Accretivity of the general second order linear differential operator

TL;DR

This paper establishes conditions under which the negative of a general second order linear differential operator with complex coefficients is accretive, ensuring non-negative real part of the associated quadratic form for all smooth compactly supported functions.

Contribution

It provides new criteria for accretivity of second order linear differential operators with complex distributional coefficients.

Findings

Derived sufficient conditions for accretivity.

Extended classical results to operators with complex coefficients.

Applicable to operators with distributional coefficients.

Abstract

For the general second order linear differential operator $$\mathcal L_0 = \sum_{j, \, k=1}^n \, a_{jk} \, \partial_j \partial_k + \sum_{j=1}^n \, b_{j} \, \partial_j + c$$ with complex-valued distributional coefficients $a_{jk}$, $b_{j}$, and $c$ in an open set $\Omega \subseteq \mathbf{R}^n$ ($n \ge 1$), we present conditions which ensure that $-\mathcal L_0$ is accretive, i.e., ${\rm Re} \, \langle -\mathcal L_0 \phi, \phi\rangle \ge 0$ for all $\phi \in C^\infty_0(\Omega).$