2016/09/09 by Tan Duc, Do, Tan Duc
Computer Science · Mathematics · #35J25 #35K65 #47B44 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1609.02650
openalex publication_date 2016/09/09 · openalex created_date 2016/09/23 · openalex updated_date 2026/07/28
Let ckl ∈ W1,∞(Ω, ℂ) for all k,l ∈ \1, …, d\ and Ω⊂ ℝd be open with Lipschitz boundary. We consider the divergence form operator Ap = - ∑k,l=1d ∂l (ckl ∂k) in Lp(Ω) when the coefficient matrix satisfies (C(x) ξ, ξ) ∈ Σθ for all x ∈ Ω and ξ∈ ℂd, where Σθ be the sector with vertex 0 and semi-angle θ in the complex plane. We show that a sectorial estimate hold for Ap for all p in a suitable range. We then apply these estimates to prove that the closure of -Ap generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.