Affiliation:
1. Division of Applied Mathematics , Thu Dau Mot University , Thu Dau Mot , Binh Duong Province , Vietnam
Abstract
Abstract
Let
Ω
⊂
ℝ
d
{\Omega\subset\mathbb{R}^{d}}
be open and
c
k
l
∈
L
∞
(
Ω
,
ℂ
)
{c_{kl}\in L_{\infty}(\Omega,\mathbb{C})}
with
Im
c
k
l
=
Im
c
l
k
{\operatorname{Im}c_{kl}=\operatorname{Im}c_{lk}}
for all
k
,
l
∈
{
1
,
…
,
d
}
{k,l\in\{1,\ldots,d\}}
.
Assume that
C
=
(
c
k
l
)
1
≤
k
,
l
≤
d
{C=(c_{kl})_{1\leq k,l\leq d}}
satisfies
(
C
(
x
)
ξ
,
ξ
)
∈
Σ
θ
{(C(x)\xi,\xi)\in\Sigma_{\theta}}
for all
x
∈
Ω
{x\in\Omega}
and
ξ
∈
ℂ
d
{\xi\in\mathbb{C}^{d}}
, where
Σ
θ
{\Sigma_{\theta}}
is the closed sector with vertex 0 and semi-angle θ in the complex plane.
We emphasize that Ω is an arbitrary domain and C need not be symmetric.
We show that C is (degenerate) p-elliptic for all
p
∈
(
1
,
∞
)
{p\in(1,\infty)}
with
|
1
-
2
p
|
<
cos
θ
{|1-\frac{2}{p}|<\cos\theta}
in the sense of Carbonaro and Dragičević.
As a consequence, we obtain the consistent holomorphic extension for the
C
0
{C_{0}}
-semigroup generated by the second-order differential operator in divergence form associated with C.
The core property for this operator is also investigated.
Subject
Applied Mathematics,General Mathematics