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

Abstract

Let $\Omega \subseteq {\mathbb{R}}^d$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty }$ coefficients. Consider the divergence-form operator ${\mathcal{L}}^A=-\mathrm{div}(A\nabla )$ with mixed boundary conditions on $\Omega$. We extend the bilinear inequality that we proved in [16] in the special case when $\Omega ={\mathbb{R}}^d$. As a consequence, we obtain that the solution to the parabolic problem $u^{\prime }(t)+{\mathcal{L}}^{A}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^p(\Omega )$, for all $p>1$ such that $A$ satisfies the $p$-ellipticity condition that we introduced in [16]. This range of exponents is optimal for the class of operators we consider. We do not impose any conditions on $\Omega$, in particular, we do not assume any regularity of $\partial \Omega$, nor the existence of a Sobolev embedding. The methods of [16] do not apply directly to the present case and a new argument is needed.

General improvements with respect to v1