Accretivity of the general second order linear differential operator

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 $\phi \in C^\infty_0(\Omega).$