2019/05/20 by V. G. Maz'ya, V. G. Maz’ya, I. E. Verbitsky
Mathematics · #Advanced Harmonic Analysis Research #Differential operator #Linear map #Linear operators #Linear system #Mathematical and Theoretical Analysis #Nonlinear Differential Equations Analysis #Operator (biology) #Order (exchange) #Pseudo-differential operator #Set (abstract data type) #math.AP #msc:35J15 #msc:42B37
paper · pdf · doi:10.1007/s10114-019-8127-9
published as Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 6, 832-852 · 17 pages. arXiv admin note: text overlap with arXiv:1804.10326
openalex created_date 2019/05/16 · openalex publication_date 2019/05/20 · arxiv created 2019/06/05 · arxiv updated 2020/11/10 · openalex updated_date 2026/08/05
Let L be the general second order differential operator with complex-valued distributional coefficients A=(ajk)j, k=1n, b=(bj)j=1n, and c in an open set Ω⊆ ℝn (n ≥ 1), with principal part either in the divergence form, L u= \rm div (A ∇ u) + b ⋅∇ u + c u, or non-divergence form, \mathcal L u= ∑j, k=1n ajk ∂j ∂k u + b ⋅∇ u + c u . We give a survey of the results by the authors which characterize the following two properties of L: (1) -L is accretive, i.e., \rm Re ⟨ -\mathcal L u, u⟩ ≥ 0; (2) \mathcal L is form bounded, i.e., \vert ⟨ \mathcal L u, u ⟩ \vert ≤ C \Vert ∇ u \VertL2(Ω)2, for all complex-valued u ∈ C^∞0(Ω).