DOI: 10.68381/jca33001 ISSN: 0944-6532
A Generalization of the Stampacchia Lemma and Applications
Hongya Gao, Jiaxiang Zhang, Hongyan Ma
We present a generalization of Stampacchia Lemma and give applications to regularity property of weak and entropy solutions of degenerate elliptic equations of the form
\begin{cases} -\text{div} (a(x,u(x)) Du (x)) =f(x), & \text { in } \Omega, \\[1mm] u(x)=0, & \text { on } \partial \Omega, \end{cases}
{
−
div
(
a
(
x
,
u
(
x
)
)
D
u
(
x
)
)
=
f
(
x
)
,
in
Ω
,
u
(
x
)
=
0
,
on
∂
Ω
,
where
\displaystyle\frac {\alpha}{(1+|u|) ^\theta} \le a(x,s)\le \beta
α
(
1
+
∣
u
∣
)
θ
≤
a
(
x
,
s
)
≤
β
with
0<\alpha \le \beta <\infty
0
<
α
≤
β
<
∞
and
0\le \theta <1
0
≤
θ
<
1
. The method to derive regularity seems to be simpler than the classical ones.