Accretivity and form boundedness of 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, $\mathcal{L}u=\mathrm{div}(A\nabla u)+\vec{b}\cdot \nabla u+cu$, or non-divergence form, $\mathcal{L}u={\sum }_{j,k=1}^n{a}_{jk}{\partial }_j{\partial }_{k}u+\vec{b}\cdot \nabla u+cu$. 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., $\mathrm{Re}⟨-\mathcal{L}u,u⟩\ge 0$; (2) $\mathcal{L}$ is form bounded, i.e., $|⟨\mathcal{L}u,u⟩|\le C\Vert \nabla u{\Vert }_{L^2(\Omega )}^2$, for all complex-valued $u\in C_0^{\infty }(\Omega )$.

17 pages. arXiv admin note: text overlap with arXiv:1804.10326

References in corpus (2)

• Infinitesimal form boundedness and Trudinger's subordination for the Schrödinger operator
• Global estimates for kernels of Neumann series and Green's functions