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 $k,l\in \{1,\dots ,d\}$. 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.

40 pages

References in corpus (1)

• Flows and invariance for elliptic operators