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

Abstract

Let $\hat{I}\text{c◯}\subseteq {\mathbb{R}}^d$ be open, $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\hat{I}\text{c◯}$ with $L^{\infty }$ coefficients, $b$ and $c$ two $d$-dimensional vector-valued functions on $\hat{I}\text{c◯}$ with $L^{\infty }$ coefficients and $V$ a locally integrable nonegative function on $\hat{I}\text{c◯}$. Consider the operator ${\mathcal{L}}^{A,b,c,V}=-\mathrm{div}(A\nabla )+⟨\nabla ,b⟩-\mathrm{div}(c\cdot )+V$ with mixed boundary conditions on $\hat{I}\text{c◯}$. 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 $u^{\prime }(t)+{\mathcal{L}}^{A,b,c,V}u(t)=f(t)$, $u(0)=0$, has maximal regularity in $L^p(\hat{I}\text{c◯})$, 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 $\hat{I}\text{c◯}$, in particular, we do not assume any regularity of $\partial \hat{I}\text{c◯}$, nor the existence of a Sobolev embedding.

45 pages. arXiv admin note: text overlap with arXiv:1905.01374 by other authors