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

TL;DR

This paper extends bilinear inequalities for divergence-form operators with complex coefficients on irregular domains, leading to maximal regularity results for associated parabolic problems under new conditions.

Contribution

It generalizes bilinear inequalities to operators with complex coefficients and irregular domains, establishing maximal regularity for parabolic equations with minimal domain regularity assumptions.

Findings

Extended bilinear inequality to complex coefficients and irregular domains.

Proved maximal regularity for parabolic problems under new conditions.

No regularity assumptions on the domain boundary.

Abstract

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