Abstract
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.
On p-elliptic divergence form operators and holomorphic semigroups
