On sectoriality of degenerate elliptic operators

Abstract

Let $c_{kl}\in W^{1,\infty }(\Omega ,\mathbb{C})$ for all $k,l\in \{1,\dots ,d\}$ and $\Omega \subset {\mathbb{R}}^d$ be open with Lipschitz boundary. We consider the divergence form operator $A_p=-{\sum }_{k,l=1}^d{\partial }_l(c_{kl}{\partial }_k)$ 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 hold 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.

18 pages