# Accretivity and form boundedness of second order differential operators

## TL;DR

This paper surveys conditions under which second order differential operators with complex coefficients are accretive or form bounded, providing a comprehensive overview of recent characterizations in divergence and non-divergence forms.

## Contribution

It consolidates and reviews recent results characterizing accretivity and form boundedness of complex second order differential operators in various forms.

## Key findings

- Characterization of accretivity for divergence and non-divergence operators.
- Criteria for form boundedness involving complex coefficients.
- Unified framework for analyzing second order differential operators.

## Abstract

Let $\mathcal{L}$ be the general second order differential operator with complex-valued distributional coefficients $A=(a_{jk})_{j, k=1}^n$, $\vec{b}=(b_{j})_{j=1}^n$, and $c$ in an open set $\Omega \subseteq \mathbb{R}^n$ ($n \ge 1$), with principal part either in the divergence form, $\mathcal{L} u= {\rm div} \, (A \nabla u) + \vec{b} \cdot\nabla u + c \, u$, or non-divergence form, $ \mathcal L u= \sum_{j, \, k=1}^n \, a_{jk} \, \partial_j \partial_k u + \vec{b} \cdot\nabla u + c \, u $. We give a survey of the results by the authors which characterize the following two properties of $\mathcal{L}$: (1) $-\mathcal{L}$ is accretive, i.e., ${\rm Re} \, \langle -\mathcal L u, \, u\rangle \ge 0$; (2) $\mathcal L$ is form bounded, i.e., $\vert \langle \mathcal L u, u \rangle \vert \le C \, \Vert \nabla u \Vert_{L^2(\Omega)}^2$, for all complex-valued $u \in C^\infty_0(\Omega)$.