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Miklós Schweitzer
2020 Miklós Schweitzer
11
11
Part of
2020 Miklós Schweitzer
Problems
(1)
inequality with continuous and smooth vector field
Source: Miklos Schweitzer 2020, Problem 11
12/1/2020
Given a real number
p
>
1
p>1
p
>
1
, a continuous function
h
:
[
0
,
∞
)
→
[
0
,
∞
)
h\colon [0,\infty)\to [0,\infty)
h
:
[
0
,
∞
)
→
[
0
,
∞
)
, and a smooth vector field
Y
:
R
n
→
R
n
Y\colon \mathbb{R}^n \to \mathbb{R}^n
Y
:
R
n
→
R
n
with
d
i
v
Y
=
0
\mathrm{div}~Y=0
div
Y
=
0
, prove the following inequality
∫
R
n
h
(
∣
x
∣
)
∣
x
∣
p
≤
∫
R
n
h
(
∣
x
∣
)
∣
x
+
Y
(
x
)
∣
p
.
\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.
∫
R
n
h
(
∣
x
∣
)
∣
x
∣
p
≤
∫
R
n
h
(
∣
x
∣
)
∣
x
+
Y
(
x
)
∣
p
.
inequalities
vector
real analysis
function