Sectoriality of degenerate elliptic operators via p-ellipticity

Abstract

Abstract Let Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} be open and c k ⁢ l ∈ L ∞ ⁢ ( Ω , ℂ ) {c_{kl}\in L_{\infty}(\Omega,\mathbb{C})} with Im ⁡ c k ⁢ l = Im ⁡ c l ⁢ k {\operatorname{Im}c_{kl}=\operatorname{Im}c_{lk}} for all k , l ∈ { 1 , … , d } {k,l\in\{1,\ldots,d\}} . Assume that C = ( c k ⁢ l ) 1 ≤ k , l ≤ d {C=(c_{kl})_{1\leq k,l\leq d}} satisfies ( C ⁢ ( x ) ⁢ ξ , ξ ) ∈ Σ θ {(C(x)\xi,\xi)\in\Sigma_{\theta}} for all x ∈ Ω {x\in\Omega} and ξ ∈ ℂ d {\xi\in\mathbb{C}^{d}} , where Σ θ {\Sigma_{\theta}} is the closed sector with vertex 0 and semi-angle θ in the complex plane. We emphasize that Ω is an arbitrary domain and C need not be symmetric. We show that C is (degenerate) p-elliptic for all p ∈ ( 1 , ∞ ) {p\in(1,\infty)} with | 1 - 2 p | < cos ⁡ θ {|1-\frac{2}{p}|<\cos\theta} in the sense of Carbonaro and Dragičević. As a consequence, we obtain the consistent holomorphic extension for the C 0 {C_{0}} -semigroup generated by the second-order differential operator in divergence form associated with C. The core property for this operator is also investigated.