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Regularity estimates for singular parabolic measure data problems with sharp growth
Authors
Jung-Tae Park
Pilsoo Shin
Publication date
8 April 2020
Publisher
View
on
arXiv
Abstract
We prove global gradient estimates for parabolic
p
p
p
-Laplace type equations with measure data, whose model is
u
t
−
div
(
∣
D
u
∣
p
−
2
D
u
)
=
μ
inÂ
Ω
×
(
0
,
T
)
⊂
R
n
×
R
,
u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = \mu \quad \textrm{in} \ \Omega \times (0,T) \subset \mathbb{R}^n \times \mathbb{R},
u
t
​
−
div
(
∣
D
u
∣
p
−
2
D
u
)
=
μ
in
Â
Ω
×
(
0
,
T
)
⊂
R
n
×
R
,
where
μ
\mu
μ
is a signed Radon measure with finite total mass. We consider the singular case
2
n
n
+
1
<
p
≤
2
−
1
n
+
1
\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}
n
+
1
2
n
​
<
p
≤
2
−
n
+
1
1
​
and give possibly minimal conditions on the nonlinearity and the boundary of
Ω
\Omega
Ω
, which guarantee the regularity results for such measure data problems.Comment: 28 page
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oai:arXiv.org:2004.03889
Last time updated on 12/10/2020