Affiliation:
1. Department of Mathematical Sciences ; and Research Institute of Mathematics , Seoul National University , Seoul 08826 , Korea
2. Department of Mathematical Sciences , Seoul National University , Seoul 08826 , Korea
Abstract
Abstract
We establish a sharp higher integrability near the initial boundary for a weak solution to the following p-Laplacian type system:
{
u
t
-
div
𝒜
(
x
,
t
,
∇
u
)
=
div
|
F
|
p
-
2
F
+
f
in
Ω
T
,
u
=
u
0
on
Ω
×
{
0
}
,
\left\{\begin{aligned} \displaystyle{}u_{t}-\operatorname{div}\mathcal{A}(x,t,%
\nabla u)&\displaystyle=\operatorname{div}\lvert F\rvert^{p-2}F+f&&%
\displaystyle\phantom{}\text{in}\ \Omega_{T},\\
\displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on}\ \Omega%
\times\{0\},\end{aligned}\right.
by proving that, for given
δ
∈
(
0
,
1
)
{\delta\in(0,1)}
, there exists
ε
>
0
{\varepsilon>0}
depending on δ and the structural data such that
|
∇
u
0
|
p
+
ε
∈
L
loc
1
(
Ω
)
and
|
F
|
p
+
ε
,
|
f
|
(
δ
p
(
n
+
2
)
n
)
′
+
ε
∈
L
1
(
0
,
T
;
L
loc
1
(
Ω
)
)
⟹
|
∇
u
|
p
+
ε
∈
L
1
(
0
,
T
;
L
loc
1
(
Ω
)
)
.
\lvert\nabla u_{0}\rvert^{p+\varepsilon}\in L^{1}_{\operatorname{loc}}(\Omega)%
\quad\text{and}\quad\lvert F\rvert^{p+\varepsilon},\lvert f\rvert^{(\frac{%
\delta p(n+2)}{n})^{\prime}+\varepsilon}\in L^{1}(0,T;L^{1}_{\operatorname{loc%
}}(\Omega))\implies\lvert\nabla u\rvert^{p+\varepsilon}\in L^{1}(0,T;L^{1}_{%
\operatorname{loc}}(\Omega)).
Our regularity results complement established higher regularity theories near the initial boundary for such a nonhomogeneous problem with
f
≢
0
{f\not\equiv 0}
and we provide an optimal regularity theory in the literature.
Funder
National Research Foundation of Korea
Subject
Applied Mathematics,General Mathematics
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