This article was downloaded by:[Library of the Mathematical Seminary] On: 2 August 2008 Access Details: [subscription number 789755697] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Communications in Partial Differential Equations Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713597240 Sobolev Inequalities for Weighted Gradients Roberto Monti a a Department of Pure and Applied Mathematics, University of Padova, Padova, Italy Online Publication Date: 01 October 2006 To cite this Article: Monti, Roberto (2006) 'Sobolev Inequalities for Weighted Gradients', Communications in Partial Differential Equations, 31:10, 1479 — 1504 To link to this article: DOI: 10.1080/03605300500361594 URL: http://dx.doi.org/10.1080/03605300500361594 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Transcript
This article was downloaded by:[Library of the Mathematical Seminary]On: 2 August 2008Access Details: [subscription number 789755697]Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Communications in PartialDifferential EquationsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713597240
Sobolev Inequalities for Weighted GradientsRoberto Monti aa Department of Pure and Applied Mathematics, University of Padova, Padova, Italy
Online Publication Date: 01 October 2006
To cite this Article: Monti, Roberto (2006) 'Sobolev Inequalities for WeightedGradients', Communications in Partial Differential Equations, 31:10, 1479 — 1504
To link to this article: DOI: 10.1080/03605300500361594URL: http://dx.doi.org/10.1080/03605300500361594
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf
This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction,re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expresslyforbidden.
The publisher does not give any warranty express or implied or make any representation that the contents will becomplete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should beindependently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings,demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with orarising out of the use of this material.
Let x ∈ �m, y ∈ �k, and n = m+ k, with m� k ≥ 1, introduce the weight functionw�x� = ��+ 1��x��, where � > 0 is a positive real number, and consider the“homogeneous dimension” Q = m+ k��+ 1�. We study the weighted Sobolevinequality
( ∫�n
�u� 2QQ−2dx dy
)Q−2Q
≤ cm�k��∫�n
{��xu�2 + w2��yu�2}dx dy (1.1)
for functions u in an appropriate Sobolev space. Here, �xu and �yu denote thegradients of u in the variables x and y, respectively. We are interested in symmetry,existence, and uniqueness properties of extremal functions for (1.1), i.e., functionsachieving the best constant cm�k�� > 0. This article continues the research initiated inMonti and Morbidelli (2006), where Morbidelli and the author studied symmetryand uniqueness properties of non-negative, entire solutions to the critical semilinear
Received April 7, 2005; Accepted June 15, 2005Address correspondence to Roberto Monti, Department of Pure and Applied
Mathematics, University of Padova, via Belzoni 7, Padova I-35131, Italy; E-mail: [email protected]
1479
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1480 Monti
equation
�xu+ w2�yu = −uQ+2Q−2 in �n� (1.2)
This is the Euler equation associated with (1.1). When � = 0, we have the Yamabeequation and positive, entire solutions are extremal functions for the Sobolevinequality (1.1) with Q = n and w = 1. For � > 0, the inequality is known asSobolev inequality for the Grushin operator and it can be proven by generaltechniques in doubling metric spaces with Poincaré inequality. A general referenceto the topic is Hajłasz and Koskela (2000) (see also Franchi et al., 1994 andGarofalo and Nhieu, 1996).
For � = 0, extremal functions are radially symmetric and decreasing about somepoint in �n. This is no longer true for � > 0, because inequality (1.1) is not invariantfor the full group of Euclidean translations and orthogonal transformations. Let usintroduce the following notion of symmetry. We say that a non-negative functionu � �n → 0�+�� belongs to the class � if:
i) there is a function v � �m → 0�+�� such that
u�x� y� = �x� 2−Q2 v(�1− �x��+1�2 + �y�2
4�x��+1
x
�x�)� x �= 0 (1.3)
ii) the function t �→ tQ−22 u�t�� 0� is nondecreasing on 0� 1� for all � ∈ �m−1, i.e.,
�∈�m and ��� = 1.
In Monti and Morbidelli (2006), any positive function u ∈ C2��n� solving equation(1.2) is proved to be of the form (1.3), up to translation in the variable y andscaling u �x� y� = Q−2
2 u� x� �+1y� for some positive number . The proof relies onthe Kelvin transform for Grushin operators and on the moving spheres method.In this work, we take a different point of view and we show by a rearrangementargument that the family of competing functions for inequality (1.1) with sharpconstant can be reduced to the class � .
Let D1��n� be the Sobolev space obtained as completion of C�0 ��
n� withrespect to the norm
uD1��n� =( ∫
�n
{��xu�2 + w2��yu�2}dx dy
)1/2
�
The space D1��n� is the natural framework for inequality (1.1). It can beequivalently defined as the space of functions u ∈ L 2Q
Q−2 ��n� with weak derivativessatisfying uD1��n� < +�. Any function in C�
0 ��n� can be approximated uniformly
and in D1��n� by compactly supported, piecewise affine functions u satisfying theadditional condition
��yu� �= 0 on the support of u� (1.4)
We denote by � the family of all such functions. The set � is dense in D1��n� andit is sufficient to work in this class.
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1481
Theorem 1. Let m� k ∈ �, � ≥ 0, and Q > 2. For any non-negative function u ∈ �,there exists a non-negative function u∗ ∈ D1��n� in the class � such that
∫�n
{��xu∗�2 + w2��yu∗�2}dx dy ≤
∫�n
{��xu�2 + w2��yu�2�dx dy� (1.5)
Moreover, ∫�nu∗
2QQ−2dx dy =
∫�nu
2QQ−2dx dy� (1.6)
The proof of Theorem 1 relies on a rearrangement argument in the hyperbolicspace. We decompose �n as �m−1 ×�k+1, where �m−1 is the unit Euclidean spherein �m and �k+1 is the �k+ 1�-dimensional hyperbolic space. After a suitablefunctional change, functions are rearranged in each slice �k+1 with respect tohyperbolic volume and distance. The rearranged function is radially symmetric ineach slice �k+1 about some fixed origin. Inverting the functional change, we get arearranged function in the class � . The procedure is introduced in Section 2, andTheorem 1 is proven in Section 3. Beckner (2001) recognized the connection betweenGrushin operators and hyperbolic symmetry: In the case of the plane (m = k = 1and also � = 1) he was able to determine extremal functions, controlling inequality(1.1) by the sharp Sobolev inequality on the sphere.
As functions in the class � are determined by their values in the “disk” ��x� 0� ∈�m ×�k � �x� ≤ 1�, inequality (1.1) is related to a Sobolev–Poincaré inequality inthe unit ball B = �x ∈ �m �x� < 1�. Let us introduce the weights
Both p and q are radial and non-negative on B. We omit reference to the relevantparameters m� k� �.
Theorem 2. Let m� k ∈ �, � ≥ 0, Q > 2, and p� q as in (1.7). Any non-negativefunction v ∈ C1��B� satisfies the inequality
( ∫Bv
2QQ−2 p dx
)Q−2Q
≤ c′m�k��∫B
{��xv�2p+ v2q}dx (1.8)
with the sharp constant c′m�k�� = ��k2−k�2Q cm�k��. Here, �k is the area of the unit sphere
in �k+1 and cm�k�� is the sharp constant in (1.1).
The vanishing of p and q at the boundary �B is, even for � = 0, the maindifference between inequality (1.8) and the standard Sobolev–Poincaré inequalityin the ball. Its proof relies on the fact that a function in C1��B� can be extendedto a function in �n of the form (1.3) satisfying the Sobolev inequality (1.1).The reduction procedure from �n to B, which produces the weights p and q, isperformed in Theorem 3.2.
The Euler equation for (1.8) is
divx�p�xv�− qv = −pvQ+2Q−2 � �x� < 1�
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1482 Monti
This equation, which can also be obtained from (1.2) assuming u of the form (1.3),was studied in Monti and Morbidelli (2006). We believe that the positive solution vof this equation with mixed boundary condition
�v
��+
(Q2− 1
)v = 0� �x� = 1�
is unique. Equivalently, we believe that extremal functions for inequality (1.8) areradial in the variable x. We are able to prove this conjecture for m = 1. In this case,the inequality is one-dimensional
( ∫ +1
−1v
2QQ−2 p dx
) Q−2Q
≤ c′1�k��∫ +1
−1
{v′2p+ v2q}dx� (1.9)
where now Q = 1+ k��+ 1�. We prove that (1.9) is optimized by even functions andwe establish an existence result of extremals. As a corollary we have the followingtheorem.
Theorem 3. Let m = 1, k ∈ �, and � > 0. There exist extremal functions in D1��n�for inequality (1.1), they are even in x and belong to the class � (up to translation iny and scaling).
Existence of extremals can be proven elementarly, because the loss ofcompactness for (1.1) is removed in the class � (see Theorem 4.2). Thesymmetry property u�−x� y� = u�x� y� is the main point. The proof is based ona rearrangement argument for (1.9) combined with a uniqueness result for thecorresponding Euler equation, which is proven in Monti and Morbidelli (2006)(see Theorem 4.5). The premise for Theorem 3 is a complicate functional changediscussed at the beginning of Section 4 (see Proposition 4.1).
In the plane m = k = 1, we have uniqueness of extremals. In this case, a positiveextremal of (1.9) solves the problem (see Lemma 4.4)
(pv′
)′ − qv = −pvQ+2Q−2 in �−1� 1�
2v′�x�+ �Q− 2�xv�x� = 0 x = ±1�(1.10)
Here Q = �+ 2. By Theorem 6 in Monti and Morbidelli (2006), the positive solutionto problem (1.10) is unique. In the case � = 1, the solution can be expressed in closedform and it is
v�x� = 1�1+ x2�1/2 � (1.11)
Then we have the following theorem.
Theorem 4. Let m = k = 1 and � > 0. Extremal functions for inequality (1.1) areunique, up to translation in y, scaling and multiplication by a nonzero constant. Inparticular, for � = 1 the extremal is
u�x� y� = 1��1+ x2�2 + y2�1/4 � (1.12)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1483
Beckner’s function (1.12) is obtained from (1.11) and (1.3). In the case of the plane,the sharp Sobolev inequality for the L1 norm of the Grushin gradient was establishedin Monti and Morbidelli (2004) as a corollary of a sharp isoperimetric inequality.Grushin isoperimetric sets and level sets of extremal functions (1.12) are different.
If radiality in x is assumed, existence, symmetry, and uniqueness results canbe obtained also for m > 1. Precisely, let � be the family of functions u � �m ×�k → 0�+�� such that u�x� y� = v��x�� y� for some v � 0�+��×�k → 0�+��and define c�m�k�� > 0 by
1c�m�k��
= inf{ ∫
�n
{��xu�2 + w2��yu�2}dx dy � u ∈ � ∩D1��n��
∫�n
�u� 2QQ−2dx dy = 1
}�
By Theorem 1 and Lions’s concentration-compactness principle (Lions, 1985), theconstant c�m�k�� > 0 is achieved by functions belonging to the class� (up to translationin y and scaling). In the case � = 1, this result is linked to the sharp Sobolev inequalitiesin groups of Heisenberg type proven by Garofalo and Vassilev (2001).
A short overview of the article is in order. In Section 2, we introduce the mainnotion of rearrangement and we describe its basic properties. In Section 3, we proveTheorems 1 and 2. Section 4 is devoted to the proof of Theorem 3.
2. Hyperbolic Type Rearrangement
Let u � �n → 0�+�� be a non-negative function and associate with it the newfunction U � �n → 0+�� of the variables � ∈ �m and � ∈ �k
U��� �� = ��� Q−22��+1� u
(���− ��+1 �� �
)� (2.1)
If u is of the form (1.3) and U is given by (2.1), then
U��� �� = �(�1− ����2 + ���2
4����
���)� (2.2)
For any fixed direction �/��� ∈ �m−1, a function U of the form (2.2) is radiallysymmetric for the hyperbolic metric in the �k+ 1�-dimensional halfspace “spanned”by �/��� and �.
Let �k+1 = �+ ×�k be the �k+ 1�-dimensional hyperbolic space with metricr−2�dr2 + �d��2�, r > 0, and � ∈ �k. The Riemannian volume � and the Riemannianhypersurface measure � are respectively
� = 1rk+1
�k+1 and � = 1rkk�
where �k+1 is the Lebesgue measure in �k+1 and k is the k-dimensional Hausdorffmeasure in �k+1. If A ⊂ �k+1 is an open set and B is a metric ball in �k+1, thenthe hyperbolic isoperimetric inequality states that
��B� = ��A�⇒ ���B� ≤ ���A�� (2.3)
with equality only if A is a ball (see Burago and Zalgaller, 1988, §10.2).
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1484 Monti
A ball Bs centered at �1� 0� ∈ �k+1 having radius s = log( 1+�1−�
), � ∈ �0� 1�, is the
set of points �r� �� ∈ �k+1 satisfying
�1− r�2 + ���24r
<�2
1− �2 � (2.4)
The measure of Bs is
��Bs� = �k2k+1∫ �
0
rk
�1− r2�k+1dr� (2.5)
where �k is the surface measure of the unit Euclidean sphere in �k+1.Formulae (2.4) and (2.5) can be easily proven in the ball-model of the hyperbolic
space. Endow k+1 = �z ∈ �k+1 � �z� < 1� with the metric 4�1− �z�2�−2�dz�2 and withthe Riemannian volume 2k+1�1− �z�2�−k−1dz. The Möbius map S ��k+1 → k+1
S�r� �� =(
1− r2 − ���2�1+ r�2 + ���2 �
−2��1+ r�2 + ���2
)� (2.6)
is an isometry and it maps the ball (2.4) onto the ball ��z� < ��. Formula (2.5) easilyfollows.
Now we construct a function u∗ ∈ � starting from a non-negative function u ∈C0��
n�. Let U be the function defined in (2.1) and for any � ∈ �m−1 let U��r� �� =U�r�� ��. Analogously, for any t ≥ 0 we have the superlevel sets �t = �U > t� ⊂ �n
and ���t = �U� > t� ⊂ �k+1. For fixed �, the distribution function ���t� = �����t�is decreasing in t and ���t� = ���t + 0�. Its “inverse” �� � 0� �����0��→ 0�+�� isdefined by ���s� = inf�t > 0 � ���t� ≤ s�. The rearranged function U ∗
� is by definitionradially symmetric about the point �1� 0� ∈ �k+1 and with level sets �∗
��t satisfyingthe condition
���∗��t� = �����t�� t > 0� (2.7)
Precisely, we let
U ∗� �r� �� = ��
(��Bd�r����
)� (2.8)
where
d�r� �� = log1+ �1− �� with �2 = �1− r�2 + ���2
�1+ r�2 + ���2 � (2.9)
is the hyperbolic distance of �r� �� from �1� 0�. By Cavalieri principle, for any realnumber p > 0 condition (2.7), implies the identity
∫�k+1
U ∗�pd� =
∫ +�
0���∗
��t1/p �dt =∫ +�
0�����t1/p �dt =
∫�k+1
U�pd�� (2.10)
Letting U ∗�r�� �� = U ∗� �r� ��, we can finally define u∗ by
U ∗��� �� = ��� Q−22��+1� u∗
(���− ��+1 �� �
)� (2.11)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1485
as in (2.1). We call u∗ the rearrangement of u. The function u∗ is always in the class� , but it does not necessarily have compact support even if u itself has compactsupport.
Now we discuss the properties of u∗, assuming u in �, the family of piecewiseaffine functions with compact support satisfying (1.4). In this case, the level sets����t = �U� = t� are negligible and ��� t� �→ ���t� is therefore continuous. Moreover,t �→ ���t� is strictly decreasing. We shall equivalently write ���� t� = ���t� = �t���,according to the relevant variable.
Throughout the article, we use the following notation. ��U and ��U denotethe gradients of a function U in �n in the variables � and �, respectively, �RU =���U� �/���� is the radial part of ��U and �TU = ��U − �RU�/��� is the tangentialpart of ��U . We also denote by �HU = �����RU� ��U� the hyperbolic componentof �U . Finally, if W � �m−1 → � is differentiable at � ∈ �m−1 and v ∈ T��m−1 is atangent vector, we define the tangential derivative in the direction v
�vW��� = lim�→0
W������−W����
�
where � � �−�� ��→ �m−1 is a smooth curve such that ��0� = � and ��0� = v.The next propositions describe some properties of the rearrangement u∗ and of
the distribution function �����t�. The proof is a standard application of coarea andarea formulas and it is sketched in the Appendix at the end of the article.
Proposition 2.1. Let u ∈ � be non-negative, U as in (2.1) and ���t = �U� > t�. Thefunction ���� t� = �����t� is differentiable at almost every ��� t� ∈ �m−1 ×�+ andmoreover
����t�
�t= −
∫����t
1��HU��
d�� (2.12)
and, for v ∈ T��m−1 (with m ≥ 2),
�v�t��� =∫����t
�vU�
��HU��d�� (2.13)
where �vU� denotes the tangential derivative of U� in the variable � and direction v.
Proposition 2.2. If u ∈ � is non-negative, then the rearrangement u∗ is continuous in�n minus the set �x= 0� and locally Lipschitz continuous in �n minus the set �x= 0� ∪��x� = 1� y = 0�.
In the next proposition we show that the function u∗ is actually continuouson the whole �n. The proof also provides the profile of extremals for (1.1) on thesubspace x = 0.
Proposition 2.3. If u ∈ � is a non-negative function, then u∗ can be continuouslyextended to �n. Moreover,
u∗�0� y� = C�m� k� ��( ∫
�0
s−k��+1�−1ds dy
) Q−22k��+1�
�1+ �y�2� 2−Q2��+1� � (2.14)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1486 Monti
where �0 = ��s� y� ∈ �+ ×�k � sQ−22 u�0� y� > 1� and C�m� k� �� > 0 depends on
m� k� �.
Proof. We claim that for any � > 0 there is a � > 0 such that for �x� ≤ � and y ∈ �k
we have �u∗�x� y�− u∗�0� y�� ≤ � with u∗�0� y� given by (2.14). If ��x��x� y� ∈ ��∗t =
��yu�2 = �x�2−Q���U �2�Here, �xu and �yu are evaluated at �x� y�, while U , �RU , �TU , and ��U are evaluatedat ��� �� = ��x��x� y�.
For any t > 0 let �t = �U > t� and �t = ��t�, where ��� �� = ����− �
�+1 �� ��� (3.2)
Using (3.1), we get
Et � =∫�t
���xu�2 + w2��yu�2�dx dy
=∫�t
{��+ 1�2�x�2��+1�
(��RU �2 + ���U �2)+ �x�2��+1���TU �2
+ ��+ 1��2−Q��x��+1U�RU +(1− Q
2
)2
U 2
}dx dy
�x�Q
=∫�t
{���HU �2 +
���2�
��TU �2 +2����kU�RU + !U 2
}d�d�
���n � (3.3)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1488 Monti
where we set
! = �2−Q�24��+ 1�
� � = k
2�2−Q�� � = �+ 1� (3.4)
We got the last equality in (3.3) by the change of variable �x� y� = ��� ��, whichhas Jacobian determinant �det J � = �a+ 1�−1���− m�
�+1 .Let U��r� �� = U�r�� ��, with � ∈ �m−1 and r > 0, and ���t = �U� > t� ⊂ �k+1.
By Fubini–Tonelli theorem, we split the last integral in (3.3) into a double integralin �k and �m. Then, introducing polar coordinates in �m, we obtain
Et =∫�m−1
∫���t
{���HU��2 +
r2
���TU��2 +
2�krU��RU� + !U 2
�
}dr d�
rk+1dm−1� (3.5)
where �TU��r� �� = �TU�r�� �� and �RU��r� �� = �RU�r�� �� = ��rU��r� ��. Note in the
inner integral the hyperbolic measure d� = dr d�
rk+1 .We have to estimate four terms. Let U ∗
� be the rearrangement of U� as in (2.8)and let �∗
��t = �U ∗� > t�. We claim that for all � ∈ �m−1 and for all t > 0, we have
∫�∗��t
U ∗�2d� =
∫���t
U�2d�� (3.6)
∫�∗��t
rU ∗�
�U ∗�
�rd� =
∫���t
rU��U�
�rd�� (3.7)
∫�∗��t
r2��TU ∗� �2d� ≤
∫���t
r2��TU��2d�� (3.8)
∫�∗��t
��HU ∗� �2d� ≤
∫���t
��HU��2d�� (3.9)
Relations (3.6)–(3.9) imply
Et ≥∫�∗t
{��xu∗�2 + w2��yu∗�2}dx dy�
where u∗ is the function defined in (2.11) and �∗t = ��∗
t �. Our claim (1.5) thenfollows by monotone convergence letting t→ 0.
The proof of identity (3.6) is as in line (2.10). We prove (3.7). This is the part ofthe proof where we need to work with superlevel sets �t with t > 0. For any � ∈ �k
let �����t = �r > 0 � �r� �� ∈ ���t�. Then∫���t
rU��U�
�rd� = 1
2
∫���t
�
�r�U 2
� �dr d�
rk= 1
2
∫�k
∫�����t
�
�r�U 2
� �dr
rkd��
The open set �����t ⊂ �+ is an at most countable disjoint union of open intervals
�����t =+�⋃h=1
�ah� bh�� 0 < ah ≤ bh ≤ ah+1�
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1489
(The numbers ah� bh depend on �� �� t.) Then, integrating by parts, we obtain
∫�����t
�
�r�U 2
� �dr
rk=
+�∑h=1
∫ bh
ah
�
�r�U 2
� �dr
rk
=+�∑h=1
(U��bh� ��
2
bkh− U��ah� ��
2
akh+ k
∫ bh
ah
U 2�
dr
rk+1
)
= k∫�����t
U 2�
dr
rk+1+ t2
+�∑h=1
(1bkh
− 1akh
)
= k∫�����t
�U 2� − t2�
dr
rk+1�
Hence, integrating in � and using (2.7) and (3.6), we get
∫���t
�
�r
(U 2�
)d� = k
∫���t
(U 2� − t2
)d� = k
∫�∗��t
(U ∗�2 − t2)d� =
∫�∗��t
�
�r
(U ∗�2)d��
(3.10)
This concludes the proof of (3.7).We prove (3.8) (we need this part of the argument only if m ≥ 2). First of all,
notice that Proposition 2.1 applies also to U ∗� . From (2.12), we have for a.e. t > 0
∫����t
1��HU��
d� =∫��∗
��t
1��HU ∗
� �d�� (3.11)
From (2.13), we have for a.e. � ∈ �m−1 and for every tangential direction v ∈ T��m−1
∫����t
�vU�
��HU��d� =
∫��∗
��t
�vU∗�
��HU ∗� �d�� (3.12)
By the coarea formula, as in (A.1), we have
∫���t
r2��TU��2d� =∫ +�
t
∫����s
r2��TU��2��HU��
d� ds� (3.13)
Let v1� � � � � vm−1 ∈ T��m−1 be an orthonormal basis. Then
r2��TU��2 =m−1∑i=1
��viU��2� (3.14)
where �viU� is the tangential derivative of � �→ U� in the direction vi.
By Hölder inequality, (3.11) and (3.12) we have
∫����s
��viU��2
��HU��d� ≥
( ∫����s
�viU�
��HU��d�
)2( ∫����s
1��HU��
d�
)−1
=( ∫
��∗��s
�viU ∗�
��HU ∗� �d�
)2( ∫��∗
��s
1��HU ∗
� �d�
)−1
= ��viU ∗� �2
��HU ∗� �����∗
��s��
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1490 Monti
The last equality follows from the fact that �viU ∗� and ��HU ∗
� � are constant on��∗
��s. Indeed, the length of the gradient of a radial function in �k+1 is still radial.Precisely, let V ��k+1 → � be a function of the form V�r� �� = ��d�r� ��� where dstands for the hyperbolic distance from some fixed point and � is a differentiablefunction. Then
because ��Hd�r� ��� = 1, and thus ��HV�r� ��� is radial.Now (3.8) is a consequence of (3.13), (3.14), and
∫��∗
��s
��viU ∗� �2
��HU ∗� �d� ≤
∫����s
��viU��2
��HU��d��
In order to prove (3.9), note that by the coarea formula, analogously to (A.1),we have ∫
���t
��HU��2 =∫ +�
t
∫����s
��HU��d� ds� (3.16)
and by Hölder inequality
������s� ≤( ∫
����s
��HU��d�)1/2( ∫
����s
1��HU��
d�
)1/2
� (3.17)
For ���∗��s� = �����s�, the isoperimetric inequality (2.3) implies ����∗
��s� ≤ ������s�.Thus, from (3.17) and (3.11) it follows that
∫����s
��HU��d� ≥ ����∗��s�
2
( ∫��∗
��s
1��HU ∗
� �d�
)−1
=∫��∗
��s
��HU ∗� �d��
In the last equality we used again the fact that, by (3.15), ��HU ∗� � is constant on
��∗��s. Now (3.9) follows from (3.16) and the proof of (1.5) is concluded.The proof of (1.6) follows from
∫�nu
2QQ−2dx dy = 1
�+ 1
∫�m−1
∫�k+1
U2QQ−2
� d� dm−1�
and from (2.10). �
We recall that B = �x ∈ �m � �x� < 1�. By Theorem 3.1, competing functions inthe class � optimize inequality (1.1). Theorem 2 is a consequence of (1.1) and of thefollowing theorem.
Theorem 3.2. Let m� k ∈ �, � ≥ 0, Q = m+ k��+ 1� > 2. For any non-negativefunction v∈C1�B�, the function u of the form (1.3) and such that u�x� 0�= v�x�, �x� ≤ 1,satisfies the identities∫
�n
{��xu�2 + w2��yu�2}dx dy = �k
2k
∫B
{��xv�2p+ v2q}dx� (3.18)∫�nu
2QQ−2dx dy = �k
2k
∫Bv
2QQ−2 p dx� (3.19)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1491
Here, p and q are the weights (1.7), and �k stands for the surface measure of the unitEuclidean sphere in �k+1.
Proof. Let U be the function associated with u as in (2.1), U��r� �� = U�r�� ��, �t =�U > t�, ���t = �U� > t� ⊂ �k+1, �t = ��t�, with t > 0 and � ∈ �m−1. Here, isthe mapping introduced in (3.2). In order to prove (3.18), we begin with the identity
Et �=∫�t
{��xu�2 + w2��yu�2}dx dy =
∫�m−1
E��tdm−1� (3.20)
where
E��t =∫���t
{���HU��2 +
r2
���TU��2 +
2�krU��RU� + !U 2
�
}d��
Identity (3.20) is proven in Theorem 3.1, formulae (3.1)–(3.5). The constants !� �� �are defined in (3.4). Using the identity
∫���t
rU��RU�d� = k
2
∫���t
�U 2� − t2�d��
which is proven in (3.10), we also have
E��t =∫���t
{���HU��2 +
r2
���TU��2 + �! + ��U 2
� − �t2}d�� (3.21)
The function U� is radial in the hyperbolic metric about the point �1� 0� ∈�k+1. In order to exploit this symmetry, it is useful to work with the ball modelof the hyperbolic space. Let k+1 = �z ∈ �k+1 � �z� < 1� and let S ��k+1 → k+1
be the isometry defined in (2.6). Notice that S−1 = S and then in the coordinatesz= �z0� z′� ∈ �+ ×�k the components �S1� S2� of S
−1 are
S1�z� =1− �z�2
�1+ z0�2 + �z′�2 � S2�z� =−2z′
�1+ z0�2 + �z′�2 �
We need the function
��r� = 1− r1+ r � with �′�r� = − 2
�1+ r�2 � r �= −1� (3.22)
Note that S�r� 0� = ���r�� 0�.The function V� �
k+1 → �, V��z� = U��S�z��, is radial in the variable z.Moreover, the hyperbolic gradient transforms as
��HU��S�z���2 =�1− �z�2�2
4��zV��z��2� (3.23)
We transform the tangential component r2��TU��2. Fix a point �r� �� ∈ �k+1 and letr0 ∈ �0� 1� be the solution of the equation
�1− r0�24r0
= �1− r�2 + ���24r
� (3.24)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1492 Monti
For U� is radial about �1� 0�, we have r2��TU��r� ���2 = r20 ��TU��r0� 0��2. Lettingz= S�r� ��, by (3.24) we have �z� = ��r0�, which is equivalent to r0 = ���z��. Then wefind U��r0� 0� = V��S�r0� 0�� = V����r0�� 0�, and hence
The Möbius map S transforms the measure � into the measure 2k+1dz�1−�z�2�k+1 and the
set ���t onto the set Z��t = �V� > t�. Taking into account (3.23) and (3.25), identity(3.21) becomes
E��t =∫Z��t
{��1− �z�2�2
4��zV��2 +
���z��2�
��TV��2 + �! + ��V 2� − �t2
}2k+1dz
�1− �z�2�k+1�
(3.26)
From V��z� = V���z�� 0� = U��S��z�� 0�� = U�����z��� 0�� and (3.22), we find
�1− �z�2�24
��zV��z��2 = ���z��2��RU�����z��� 0��2�
Thus, by polar coordinates we obtain from (3.26)
E��t = 2k+1∫ r��t
0
{��r�2
(���RU��2 +
1���TU��2
)+ �! + ��U 2
� − �t2}
�krk
�1− r2�k+1dr�
where r��t ∈ �0� 1� is a number such that U����r��t�� 0� = t and functions insidethe integral are evaluated at ���r�� 0�. Now the change of variable � = ��r� withd�= �′�r�dr furnishes
E��t =�k2k
∫ 1
���t
{�2(���RU��2 +
1���TU��2
)+ �! + ��U 2
� − �t2}�1− �2�k�k+1
d�� (3.27)
where ���t = ��r��t� and functions inside the integral are evaluated at ��� 0�.In the next step we write (3.27) for the function v. By (3.1) with v�x� = u�x� 0�
(recall also the constants in (3.4)) we have
1��
Q�+1 ��xv
(�
1�+1 �
)�2 = �2(���RU��2 + 1���TU��2
)+ !U 2
� +2�k�U��RU�� (3.28)
where functions in the right-hand side are evaluated at ��� 0�. Identity (3.27) thenbecomes
E��t =�k2k
∫ 1
���t
{1��
Q�+1 ��xv�� 1
�+1 ���2 + �(U 2� − t2
)− 2�k�U��RU�
}�1− �2�k�k+1
d�� (3.29)
with U� and �RU� evaluated at ��� 0�. Let us consider
−2�k
∫ 1
���t
U��RU��1− �2�k�k
d� = − �k
∫ 1
���t
�R�U2� − t2�
�1− �2�k�k
d�
= �
k
∫ 1
���t
�U 2� − t2�f���d�� (3.30)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1493
with
f��� = d
d�
�1− �2�k�k
= −2k�1− �2�k−1
�k−1− k�1− �
2�k
�k+1= f1���+ f2���� (3.31)
Integration by parts in (3.30) produces no boundary contribution becauseU�����t� 0� = t. The term ��U 2
� − t2� in (3.29) is cancelled by the part of (3.30)corresponding to f2 in (3.31).
We are therefore left with
E��t =�k2k
∫ 1
���t
{1���xv
(�
1�+1 �
)�2� m�+1−1�1− �2�k − 2�v
(�
1�+1 �
)2�m−2�+1 +1�1− �2�k−1
}d�
+ ��k2k−1
R��� t�� (3.32)
where
R��� t� = t2∫ 1
���t
�1−k�1− �2�k−1d��
We claim that R��� t� converges to zero as t→ 0, uniformly in � ∈ �m−1. ForU�����t� 0� = t and U���� 0� = �
Q−22��+1� u��
1�+1 �� 0�, by (2.1), the claim follows from the
boundedness of u and from
lims→0
sQ−2�+1
∫ 1
s�1−k�1− �2�k−1d� = 0�
Letting t tend to zero in (3.32) and performing the change of variable � = r�+1
we get (recall (3.4))
E��0 �= limt→0E��t =
�k2k
∫ 1
0
{��xv�r���2p�r�+ v�r��2q�r�}rm−1dr�
Here, consistently with (1.7), we let p�r� = �1− r2��+1��k and q�r� = �Q−m��Q− 2��1− r2��+1��k−1r2�. Integrating E��0 over �m−1 and using the coarea formula (polarcoordinates), we complete the proof of (3.18).
The proof of (3.19) is much easier and it is left to the reader. �
4. Existence and Even Symmetry of Extremals
In this section we prove Theorem 3. First, we transform inequality (1.8) intoa Sobolev inequality of the Riemannian type. Then, in Theorem 4.2, we proveexistence of extremals for inequality (1.9) and hence for inequality (1.1), in the casem = 1. Finally, in Theorem 4.5, we show that extremals are even in the variablex. Note that outside the class � there can be no extremals for (1.1), because, byTheorem 4 in Monti and Morbidelli (2006), any positive solution to equation (1.2)must be in � , up to scaling and translation in y. The important technical point inthis section is Proposition 4.1, which provides the functional transformation suitableto proving existence and symmetry of extremals. The coefficient functions and theconstants appearing in this proposition will play a fundamental role.
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1494 Monti
We introduce in M = �� ∈ �m � ��� < "� a Riemannian structure with “axialsymmetry”. The number " > 1 will be fixed later. For v ∈ C1�M� let
��Mv�2 = ��Rv�2 +1#2
��Tv�2� (4.1)
where �Rv and �Tv are the radial and tangential components of the Euclideangradient of v and # is a positive function. Let � be the measure on M
� = #m−1�m� (4.2)
where �m is Lebesgue measure in M .Now consider the unit ball B = �x ∈ �m � �x� < 1�. Given u ∈ C1�B�, define the
function v ∈ C1�M� by letting
u�x� = ���x�� v(���x�� x�x�
)� (4.3)
where ��� � 0� 1�→ 0�+�� are the functions
��t� = �1− t2��+1��−Q−2
2��+1� and ��t� =∫ t
0�1− s2��+1��−
1�+1ds� (4.4)
The function � can be continuously extended to 0� 1� and it is strictly increasing onthis interval. Now we fix " �= ��1� > 1 and we fix the function # by letting
h = �x��′
�and #��� = h��−1������� ��� < "� (4.5)
We are ready to transform the left- and right-hand side of inequality (1.8). Wedraw the reader’s attention to the function G and to the constant $ in line (4.7)below. The qualitative behaviour of G and the sign of $ will play an important role.
Proposition 4.1. Let m� k ≥ 1 and � > 0. Given a non-negative function u ∈ C1�B�, letv be defined as in (4.3). Then we have the identities∫
B
{��xu�2p+ u2q}dx = �k ∫�Bu2dm−1 +
∫M
{��Mv�2 + Fv2}d�� (4.6)
where �k = Q− 2 if k = 1 and �k = 0 otherwise, and F��� = G��−1������ with
G��x�� = $�x�2��1− �x�2��+1��
2��+1
� $ = �Q− 2��k��+ 1�− 2�−m� (4.7)
and ∫Bu
2QQ−2 pdx =
∫Mv
2QQ−2d�� (4.8)
Proof. If u and v are related as in (4.3), then by a short computation we find
��xu�2 = �2�′2��Rv�2 +1�x�2�
2�2��Tv�2 + 2��′�′v�Rv+ �′2v2� (4.9)
with v� �Rv� �Tv evaluated at ���x��x/�x�.
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1495
We explain how �, �, and h can be determined. In order that the first two termsin the right-hand sides of (4.9) have the form (4.1), we require
1�x�2�
2�2 = 1h2�2�′2� (4.10)
Expression (4.9) is multiplied by p, added to u2q and finally integrated over the unitball B. In the integral, we have to perform the change of variable � = ���x��x/�x� =��x�, which has the Jacobian determinant
�det J�� = �m−1�′
�x�m−1� (4.11)
In order to get the measure (4.2) after this change of variable, we also require
�2�′2p�x�m−1
�m−1�′ = hm−1� (4.12)
Analogously, the condition producing the measure (4.2) after the change of variable� = ��x� in (4.8) is
�2∗p�x�m−1
�m−1�′ = hm−1� (4.13)
Here and in the following, we let 2∗ = 2QQ−2 .
Now, (4.10), (4.12), and (4.13) determine �, �, and h. Indeed, comparing (4.12)and (4.13) we find
� = �′ 22∗−2 � (4.14)
From (4.10) we get
h = �x��′
�� (4.15)
Note that, with such an h, the Jacobian determinant (4.11) is
�det J�� = �′mh1−m� (4.16)
Replacing (4.15) and (4.14) into (4.12), we determine
�′��x�� = p−�2−m+4/�2∗−2��−1 = p− 1k��+1� = (
1− �x�2��+1�)− 1
�+1 � (4.17)
With ��0� = 0, � is also determined. Note that
" = ��1� =∫ 1
0
(1− t2��+1�
)− 1�+1dt (4.18)
is finite for any � > 0. From (4.14), we have
���x�� = p− Q−22k��+1� = (
1− �x�2��+1�)− Q−2
2��+1� � (4.19)
Finally, h is determined by (4.15). It is not difficult to check that h�0� = 1.
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1496 Monti
By (4.9), (4.10), (4.12), and (4.15) we obtain
E � =∫B
{��xu�2p+ u2q}dx=
∫B
{(��Rv�2 + h−2��Tv�2)�′m + p��′�R�v
2�+ v2(�′2p+ �2q)}dx� (4.20)
where �R�v2� = 2�′v�Rv and v� �Rv are evaluated at ���x��x/�x�. We are going to
By the definition of b in line (4.23), it can be checked that 2��+ 1��1− b − k� =2�+m− k��+ 1� and we have
P = $�x�2�(1− �x�2��+1�
) m+2��+1
� with $ = �Q− 2��k��+ 1�− 2�−m�� (4.27)
The function
G = P
�′m = $�x�2��1− �x�2��+1��
2��+1
is the one in (4.7). Now, by (4.21) and (4.26), (4.20) becomes
E = �k∫�x�=1
u2dm−1 +∫B
{∣∣�Rv∣∣2 + h−2∣∣�Tv∣∣2 +Gv2}�′mdx�
and after the change of variable � = ��x� with Jacobian determinant (4.16), andwith # and F as in the statement of the proposition, we get (4.6).
Finally, (4.8) easily follows from (4.3), (4.13), and (4.11). �
We begin the proof of Theorem 3. Now let m = 1. Our first task is to proveexistence of extremal functions for inequality (1.9). We introduce the following normof a function u ∈ C1�−1� 1��
u =( ∫ +1
−1
{�u′�2p+ �u�2q}dx)1/2
�
where p and q are the functions (1.7) with m = 1. Clearly, the norm · originsfrom a scalar product. Denote by H1 the Hilbert space obtained completingC1�−1� 1�� in the norm · . Vectors in this Sobolev space are measurable functionsu with measurable distributional derivative u′ satisfying u < +�.
Consider the following minimum problem
1c′1�k��
= inf{u2 � u ∈ H1�
∫ +1
−1�u� 2Q
Q−2 p dx = 1}� (4.28)
Theorem 4.2. The infimum (4.28) is attained in H1.
Proof. Let �uh�h∈� be a minimizing sequence for (4.28)
limh→�
uh2 =1c′1�k��
and∫ +1
−1�uh�
2QQ−2 p dx = 1� h ∈ �� (4.29)
We can assume without loss of generality that uh ∈ C1�−1� 1�� and uh�±1� > 0.Indeed, we can work with non-negative functions uh ≥ 0. Moreover, if uh�1� = 0 oruh�−1� = 0 we replace uh with auh + b, where b > 0 is close to zero and a ∈ �0� 1�is close to 1 as we wish and such that the integral constraint is maintained. Theerror in the norm can be made arbitrarily small. By weak compactness, there is a
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1498 Monti
subsequence—which is still denoted by uh—weakly converging to a function u ∈ H1.The norm is lower semicontinuous for the weak convergence
u ≤ lim infh→�
uh� (4.30)
Possibly extracting a new subsequence, we can also assume that uh�x�→ u�x� foralmost every x ∈ �−1� 1�. If u satisfies the integral constraint
∫ +1
−1u
2QQ−2 p dx = 1� (4.31)
i.e., there is no concentration of measure, then from (4.29)–(4.31) it follows that uis a minimizer for (4.28).
In order to prove (4.31), consider for any h ∈ � the function vh ∈ C1�−"� "�associated with uh as in (4.3). Note that vh is continuous on −"� "� and vh�±"� = 0.The number " > 1 is defined in (4.18). By Proposition 4.1, we have the identities
∫ +1
−1u
2QQ−2
h p dx =∫ +"
−"v
2QQ−2
h d�� (4.32)
and
∫ +1
−1
{u′h
2p+ u2hq
}dx = �k
{uh�−1�2 + uh�1�2
}+ ∫ +"
−"
{v′h
2 + Fv2h}d�� (4.33)
where �k = Q− 2 if k = 1 and �k = 0 otherwise. F is the function defined in thestatement of Proposition 4.1. Examining the constant $ in (4.7), we notice that withm = 1 we have
F ≤ 0 for k = 1� F ≥ 0 for k ≥ 2�
We must discuss separately the cases k = 1 and k ≥ 2.We study the case k = 1. By continuity, for any h ∈ � there is a point �h ∈
−"� "� such that vh��h� = max−"�"� vh �= Mh. Let zh be the piecewise affine functionsuch that zh�±"� = 0 and zh��h� = Mh. Define vh = max�vh� zh� and let uh beassociated with vh as in (4.3). Note that, with m = k = 1, the functions in (4.17)and (4.19) are respectively �′�x� = �1− x2��+1��−
1�+1 and ��x� = �1− x2��+1��−
�2��+1� ,
x∈ �0� 1�. Then, by Hôpital’s rule
limx→1−
vh���x��
��x�− " = limx→1−
��x�−1uh�x�
��x�− " = −�uh�1� limx→1−
(1− x2��+1�
)− �2��+1� = −��
because uh�1� > 0. It follows that vh ≥ zh in some left neighborhood of " and thisproves that uh�1� = uh�1�. The same holds at x = −1.
For the L2 norm of v′h is less than or equal to the L2 norm of v′h, we have
∫ +"
−"
{�v′h�2 + Fv2h}d� ≤ ∫ +"
−"
{�v′h�2 + Fv2h}d��
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1499
Indeed, F ≤ 0 and vh ≥ vh. Analogously,
∫ +"
−"v
2QQ−2
h d� ≥∫ +"
−"v
2QQ−2
h d��
Then, possibly renormalizing each uh by a positive factor smaller than 1 (in orderthat the integral constraint be satisfied), the sequence �uh�h∈� is a new minimizingsequence for (4.28) and we can without loss of generality assume uh = uh, i.e., vh ≥zh. Thus, letting 2∗ = 2Q
Q−2 , we have
1 =( ∫ +"
−"v2
∗h d�
)1/2∗
≥( ∫ +"
−"z2
∗h d�
)1/2∗
≥ 212∗ −1"
12∗Mh�
and then vh� is uniformly bounded. Now, (4.31) easily follows from (4.32) bydominated convergence.
In the case k ≥ 2, we have �k = 0 and F ≥ 0 in (4.33), with left-hand sideuniformly bounded in h ∈ �. It follows that the sequence �vh�h∈� is bounded inH1
0 �−"� "�, the usual Sobolev space of functions vanishing at the boundary and withderivative in L2. Then suph∈� vh� < +� and the proof of (4.31) is concluded asin the case k = 1. �
The extension of a function u ∈ H1 to a function on �n� n = 1+ k, of the form(1.3) produces a function which is in D1��n�, by Theorem 3.2. In view of theminimization problem, this function can also be assumed to satisfy condition ii)in the definition of the class � , by Theorem 3.1. Then, without loss of generality,competing functions u ∈ H1 for problem (4.28) can also be assumed to satisfy thecondition
t �→ tQ−22 u�±t� is non decreasing for t ∈ 0� 1�� (4.34)
To a minimum u ∈ H1 for problem (4.28) there corresponds an extremal u ∈ D1��n�
for inequality (1.1) solving in weak sense equation (1.2). By abuse of notation, wedenote the two functions with the same symbol u. By a result due to Serrin, weaksolutions to (1.2) are locally bounded. For our purposes, the boundedness near�x� = 1 is enough. We state the next Lemma for general m.
Lemma 4.3. Let m� k ∈ �, � > 0, Q = m+ k��+ 1� and n = m+ k. A non-negativeweak solution u ∈ D1��n� of equation (1.2) is in L�
loc���, with � = �n\�x = 0�.
Proof. Letting V = −u 4Q−2 , the function u solves the equation
�xu+ w2�yu = Vu�
If we show that for some positive � ∈ �0� 1�
V ∈ L n2−�loc ���� (4.35)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1500 Monti
then the claim follows from Theorems 1 and 2 in Serrin (1964). If n ≥ 3, we have
u ∈ L 2nn−2loc ���, by the standard Sobolev–Poincaré inequality, and (4.35) follows from
4n�Q− 2��2− �� ≤
2nn− 2
�
which is satisfied for some positive �, because Q > n for � > 0.If n = 2, i.e., m = k = 1, we have u ∈ L 2Q
Q−2 ��2�, by the Sobolev inequality (1.1),and (4.35) follows from
8�Q− 2��2− �� ≤
2QQ− 2
�
which is satisfied for some positive �, because Q > 2 for � > 0. �
The next Lemma describes the regularity and the boundary behaviour ofminimizers of (4.28).
Lemma 4.4. Let m = 1, k ∈ � and � > 0. A non-negative function u ∈ H1 satisfying(4.34) and solving (4.28) is in C2�−1� 1� ∩ C1�−1� 1�� and moreover 2u′�x�+ �Q−2�xu�x� = 0 for x = ±1.
Proof. For it is u ∈ H1, u is continuous in �−1� 1� with weak derivative inL2loc�−1� 1�. By Lemma 4.3, we also have u ∈ L��−1� 1�, and then by (4.34) u can be
continuously extended to −1� 1�. By minimality, u solves in a weak sense the Eulerequation for problem (4.28), and precisely (up to a multiplicative constant whichcan be assumed to be equal to 1 upon changing the value of the integral constraint)
�pu′�′ − qu = −puQ+2Q−2 in �−1� 1�� (4.36)
Then u′′ ∈ L2loc�−1� 1� and consequently u ∈ C1�−1� 1� and thus equation (4.36)
implies u ∈ C2�−1� 1�. Higher regularity of u depends on � > 0. The function �pu′�′
and hence the function pu′ can be continuously extended to −1� 1�. Il must be
L �= limx→1
p�x�u′�x� = 0�
If it were L> 0, then pu′>L/2 in some left neighborhood of 1 and thus u�1� = +�,that is not possible. The case L < 0 can be excluded in the same way. Integrating(4.36) we obtain
u′�x� = 1p�x�
∫ 1
x
{pu
Q+2Q−2 − qu}dt� (4.37)
and by Hôpital’s rule there exists
u′�1� = limx→1
u′�x� = limx→1
p�x�u�x�Q+2Q−2 − q�x�u�x�p′�x�
= −Q− 22
u�1�� (4.38)
Analogously, u′�−1� = Q−22 u�−1�. �
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1501
We prove the even symmetry for extremals of (1.1) when m = 1. This finishesthe proof of Theorem 3. The proof is different according to the case: k = 1 or k ≥2. If k ≥ 2, then the function F introduced in Proposition 4.1 satisfies F ≥ 0. Thismakes possible a rearrangement argument. If k = 1, then F ≤ 0, and we can use auniqueness result for problem (1.10).
Theorem 4.5. Let m = 1, k ∈ � and � > 0. Any non-negative function u ∈ H1 solving(4.28) is even.
Proof. We begin with the case k ≥ 2. We claim that for any non-negative functionu ∈ C1�−1� 1�� there exists an even function u∗ ≥ 0 such that
∫ +1
−1u∗
2QQ−2 pdx =
∫ +1
−1u
2QQ−2 p dx� (4.39)
∫ +1
−1
{�u∗′�2p+ u∗2q
}dx ≤
∫ +1
−1
{�u′�2p+ u2q
}dx� (4.40)
Moreover, if u �= u∗ the second inequality is strict.Let v be the function associated with u as in (4.3). Identity (4.6) with m = 1 and
k ≥ 2 reads
∫ +1
−1
{u′2p+ u2q}dx = ∫ +"
−"
{v′2 + Fv2}d��
where F is the function introduced in Proposition 4.1. Note that F is non-negative,because if m = 1 and k ≥ 2 it is $ > 0 in (4.7). Moreover, F is even and strictlyincreasing on 0� "�.
Given a measurable set I ⊂ �−"� "�, let I∗ = �−�I�/2� �I�/2� be the intervalcentered at the origin with length equal to �I�, the Lebesgue measure of I . Then
∫IFd� =
∫I∩I∗
Fd�+∫I\I∗Fd� ≤
∫I∩I∗
Fd�+∫I∗\IFd� =
∫I∗Fd�� (4.41)
because �I∗\I� = �I\I∗� and F on I∗\I is less or equal than F on I\I∗. Moreover,inequality in (4.41) is strict as soon as �I\I∗� > 0, because of the strict monotonicityof F .
Denote by v∗ the symmetric decreasing rearrangement of v. The function v∗ isdefined by the condition on level sets
�v∗ > t� = �v > t�∗� t > 0� (4.42)
Note that, by (4.41) and (4.42), we have
∫ +"
−"Fv∗2d� =
∫ +�
0
∫�v∗>t1/2�
Fd� dt ≤∫ +�
0
∫�v>t1/2�
F d� dt =∫ +"
−"Fv2d� (4.43)
and the inequality is strict if v �= v∗. Moreover, the following well-known facts hold
∫ +"
−"�v∗′�2d� ≤
∫ +"
−"�v′�2d��
∫ +"
−"v∗
2QQ−2d� =
∫ +"
−"v
2QQ−2d�� (4.44)
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1502 Monti
Let u∗ be defined by v∗ as in (4.3). Then our claims (4.39)–(4.40) follow from(4.43) and (4.44), as soon as the integration by parts in (4.21) involving v∗ and u∗
produces no boundary contribution in (4.22), and precisely we need
limx→±1
p��′v∗��� = 0� (4.45)
where � and � are the functions in (4.4) with m = 1. This holds for v (this is (4.26)with k ≥ 2). By the rearrangement property (4.42), for any x ∈ �0� 1� there is anx ∈ �−1� 1� such that �x� ∈ �x�� 1� and v∗���x�� ≤ v���x��. For the function �p��′�is even and increasing on �0� 1�, we have �p��′v∗����x�� ≤ �p��′v∗����x�� and theclaim (4.45) is proven.
If k = 1, we argue in the following way. By Lemma 4.4, a minimizer u ofproblem (4.28) is in C2�−1� 1� ∩ C1�−1� 1�� and solves the problem
�pu′�′ − qu+ puQ+2
Q−2 = 0 in �−1� 1�
u > 0 in −1� 1�
2u′�x�+ �Q− 2�xu�x� = 0 x = ±1�
(4.46)
By Theorem 6 in Monti and Morbidelli (2006), the solution u ∈ C1�−1� 1�� ∩C2�−1� 1� of problem (4.46) is unique and thus even. The proof of the theorem iscomplete. �
Appendix
Proof of Proposition 2.1. We begin with (2.12). By the coarea formula (see Federer,1969, 3.2.12)
�����t� =∫���t
1rk+1
dr d� =∫ �
t
∫����s
1��r�U��
dk
rk+1ds =
∫ �
t
∫����s
1��HU��
d� ds�
(A.1)
Here, �r�U� denotes the gradient of U� in the variables r and �. Indeed, note thatr��r�U�� = ��HU�� and r−kk = �. Since u ∈ � satisfies (1.4), ��HU�� is bounded frombelow by a positive constant and from (A.1) it follows that t �→ ���t� is locallyLipschitz for t > 0. Upon differentiating (A.1), we get (2.12).
We prove (2.13). Without loss of generality, assume m = 2� � = �1� 0� and v =�0� 1�. Let �$ = �cos$� sin$��$ ∈ 0� 2%�, and write—with abuse of notation—�$ = ��$�t. Then we have
and denote by N the exterior unit normal to ��$ . The vector field Y��� �� =���−k−2�−�2� �1� 0� is divergence free and �Y�N� = ���−k−1 on the part of ��$
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
Sobolev Inequalities for Weighted Gradients 1503
corresponding to �$ and �Y�N� = −���−k−1 on the part of ��$ corresponding to�0. The divergence theorem therefore implies
���$�− ���0� = −∫S$
�Y�N�dk+1�
where k+1 is the �k+ 1�-dimensional Hausdorff measure in �k+2. From thisformula we see that $ �→ ���$� is Lipschitz. Then, for general m > 1� � �→ �t��� isLipschitz on �m−1 for any t > 0. By Rademacher theorem, � is differentiable almosteverywhere on �m−1 ×�+.
We claim that
lim$→0
1$
∫S$
�Y�N�dk+1 = −∫����t
�vU�
��HU��d�� (A.2)
First of all, note that on S$ we have N = −�U/��U � and thus
�Y�N� = −��−�2� �1�� ��U����k+2��U � �
It is enough to prove (A.2) in a local form. Consider a �k+ 1�-dimensional surfaceS$ parameterized by a Lipschitz function f � 0� $�× 0� 1�k → �k+2 of the formf = �r cos s� r sin s� �� where r = r�s� �� is Lipschitz. Let T be the k-dimensionalsurface parameterized by g��� = f�0� ��� � ∈ 0� 1�k. Applying the area formula tothe integral on S$ (see Federer, 1969, 3.2.3), taking the limit in $, and finallyapplying the area formula to the integral over T below, we get
lim$→0
1$
∫S$
�Y�N�dk+1 = −∫T
�vU�
��U �(det�JfTJf�det�JgTJg�
)1/2
���−k−2dk�
where Jf and Jg are the differential matrices of f and g, and JfT stands for thetransposed matrix. Here, Jf is evaluated on T .
A short computation furnishes
det(JfTJf
) = ∣∣∣∣�r�s∣∣∣∣2
+ r2(1+
∣∣∣∣�r��∣∣∣∣2)� det
(JgTJg
) = 1+∣∣∣∣�r��
∣∣∣∣2
�
and moreover, differentiating the identity U�r cos s� r sin s� �� = 0, we find
∣∣∣∣�r��∣∣∣∣2
= ���U �2��RU �2
�
∣∣∣∣�r�s∣∣∣∣2
= r2 ��TU �2
��RU �2�
Thus (det
(JfTJf
)det
(JgTJg
) )1/2
= ���2 ��U ���HU �
and the claim (A.2) follows. �
Proof of Proposition 2.2. Without loss of generality, we assume �����0� = +� forall � ∈ �m−1. The distribution function ���� t� = �����t� is continuous and strictly
Dow
nloa
ded
By:
[Lib
rary
of t
he M
athe
mat
ical
Sem
inar
y] A
t: 09
:37
2 A
ugus
t 200
8
1504 Monti
decreasing in t. The inverse function �� � 0�+��→ �0�maxU�� is defined by theidentity
���� ���� t�� = t�
We write ���s� = ���� s�. Note that from (A.1) it follows that for any � > 0 there isa constant C > 0 such that for � < t1 < t2 < maxU� − �
���t1�− ���t2� ≥ C�t2 − t1��
and thus also �� is locally Lipschitz for s ∈ �0�+��.Now, from the Lipschitz continuity of �, it easily follows that � � �m−1 ×�+ →
� is locally Lipschitz continuous. The function �r� �� �� �→ U ∗� �r� �� = ��
(��Bd�r����
)is continuous for r > 0 and is locally Lipschitz continuous for r > 0 and �r� �� �=�1� 0�. The statement of Proposition 2.2 immediately follows from (2.11). �
Acknowledgment
It is a pleasure to acknowledge the debt I owe Daniele Morbidelli. The problems andthe ideas contained in this article originated in our discussions and in our commonresearch activity.
References
Beckner, W. (2001). On the Grushin operator and hyperbolic symmetry. Proc. Amer. Math.Soc. 129:1233–1246.
Burago, Yu. D., Zalgaller, V. A. (1988). Geometric Inequalities. Springer.Federer, H. (1969). Geometric Measure Theory. Springer.Franchi, B., Gutiérrez, C. E., Wheeden, R. L. (1994). Weighted Sobolev–Poincaré inequalities
for Grushin type operators. Comm. Partial Differential Equations 19(3–4):523–604.Garofalo, N., Nhieu, D. M. (1996). Isoperimetric and Sobolev inequalities for Carnot-
Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math.49:1081–1144.
Garofalo, N., Vassilev, D. (2001). Symmetry properties of positive entire solutions ofYamabe-type equations on groups of Heisenberg type. Duke Math. J. 106(3):411–448.
Hajłasz, P., Koskela, P. (2000). Sobolev met Poincaré. Memoirs of the Amer. Math. Soc. 688.Lions, P.-L. (1985). The concentration-compactness principle in the calculus of variations.
The limit case. I and II. Rev. Mat. Iberoamericana 1(1):145–201, and (2):45–121.Monti, R., Morbidelli, D. (2004). Isoperimetric inequality in the Grushin plane. J. Geom.
Anal. 14(2):355–368.Monti, R., Morbidelli, D. (2006). Kelvin transform for Grushin operators and critical
semilinear equations. Duke Math. J. 131(1):167–202.Serrin, J. (1964). Local behavior of solutions of quasi-linear equations. Acta Math.
