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On sectoriality of degenerate elliptic operators
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 3 / August 2021
- Published online by Cambridge University Press:
- 18 August 2021, pp. 689-710
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- Article
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Let
$c_{kl} \in W^{1,\infty }(\Omega , \mathbb{C})$ for all 
$k,l \in \{1, \ldots , d\};$ and 
$\Omega \subset \mathbb{R}^{d}$ be open with uniformly 
$C^{2}$ boundary. We consider the divergence form operator 
$A_p = - \sum \nolimits _{k,l=1}^{d} \partial _l (c_{kl} \partial _k)$ in 
$L_p(\Omega )$ when the coefficient matrix satisfies 
$(C(x) \xi , \xi ) \in \Sigma _\theta$ for all 
$x \in \Omega$ and 
$\xi \in \mathbb{C}^{d}$, where 
$\Sigma _\theta$ be the sector with vertex 0 and semi-angle 
$\theta$ in the complex plane. We show that a sectorial estimate holds for 
$A_p$ for all 
$p$ in a suitable range. We then apply these estimates to prove that the closure of 
$-A_p$ generates a holomorphic semigroup under further assumptions on the coefficients. The contractivity and consistency properties of these holomorphic semigroups are also considered.
