Criterion for the L^p-dissipativity of second order differential operators with complex coefficients

2004/12/11 by Alberto Cialdea, Cialdea, Alberto, Vladimir Maz’ya +2 · 3 citations
Engineering · Mathematics · #47B44 (Secondary) #47D03 (Primary) 47D06 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #math.AP #msc:47B44 #msc:47D03 #msc:47D06

paper · pdf · doi:10.48550/arxiv.math/0412225

37 pages, LaTeX, no figures

arxiv created 2004/12/11 · openalex publication_date 2004/12/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the algebraic condition |p-2| |< \mathscr Im\mathscr Aξ,ξ>| ≤ 2 √(p-1) < \mathscr Re\mathscr Aξ,ξ> (for any ξ∈ℝⁿ) is necessary and sufficient for the L^p-dissipativity of the Dirichlet problem for the differential operator ∇^t(\mathscr A∇), where \mathscr A is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the L^p-contractivity of the corresponding semigroup. We consider also the operator ∇^t(\mathscr A∇)+\bf b∇ +a, where the coefficients are smooth and \mathscr Im\mathscr A may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the L^p-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the L^p-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the L^p-dissipativity in ℝⁿ of the operator ∇^t(\mathscr A∇)+\bf b∇ +a with constant coefficients.

Cited by

• Calderón-Zygmund gradient estimates for p-Laplace systems with BMO complex coefficients
• Sharp angle estimates for second order divergence operators
• L^p-quasicontractiveness and Kernel estimates for semigroups generated by systems of elliptic operators

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