Abstract
AbstractIn this paper we state the weighted Hardy inequality $$\begin{aligned} c\int _{{{\mathbb {R}}}^N}\sum _{i=1}^n \frac{\varphi ^2 }{|x-a_i|^2}\, \mu (x)dx\le \int _{{{\mathbb {R}}}^N} |\nabla \varphi |^2 \, \mu (x)dx +k \int _{\mathbb {R}^N}\varphi ^2 \, \mu (x)dx \end{aligned}$$
c
∫
R
N
∑
i
=
1
n
φ
2
|
x
-
a
i
|
2
μ
(
x
)
d
x
≤
∫
R
N
|
∇
φ
|
2
μ
(
x
)
d
x
+
k
∫
R
N
φ
2
μ
(
x
)
d
x
for any $$ \varphi $$
φ
in a weighted Sobolev spaces, with $$c\in ]0,c_o[$$
c
∈
]
0
,
c
o
[
where $$c_o=c_o(N,\mu )$$
c
o
=
c
o
(
N
,
μ
)
is the optimal constant, $$a_1,\ldots ,a_n\in \mathbb {R}^N$$
a
1
,
…
,
a
n
∈
R
N
, k is a constant depending on $$\mu $$
μ
. We show the relation between c and the closeness to the single pole. To this aim we analyze in detail the difficulties to be overcome to get the inequality.
Funder
Università degli Studi di Salerno
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,General Mathematics