficients and V a locally integrable nonegative function on Ω. Consider the operator LA,b,c,V=- div\,(A∇) + ∇ , b - div\,(c \, ·) + V with mixed boundary conditions on Ω. We extend the bilinear inequality that Carbonaro and Dragi{c}evi\'c proved in the special cases when $b=c = 0$. As a consequence, we obtain that the solution to the parabolic problem u^(t)+ LA,b,c,Vu(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 Carbonaro and Dragi{c}evi\'c introduced in arXiv:1611.00653 and $b,c,V$ satisfy another condition that we introduce in this paper. Roughly speaking, V has to be ``big'' with respect to b and c. 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.