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Symmetrization of Functions and the Best Constant of 1-DIM
Sobolev Inequality
Journal of Inequalities and Applications volume 2009, Article number: 874631 (2009)
Abstract
The best constants of Sobolev embedding of
into
for
and
are obtained. A lemma concerning the symmetrization of functions plays an important role in the proof.
1. Introduction
Let be a Sobolev space which consists of the functions whose derivatives up to
vanish at
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ1_HTML.gif)
where denotes
th derivative of
in a distributional sense. The purpose of this paper is to investigate the best constant
of
Sobolev inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ2_HTML.gif)
where and
. The result is as follows.
Theorem 1.1.
The best constant of inequality (1.2) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ4_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ5_HTML.gif)
where satisfies
and
in (1.5) is the unique solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ6_HTML.gif)
satisfying .
For special values of , the solution of (1.6) can be written explicitly, and
is expressed as follows.
Corollary 1.2.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ7_HTML.gif)
The best constants and
were recently obtained by Oshime [1]. This paper gives an alternative proof which simplifies the derivation process of
and
and computes further constant
. To compute these constants, the following lemma with respect to the symmetrization of functions plays an important role.
Lemma 1.3.
Let be an integer satisfying
and let us define the functional
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ8_HTML.gif)
Then, for an arbitrary , there exists an element
which belongs to the following sub-space
of
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ9_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ10_HTML.gif)
Remark 1.4.
The proof of this lemma (see Section 3) does not apply to the case , since the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ11_HTML.gif)
may fail to hold for (see (3.4)–(3.6)). Hence the problem to obtain
for
(essentially) still remains.
The existence of the maximizer of can be seen in the proof of Theorem 1.1, where we construct it concretely, but here we would like to see this briefly though the proof of the following lemma.
Lemma 1.5.
Assume that the assertion of Lemma 1.3 holds, then the maximizer of exists.
Proof.
Let be sufficiently large, and let
and
be as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ12_HTML.gif)
From Lemma 1.3, we see that if the maximizer exists, it is the element of . So, we can restrict the definition domain of
to
. Since
is convex and strongly closed (by Sobolev inequality) in
, it is weakly closed. In addition,
is weakly compact, so
is also weakly compact. Moreover,
is weakly lower-semicontinuous in
, and hence
attains its minimum in
. This proves the lemma.
Finally, we introduce some studies related to the present paper. When (Hilbertian Sobolev space case), the best constants for the embeddings of
into
for various conditions were treated in Richardson [2], Kalyabin [3], and [4–8]; see also references of these literatures. On the other hand, for the case
, few literature seems to be available. In [9], Kametaka, Oshime, Watanabe, Yamagishi, Nagai, and Takemura obtained the best constant of (1.2) when
belongs to a subspace
of
which consists of periodic functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ13_HTML.gif)
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ14_HTML.gif)
where is a Bernoulli polynomial,
and
is an unique solution of the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ15_HTML.gif)
in the interval . Moreover, in [1], Oshime obtained the best constant
and
. Other topics on this subject, especially the best constant of Sobolev inequalities on Riemannian manifolds, are seen in Hebey [10].
2. Proof of Theorem 1.1
First, we prepare the following lemma.
Lemma 2.1.
Let and
be a function satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ16_HTML.gif)
then, it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ17_HTML.gif)
where is an arbitrary constant (later, in Lemma 2.3, one fixes the value of
to satisfy (1.6)).
Proof.
By integration by parts, we obtain the result.
From Lemma 1.3, to obtain the best constant of (1.2), we can restrict the definition domain of the functional to the nonzero element of
. Now, let
, then from Lemma 2.1 and Hölder's inequality, we have for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ18_HTML.gif)
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ19_HTML.gif)
and the equality holds in (2.4) if and only if there exists satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ20_HTML.gif)
To confirm the existence of such , we use the following lemmas.
Lemma 2.2.
Let satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ21_HTML.gif)
then the solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ22_HTML.gif)
exists in .
Proof.
Let us define as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ23_HTML.gif)
Clearly is
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ24_HTML.gif)
Moreover, from the assumption, it holds that .
Lemma 2.3.
The solution of (1.6) uniquely exists.
Proof.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ25_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ26_HTML.gif)
the assertion is proved.
Using Lemma 2.2 and 5, we obtain the following lemma.
Lemma 2.4.
Let be a solution of (1.6) (when
), then the solution of (2.5) belongs to
for
.
Proof.
First, we prove the case and
. For simplicity, let us put
. Note that in these cases
is a continuous function on
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_IEq95_HTML.gif)
In the case ,
is an even function, so integration of
over the interval
vanishes. In addition,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ27_HTML.gif)
holds, so the integration of over the interval
also vanishes. Hence, by Lemma 2.2, the solution
of (2.5) belongs to
. Properties
and
follow from the fact that
is an even function and (2.12). So, we have proven
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_IEq108_HTML.gif)
In the case,
is an odd function, so integrations of
and
over the interval
vanish. Moreover, from (1.6), the integration of
over the interval
also vanishes. Hence, again by Lemma 2.2, we have the solution of (2.5) which belongs to
. The remaining part is the same as case (i).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_IEq117_HTML.gif)
In the case , let us define
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ28_HTML.gif)
Clearly it holds that satisfies (2.5) (a.e.),
,
and
. To see
, let
be an arbitrary element of
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ29_HTML.gif)
we have, in a distributional sense,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ30_HTML.gif)
Therefore, . This proves the case
.
Proof of Theorem 1.1.
From Lemma 1.3, and the argument of this section, especially Lemma 2.4, . This proves Theorem 1.1.
Proof.
In this case, (1.6) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ31_HTML.gif)
So, we can explicitly solve this equation with respect to . Substituting to (1.5), we obtain the result.
3. Proof of Lemma 1.3
Now, all we have to do is to prove Lemma 1.3.
Proof of Lemma 1.3.
To avoid the complexity of notation, in the follwings, we fix . Let
be an arbitrary element of
(
), and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ32_HTML.gif)
Here, we can assume , since if it does not,
can be replaced by
.
-
(i)
There is the case
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ33_HTML.gif)
Let us define as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ34_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ35_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ36_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ37_HTML.gif)
when , (3.4) and (3.5) when
, and (3.4) when
. Further, let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ38_HTML.gif)
Then , since
,
. Moreover, from (3.2), we have
. In addition, clearly
. So, in the case (i), we have proven the lemma.
-
(ii)
There is the case
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ39_HTML.gif)
Let be an element satisfying
, and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ40_HTML.gif)
Further, let be
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ41_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ42_HTML.gif)
and hence we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ43_HTML.gif)
By putting the right-hand side of (3.12) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ44_HTML.gif)
Similarly, let us put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ45_HTML.gif)
and define as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ46_HTML.gif)
So,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ47_HTML.gif)
and hence we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ48_HTML.gif)
By putting the right-hand side of (3.17) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ49_HTML.gif)
Let us put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ50_HTML.gif)
and define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ51_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ52_HTML.gif)
The derivative of is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ53_HTML.gif)
-
(a)
The case
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ54_HTML.gif)
In this case, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ55_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ56_HTML.gif)
from the assumption (3.23), is monotone increasing. So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ57_HTML.gif)
But, from (3.23), it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ58_HTML.gif)
So, if we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ59_HTML.gif)
as case (i), we have and
. In addition,
=
. So, we have proven the case (ii)-(a).
-
(b)
The case
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ60_HTML.gif)
In this case, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ61_HTML.gif)
Moreover
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ62_HTML.gif)
since we have (3.29) and the assumption (3.8) (), respectively. Therefore, there exists
such that
. Let us define the constant
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ63_HTML.gif)
then . Now we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ64_HTML.gif)
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ65_HTML.gif)
Let us define as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F874631/MediaObjects/13660_2009_Article_2024_Equ66_HTML.gif)
Then, again as case (i), we have and by (3.33),
. In addition,
. So, we have proven the case (ii)-(b). This completes the proof.
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Watanabe, K., Kametaka, Y., Nagai, A. et al. Symmetrization of Functions and the Best Constant of 1-DIM Sobolev Inequality.
J Inequal Appl 2009, 874631 (2009). https://doi.org/10.1155/2009/874631
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DOI: https://doi.org/10.1155/2009/874631