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

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\mathrm{\Omega }$ with $L^{\mathrm{\infty }}$ coefficients, $b$ and $c$ two $d$-dimensional vector-valued functions on $\mathrm{\Omega }$ with $L^{\mathrm{\infty }}$ coefficients and $V$ a locally integrable nonegative function on $\mathrm{\Omega }$. Consider the operator ${\mathcal{L}}^{A,b,c,V}=-\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla })+⟨\mathrm{\nabla },b⟩-\mathrm{d}\mathrm{i}\mathrm{v}(c\cdot )+V$ with mixed boundary conditions on $\mathrm{\Omega }$. We extend the bilinear inequality that Carbonaro and Dragičević proved in the special cases when $b=c=0$. As a consequence, we obtain that the solution to the parabolic problem $u^{\mathrm{\prime }}(t)+{\mathcal{L}}^{A,b,c,V}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^p(\mathrm{\Omega })$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity condition that Carbonaro and Dragičević 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 $\mathrm{\Omega }$, in particular, we do not assume any regularity of $\mathrm{\partial }\mathrm{\Omega }$, nor the existence of a Sobolev embedding.