Degenerate elliptic operators in $L_p$-spaces with complex $W^{2,\infty}$-coefficients

Abstract

Let $c_{kl} \in W^{2,\infty}(\mathbb{R}^d, \mathbb{C})$ for all $k,l \in \{1, \ldots, d\}$. We consider the divergence form operator $A = - \sum_{k,l=1}^d \partial_l (c_{kl} \, \partial_k) $in $L_2(\mathbb{R}^d)$ when the coefficient matrix satisfies $(C(x) \, \xi, \xi) \in \Sigma_\theta$ for all $x \in \mathbb{R}^d$ 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 for all $p$ in a suitable interval the contraction semigroup generated by $-A$ extends consistently to a contraction semigroup on $L_p(\mathbb{R}^d)$. For those values of $p$ we present a condition on the coefficients such that the space $C_c^\infty(\mathbb{R}^d)$ of test functions is a core for the generator on $L_p(\mathbb{R}^d)$. We also examine the operator $A$ separately in the more special Hilbert space $L_2(\mathbb{R}^d)$ setting and provide more sufficient conditions such that $C_c^\infty(\mathbb{R}^d)$ is a core.