Institute of Exact Sciences - ICEx
Department of Mathematics
PhD Thesis
About a Class of Optimal Sobolev Vector Inequalities
of Second Order
Aldo Peres Campos e Lopes
Advisor : Prof. Ezequiel Rodrigues Barbosa Co-Advisor: Prof. Marcos Montenegro
My supervisor Ezequiel Rodrigues Barbosa and my co-supervisor Marcos Montenegro. I appreciate the encouragement and support, and by the beautiful theme proposed. The discussions always so enlightening and instructive. I am grateful to my supervisor for his patience and kindness to address several questions that have arisen, by having their time even on Saturdays and other non-school days.
Members of the bank examiner who kindly agreed to participate in the completion of this work, professors Ezequiel Rodrigues Barbosa, Marcos Montenegro, Emerson Jurandir and Jo˜ao Marcos Bezerra do ´O.
To my parents, Alfreu and Eunice, by ample incentive to the “domain of knowledge”. Even before entering the University, the great desire of my father was watching me with the title of Doctor, “Dr. Aldo”,as always told me.
To my many colleagues in the UFMG who helped me in various ways.
To Professor Francisco Dutenhefner by the orientation in the MSc and for under-standing and flexibilitye. To Susana C. Fornari for her patience and dedication during my undergraduate research and encouraging academic research.
To the professor Sylvie Marie who without knowing it helped me get into the math career. I do not forget a phrase she said.: “Many people choose to do engineering because they like math. Why not choose mathematics since they like math?”
To many professors of UFMG math department who helped answering questions and teaching courses I attended as a student. In this regard, I am grateful to professors like: Gast˜ao, M´ario Jorge, Rog´erio Mol, Marcelo T. Cunha, among several other.
To UFMG math department that helped me, as much as possible, in several events that I participated . It is certainly well deserved the concept 6 by CAPES..
UNIFEI colleagues who helped in the reduction of my working hours for a few semesters. To other colleagues, friends and professors who should be included here. However, do not do a list with names so I could forget to mention someone.
trying to solve.
Contents
General Introduction 7
0.1 Historical Overview . . . 7
0.2 Proposal and Relevance . . . 13
0.3 Organization and Ideas . . . 15
1 Preliminary Mathematical Material 17 1.1 Curvatures in a Riemannian manifold . . . 17
1.2 The Musical Isomorphism and Divergence of Tensors . . . 21
1.3 Homogeneous Functions . . . 23
1.4 Sobolev Spaces of Vector Valued Maps of Second Order . . . 26
1.5 Coercivity . . . 30
1.6 The Scalar AB Program . . . 33
1.6.1 Partial Answers . . . 36
1.7 AB Vector Program . . . 37
1.7.1 Partial Answers . . . 41
2 Elliptic Systems of Fourth Order 45 2.1 Existence . . . 46
2.2 Regularidade . . . 55
2.3 Bubbles Decomposition . . . 62
2.4 Pointwise Estimates . . . 66
2.5 Concentra¸c˜ao L2 . . . 72
2.6 Compacidade . . . 85
3 Sharp Sobolev Vetorial Inequality of Second Order 99 3.1 Extremal . . . 99
General Introduction
0.1
Historical Overview
In 1983, Paneitz [27] introduced the fourth order operator P4
g : C4(M) → C0(M), defined by
P4
gu:= ∆2gu−divg
2
3Rgg−2 Ricg
(∇u)#
,
for all u∈C4(M), where (M, g) is a Riemannian manifold of dimension n= 4, Ric
g is the
Ricci tensor Ricci,Rg is the scalar curvature, divg is the divergente and ∆g is the
Laplace-Beltrami operator with respect to the metric g. This operatorP4
g has some properties of conformal invariance. Accurately, if ˜g =e2ϕg is conformal to the metric g, ϕ
∈ C∞(M),
then
Pg˜4 =e−4 ϕ
Pg4.
Associated with this operator, we have the notion ofQ-curvatura, a curvature which also has conformal properties. For this case n = 4, aQ-curvature is given by
Q4g = 1
6 ∆gRg −3|Ricg|
2
g+Rg2
.
Beyond this conformal invariance, the operator P4
g appears in the following relation be-tween the curvatures Q4
g and Q4˜g:
Pg4ϕ+Q4g =Q4g˜e4ϕ.
It is noteworthy that theQ-curvature in the dimensionn = 4, and for locally conformally flat manifold, is inside the integral in the Gauss-Bonnet formula for Euler characteris-tic, and thus has a very important role in the study of topology and geometry of the Riemannian manifold of dimension 4. We have the following integral identity
4π2χ(M) =
Z
M
Qg+ 1
8|Weylg|
2
dvg, (1)
where χ(M) is the Euler characteristic of the manifold M and Weylg denotes the Weyl
tensor with respect to the metricg. As the|Weylg|2dvgis an punctual conformal invariant, we obtain that the integral of Q-curvatura R Qg dvg is a conformal invariant. For more details on this, see the articles by Chang [8] and Chang-Yang [7].
The generalization for the casen≥5 was made by Branson [6] in 1987. Let (M, g) be a Riemannian manifold of dimensionn ≥5. We define the operator Pn
Pgnu:= ∆2gu−divg (anRgg+bnRicg) (∇u)#
+n−4
2 Q
n gu ,
where
an=
(n−2)2 + 4
2(n−1)(n−2), bn=− 4
n−2, and
Qng = 1
2(n−1)∆gRg +
n3 −4n2+ 16n−16
8(n−1)2(n−2)2 Rg−
2
(n−2)2 |Ricg| 2
g
is theQ-curvatura for dimensionn ≥5. We denotePn
g also byPg. This operator also has conformal invariance properties. That is, consideringu∈ C∞(M), u >0, and the metric
˜
g =un−44g wich is conformal to the metric g, we have
P˜gnϕ=u−
n+4
n−4Pn
g(uϕ), (2)
for allϕ∈C∞(M). In particular, by takingϕ≡1, we get the following elliptic semilinear
diferencial equation satisfied by the conformal factor u:
Pgnu=
n−4
2 Q
n
˜
gu
n+4
n−4 , u >0.
It is interesting to compare this relationship between the Paneitz-Branson operator and the Q-curvatura of the metric g with the scalar curvature Rg. Let’s see what we mean. Initially, we have following
Lgu=R˜gu
n+2
n−2 ,
whereLg = 4nn−−12∆g+Rg; ˜g =u
4
n−2g and n≥3. We have also a equivalent identity in the
case n = 2. The operator Lg is a conformal operator. An important result in conformal differential geometry is the resolution of the Yamabe problem , where the operator Lg plays a key role. That is, we can always find a metric g, in the conformal class of g, such that the scalar curvature Rg is constant, considering that the manifold is smooth, closed and dimension n≥2.
Because of similarities of the properties, in the conformal differential geometry, be-tween the operators Lg e Pg, it is natural to ask whether the Q-curvatura has the same property, that is, how it would be the Yamabe problem for Q-curvature. Partial results have been established in response to this question for manifolds of dimensions ≥ 5 (see [15], [22], [23], [28]).
the conformal covariance property of the Paneitz-Branson operator, that is, by (2). This property tells us thatuv is a solution for the Paneitz-Branson equation in the geometry of the metric ˜g ifuis a solution in the geometry ofg. This condition plays an important role in the type of functions which can be a local minimum of a functional which is naturally associated with the Yamabe problem for the Q-curvature.
There is another problem, an analytical problem, when we consider the Yamabe prob-lem for theQ-curvature. As in the case of the classical Yamabe problem, we have problems when we try to use variational methods to find solutions of Paneitz-Branson equation be-cause of the exponent of the nonlinearity, n+4
n−4, wheren is the dimension of the manifold,
which is the critical Sobolev exponentW2,2(M) less one, because the immersion W2,2(M)
in Ln2−n4 is not compact.
One idea contained in [10], by Hebey-Djadli-Ledoux, is that the Sobolev inequalities of second order can be used to deal with the problem of concentration in approximating sequences for the solution, since the infimum of the functional associated with the problem, the Paneitz-Branson functional, is smaller than a critical value. In this method, it is not clear when the infimum is positive and the Yamabe constant is less than or equal to the Yamabe constant of the sphere.
A partial solution to this problem was made by F. Robert and P. Esposito in 2002, see [15]. It has been shown that ifn≥8 and the manifold is locally conformally flat, then there exists a minimizer for the Paneitz-Branson functional.
The effect of this result is that the part of existence of the Yamabe problem for the
Q-curvature is brought to a point analogous to that Aubin led the Yamabe problem for the scalar curvature. But, is not yet clear when the Green function of the Paneitz Branson operator is positive, in case the operator be coercive. This excludes try to use the methods of Schoen to complete the problem.
It was shown by D. Raske (see [29]) that there exists a metric in the conformal class of the arbitrary metric in a smooth closed Riemannian manifold, of dimension n ≥ 5, such that the Q-curvature of the metric is constant. Existence of solutions is obtained through the combination of variational methods, Sobolev inequalities of second order and the blow-up theory of W2,2(M). Below is the result.
We define the Paneitz Branson constant as
λg(M) := inf w∈C∞
+(M)
R
M wPgw dvg
kwk2
2n n−4
,
where Pg is the Paneitz-Bransonoperator. Let λ(Sn) be the Paneitz-Branson constant of the unit sphere with the canonical metric. We have the following result (see [29]):
Theorem 1 (David Raske, 2011). Let (M, g) be a Riemannian manifold of dimension n ≥5. Suppose that at least one of the following conditions is valid:
ii. The Yamabe constant of g is greater than or equal to the inverse of the Yamabe constant of the n-sphere;
iii. n≥8 and g is not locally conformally flat.
then exists a smooth minimizer, positive of Paneitz-Branson functional and there exists a metric h in the conformal class of g such that Qh = λ, where λ is the Paneitz-Branson
constant of g.
thus, the study of operators like the Paneitz-Branson is very important in the analysis of geometrical problems as the problem of the prescribed Q-curvature, where the Yamabe problem for the Q-curvature is a particular case.
Consider the system of equations
−∆2gui+ divg Ai(∇ui)#
+ k
X
j=1
Aij(x)uj =u2
#−1
i , (3)
where 2# = 2n
n−4, U = (u1, . . . , uk), A = (Aij) is a continuous map of M to M
s
k(R) such thatA(x) is positive definit for allx∈ M,Ms
k(R) is the space of real symmetric matrices
k× k, and the Ai are symmetric tensor fields of type (2,0). In this context, we consider
ui > 0 for all i. This system can be seen as a natural generalization of the equations involving operators of type Paneitz Branson. Thus, considering the scalar case, that is
k = 1, and the tensor Ai = f ·g, where f is a smooth function, the system (3) can be rewritten as
−∆2gu+bα∆gu+cαu=u2
#−1
. (4)
Assume that the constantsbαandcα are converging sequences of real positive numbers, satisfying cα ≤ b
2
α
4 . From 2000 to 2004, many authors studied the case (4) above, for
example F. Robert, E. Hebey, Z. Djadli and M. Ledoux in [10, 15, 22] and [24]. Hebey-Robert-Wen in [24] discussed the compactness of solutions of (4), precisely when bα and
cα converge respectively to b0 and c0 and the solutions uα converge weakly in H2,2(M). They found conditions such that the limituα is nontrivial.
The following theorems have been proved by Hebey-Robert-Wen in 2004 (see [24]). Let
Ag =
(n−2)2 + 4
2(n−1)(n−2)Rgg− 4
n−2 Ricg (5)
be field of (2,0)-tensor. We denote by λi(Ag)x, 1, ..., k, the g-eigenvalues of Ag(x) and define λ1 as infimum of the i and x of theλi(Ag)x and λ2 as the supremum of the i and
x of the λi(Ag)x. We denote por Sc the critical set defined by
We have the following results:
Theorem 2 (Hebey-Robert-Wen, 2004). Let(M, g)be a compact manifold locally confor-mally flat of dimension n and (bα)α, (cα)α converging sequences of positive real numbers with positive limits b∞ e c∞ such that cα ≤ b
2
α
4 for all α. We consider equations of type
∆2gu+bα∆gu+cαu=u2
#−1
, (7)
and we assume that b∞ ∈ S/ c, where b∞ is the limit of bα and Sc is the critical set given
by (6). Then the family (7) is pseudo-compact when n ≥6 and compact when n ≥9.
We say that a family of equations are solutions of (7) is pseudo-compact if, for any sequences (uα) in H2,2(M) of positive solutions that converges weakly in H2,2(M), the
weak limit u0 of u
α is nonzero.
The following theorem is a complement of the above theorem when the dimension is
n = 6,7 or 8 and b∞ is is below the lower limit λ1 deSc.
Theorem 3 (Hebey-Robert-Wen, 2004). Let (M, g) be a compact manifold locally con-formally flat of dimension n = 6,7 or 8 and (bα)α, (cα)α converging sequences of real positive numbers with positive limit b∞ and c∞ such that cα ≤ b
2
α
4 for all α. We consider
equations like
∆2gu+bα∆gu+cαu=u2
#−1
,
and we assume that b∞ < λ1 = min Sc, where b∞ is the limit of bα and Sc is the critical
set given by (6). Then the family (7) is compact.
Pseudo-compactness has a traditional interest because it seeks nontrivial solutions of limit equation we obtain from (7) making α→+∞.
On the other hand, we say that the family of equations (7) is compact if any sequence (uα)α in H2,2(M) of positive solutions of (7) is limited in C4,θ(M), 0 < θ < 1 and then converges, if necessary take a subsequence, on C4(M) for some function u0.
Compactness is a concept clearly stronger than pseudo-compactness.
Pseudo-compactness for elliptic equations of second order the type of Yamabe has been widely studied. Compactness for Yamabe equations of second order was studied by Schoen from 1988 to 1991 (see [32, 33, 34, 35]).
As application of the results of compactness, we can study also the existence of ex-tremal in sharp Sobolev scalar inequalities of second order.
From the continuous immersionH2,2(M)֒→ L2#
(M), then there are constantsA, B >
0 such that:
kuk22# ≤Ak∆guk22+BkukH21,2(M) , ∀u∈ H2,2(M). (8)
A0 = inf
A: exists B such that the above inequality (8) is valid ∀u∈ H2,2(M) . (9)
A natural question that arises is: the infimum A0 is achieved? The answer is yes, the
infimum A0 is achieved in (8). In 2000, Djadli-Hebey-Ledoux [10] proved the result with
the restriction that the metric g is conformally flat. Later, in 2003, Emmanuel Hebey proved the result for any Riemannian manifold in [25] (see also [26]).
Follow from the Sobolev theorem that the constant A0 is well defined. This constant
was calculated by Lieb [25], Lions [26], Edmunds-Fortunato-Jannelli [14] and Swason [37]. Precisely, we have:
1
A0
= n(n
2−4)(n−4)wn4
n
16 ,
wherewn is the volume of the sphere Sn inRn+1.
The infimum in (9) is achieved and there exists a constant B >0 such that:
Z
M|
u|2#dvg
2 2#
≤ A0
Z
M
(∆gu)2dvg+B
Z
M |∇
u|2g+u2
dvg, (10)
for all u ∈ H2,2(M).
The second best Sobolev constant associated with (8) is defined by:
B0 = inf{B ∈ R; (10) is valid } . (11)
The second Sobolev Riemannian inequality states that, for anyu∈ H2,2(M), we have
Z
M|
u|2#dvg
2 2#
≤ A0
Z
M
(∆gu)2 dvg+B0
Z
M |∇
gu|2+|u|2
dvg. (12)
A nonzero function u0 ∈ H2,2(M) is saidextremal for the inequality (12) if
Z
M|
u0|2#dvg
2 2#
=A0
Z
M
(∆gu0)2 dvg+B0
Z
M |∇
gu0|2+|u0|2
dvg.
You can find some comments about the second best constant in [10] (Djadli, Hebey, Ledoux, 2000). But for the extremal of the sharp inequalities there are studies only for the case of first order. To this reasoning, see [11] also [5, 4].
0.2
Proposal and Relevance
One of the main goals of this thesis is extend the results on compactness of solutions of equations involving Paneitz-Branson type operators for systems and apply these results to obtain results of existence of extremal for a class of sharp Sobolev inequalities of second order. This is an important question, both the mathematical point of view, for involve a larger structure and his understanding, as from the point of view of analytical applications, by allowing the study of several elliptic PDE systems of second order on Riemannian manifold. Let me be a bit more clear..
Let (M, g) be a compact Riemannian manifold, of dimension n ≥ 5, whose volume element is dvg. We consider here maps U ∈ Hk2,2(M) that are solutions the following model equation:
−∆2gU + divg A(∇U)#
+∇UG(x, U) = ∇F(U). (13) That is,
−∆2gui+ divg Ai(∇ui)#
+∂iG(x, U) =∂iF(U), (14) for eachi= 1, ..., k, where 2#= 2n
n−4 is the Sobolev critical exponent for the immersions of
H2,2(M) inLp(M) spaces. Note that the operatorP
g :=−∆2g+ divg Ai(∇·)#
+∂iG(x,·) is Paneitz-Branson type. Here arise some questions, such as:
• There is a nonzero solution U for (13)?
• If there is a nonzero solution U, which regularity we can get??
• The set of the solutions of (13) is compact in some topology?
We will answer these questions in the following chapters for a class of homogeneous functions F and G.
Answering the above questions, we continue with the study of existence of extremal in sharp Sobolev inequalities of second order in the vector case. This study, for the case of second order, is also motivated by the study of the best constants for first order inequalities in the vector case which was done by E. Barbosa e M. Montenegro (see [5]). E. Barbosa e M. Montenegro studied inequalities like:
Z
M
F(U)dvg
p p∗
≤ A0(p, F, G, g)
Z
M|∇
gU|pdvg+B0(p, F, G, g)
Z
M
G(x, U)dvg, (15)
where A0 and B0 are the best constants, 1 ≤ p < n, U = (u1,· · · , uk), F : Rk → R is a
constants developed (in [5]) considers a number of questions involving the constants A0
and B0. Some follow directly from scalar theory, others are more complex, for example,
the behavior of B0 for all the parameters involved and problem of existence and C0
compactness of the extremal. A map U is extremal when it achieves equality in (15). When F and Gare C1 maps a extremal is weak solution of the system:
−A0(p, F, G, g)∆p,gui+ 1
pB0(p, F, G, g)
∂G(x, U)
∂ti
= 1
p∗
∂F(U)
∂ti
, i= 1,· · · , k ,
where ∆p,gu = divg(|∇gu|p−2∇gu) denotes the p-Laplacian operator associated with the metricg.
Now considerUα ∈ H1,p(M,Rk) weak solutions of the system:
−∆p,gU +1
p∇UGα(x,U) =
1
p∗∇Fα(U) em M ,
whereFα :Rk→Rare positiveC1 functions andp∗-homogeneousand theGα :M×Rk → R are C1 functions and p-homogeneous in the second variable. Consider sequences of limited solutions (Uα)α whose limite isU ≡ 0, taking a subsequence if it is necessary, we have the following bubbles decomposition:
Uα = l
X
j=1
Bj,α+Rα,
for all α > 0, where (Bj,α)α, j = 1,· · · , l, are k-bubbles and (Rα)α ⊂ H1,p(M,Rk) is such that Rα → 0 in H1,p(M,Rk) when α → +∞. With this result we can add more properties to the sequences of limited solutions such thatUα ⇀0 inH1,p(M,Rk). We have also pointwise estimates or C0 for (Uα)α (see [9]). The blow-up points of the sequences
(Uα)αhave much of the information of the sequences, this is theLp concentration property. Concentration properties were studied by Druet, Hebey and Robert (see [17] and [12]) and extensions of these works were made by G. Souza [9].
In this work, we follow similar ideas of E. Barbosa and M. Montenegro in [5]. We consider the sharp Sobolev inequalities of second order:
Z
M
F(U)dvg
2 2#
≤ A0
Z
M
(∆gU)2dvg + + B0
Z
M
(A (∇gU)#,(∇gU)#
+G(x, U) dvg, (16)
where F is a 2#-homogeneous map and Gis 2-homogeneous on the second variable, and
−∆2gU+ divg A(∇U)#
+1
2∇UG(x, U) = 1
2#∇F(U). (17)
We consider sequences of limited solutions (Uα)α whose limite isU ≡0. If necessary take a subsequence, we obtain the following bubbles decomposition :
Uα = l
X
j=1
Bj,α+Rα,
for all α > 0, where (Bj,α)α, j = 1,· · · , l, k-bubbles and (Rα)α ⊂ H2,2(M,Rk) is such that Rα → 0 in H2,2(M,Rk) when α → +∞. Added to this result, we have pointwise estimates or C0 for (Uα)α. The blow-up points or concentration of the sequences (Uα)α
have a lot of information about the sequences and generatingL2 concentration properties.
We apply all results for the study of existence of extremal for the inequalities (16).
0.3
Organization and Ideas
This work has three chapters. In Chapter 1, we show some definitions and some basic results that will be used in the others chapters. In special, we show the Sobolev spaces of second order and we make fazemos um panorama detalhado da scalar theory de best constants on sharp Sobolev inequalities of second ordem e destacamos alguns problemas em aberto. Morover, we describe some important problems of the vector theory of best constants and we state our main contributions. We include in this chapter, some basic results about Euclidean and Riemannian Sobolev vector inequalities. Precisely, we show that the best constant associated to the inequality
Z
Rn
F(U)dx
2 2#
≤ A
Z
Rn
(∆U)2 dx
is given by
A0(F, n) = M
2 2#
F A0(n) where MF = maxSk−1
2 and S
k−1
2 =
n
t∈Rk;Pk
i=1|ti|2 = 1
o
. Morover, characterized the associated extremal as the type of U0 =u0t0, where u0 is a extremal function associated
to the Euclidean Lp-Sobolev scalar inequality and t
0 is a maximum point of F inS2k−1.
Chapter 2 has the largest number of contributions made. Turn our attentionto the following system:
−∆2gU + divg A(∇U)#
of sequences (Uα)α, given by the solutions of systems (18), as function of the behavior of
Fα and Gα.
With the hypothesis that the (Uα)α is limited, we obtain that these sequences is a Palais-Smale sequences of the functional J associated to (18). That is, the sequences Uα is such that:
(J(Uα))α is limited and DJ(Uα)→0 in Hk2,2(M)′ .
From the limitation of Uα we get the existence of a weak limit U0 in Hk2,2(M). Working with the case where U0 ≡ 0, the hypotheses about F
α and Gα let us decompose the functionalJ in terms of others functionalLi
ksuch that the components ofUα form Palais-Smale sequences for the functional Li
k. This decomposition in functional Lik allow us to use the results obtained by F. Robert (see [30]) to obtain bubbles decomposition for the components ofUα, which concludes this part.
Using bubbles decomposition we have the pointwise estimates for a sequences of solu-tions of (18) that have 0 as a weak limit.
Then, we have the L2 concentration, where we extended for the vector case some
results. In some of these, we assume thatG(x, U) =Pki,j=1Aij(x)ui(x)uj(x). We use some estimates and the Bochner-Linerowicz-Weitzenbook formula for the mainL2concentration
result.
In the main part of chapter 2, we proof the compactness. For this, we use several previous results, as bubbles decomposition andL2concentration.We also use an important
and useful tool that is the Pohozaev type identity. Our contributions at this point extend some results ound in studies of E. Hebey, F. Robert, Z. Djadli, M. Ledoux and V. Felli, made in the scalar case (fourth order equations), for the vector case (systems of fourth order).
In the chapter 3, we study the existence of etremal maps. The main ideia here is the following. Suppose that the second order vector Sobolev sharp inequality has no extremal map. We consider then a numerical sequence (α) such that 0 < α < B0 and α → B0. Associated to this sequence, we construct functions Fα and Gα and also solutions Uα of systems of type studied on the previous chapters. The sequence (Uα) converges for U
Chapter
1
Preliminary Mathematical Material
In this chapter, introduce some basic notations and definitions we will use throughout the remainder of this paper. We remember some basic facts of Riemannian geometry.
Then, there is the Sobolev space theory in Riemannian manifolds. In this section and in the rest of this paper, we assume that the manifold is compact. We show the norms that we use, and some properties of these Sobolev spaces.
Some definitions are more usual. But present below the definitions which will be useful in the following work.
Over the years, about forty years, much attention has been given to the sharp Sobolev Riemannian inequalities. There is a vast literature with a wide theory of the best constants that is connected with areas such as analysis, geometry and topology. These inequalities play an important role in geometric analysis, especially in the study of the existence and multiplicity of solutions to the Yamabe problem (see [2], [20], [34]), Riemannian isoperimetric inequalities (see [13]), among other applications. Most of these results shows the influence of geometry and topology.
Some efforts were made in the study of shapr Sobolev Riemannian inequalities in some decades. Part of the obtained results is kown as theAB program. Sharp inequalities and first order Sobolev best constants in the scalar case were studied by Aubin, Druet, Hebey, Vaugon, among others (see [3], [13], [19], [21]). But sharp inequalities and best Sobolev constants in the vector case were studied by Hebey, E. R. Barbosa and M. S. Montenegro (see [17], [18] and [5], [4]). For sharp Sobolev inequalities of second order in the scalar case, the paper made by Djadli-Hebey-Ledoux show a introduction to study the first constant. In this chapter, we show the AB program for scalar sharp Sobolev inequalities, and after in the vector case, of second order. We show here some contributions, that is, we answer some of the questions of AB program of the inequalities of second order.
1.1
Curvatures in a Riemannian manifold
R(X, Y)(x)Z =∇X˜(x)(∇Y˜(x)Z˜)− ∇Y˜(x)(∇X˜(x)Z˜)− ∇[X,˜Y˜](x)Z ,˜
where ˜X,Y ,˜ Z˜ are vector fields on M such that ˜X(x) = X, Y˜(x) = Y, Z˜(x) = Z. This definition is independant of the choice of the extensions ˜X, Y ,˜ Z˜. Given x ∈ M and
X, Y, Z ∈ TxM and η∈(TxM)∗, we define thecurvature tensor as follows:
R(x)(X, Y, Z, η) =η(R(Y, Z)(x).X) =hR(X, Y)Z, Wi.
The function R is a (3,1)-tensor. The coordinates of R in a chart are given by:
R(x)l ijk=
∂Γl ki
∂xj
x
− ∂Γ
l ji
∂xk
!
x + Γl
jα(x)Γ α
ki(x)−Γ l kα(x)Γ
α
ji(x), (1.1) where Γk
ij are the Christoffel symbols.
The Riemman tensor is a (4,0)-tensor and the coordinates in a chart are Rijkl :=
giαRαjkl.
The curvature operator R inx∈ M, R: Λ2
x →Λ2x is defined by the relation
hR(X∧Y), W ∧Zi=hR(X, Y)Z, Wi,
where Λ2
x is the space generated by the 2-forms X∧Y and
(X∧Y) (Z, W) =hY, ZihX, Wi − hX, ZihY, Wi.
Note that, if{Xi}ni=1is a orthonormal basis ofTpM, then{Xi∧Yi}i<j is a orthonormal basis of Λ2
x. Thus, the symmetric bilinear function R is well defined.
Now let σ ⊂ TpM be a bidimensional space of TpM and {X, Y} a basis for σ. The
sectional curvature of σ in x is given by:
K(σ) =K(X, Y) = R(X, Y, Y, X)
kX∧Yk2 ,
wherekX∧Yk2 =kXk2kYk2− hX, Yi2.
The Ricci tensor which we denote by Ricg or Ric is a symmetric (2,0)-tensor defined
as the following bilinear function:
Ric :T M × T M → R,
that associates to each pair of fields (X, Y) the trace of the function Z 7→ R(X, Z)Y. Then we have:
Ric (X, Y) = n
X
i=1
hR(Xi, X)Y, Xii,
where{Xi} n
The Ricci curvature in the direction of X ∈ T M, with |X|= 1, is defined as
Ric (X) = Ric (X, X).
If X ∈ T M is unitary and X(p) = v, p ∈ M and v ∈ TpM, then we write Ricp(v) instead of Ric(X, X). Let {e1,· · · , en} be a orthonormal basis with v = ei for some i. Then:
Ricp(v) = X j6=i
hR(v, ei)v, eii
= X
j6=i
K(v, ei)
The scalar curvature we denote by Rg or Scalg is the trace of the Ricci tensor Rg :=
gijR
ij, where Rij are the components of Ricg in a chart.
The Riemann curvatureRmg is defined by
Rmg(x)(X, Y, Z, T) =g(x) (X, R(Z, T)Y).
The components of Rmg in a chart are given by Rijkl = giαRαjkl where Rαjkl are the components of the curvature R as described above in (1.1).
Now let (M, g) be a Riemannian manifold of dimension n ≥ 3. The Weyl curvature of g, denoted by Weylg, is a field of tensor of class C∞ four times covariant on M. Such
curvature is defined by:
Weylg = Rmg−
1
n−2Ricg⊙g+
Rg
2(n−1)(n−2)g⊙g ,
where ⊙ is the Kulkarmi-Nomizu product, define below. For all x∈ M we have:
Weylg(x) = Rmg(x)−
1
n−2Ricg(x)⊙g(x) +
Rg(x)
2(n−1)(n−2)g(x)⊙g(x). The components Wijkl of Weylg are given by the relation:
Wijkl = Rijkl− 1
n−2(Rikgjl+Rjlgik−Rilgjk−Rjkgil)
+ Rg
(n−1)(n−2)(gikgjl−gilgjk)
h(X, Y) =h(Y, X) and k(X, Y) = k(Y, X).
We define the Kulkarni-Nomizu product of h and k, we denote by h⊙k, as the four times covariant tensor in E, defined for allX, Y, Z, T ∈ E by
h⊙k(X, Y, Z, T) = h(X, Z)k(Y, T) +h(Y, T)k(X, Z)−h(X, T)k(Y, Z)−h(Y, Z)k(X, T).
We see that the Kulkarmi-Nomizu product is symmetric in the sense that, for all h
and k, we have h⊙k =k⊙h. We also see that the product is distributive with respect to addition, in that, for all h, k1 and k2 we have h⊙(k1+k2) =h⊙k1+h⊙k2.
A Riemannian manifold is locally conformally flat if, in each point of M there exists a neighborhood conformally equivalent to the Rm. That is, if x ∈ M, there exists a neighborhood U of x and a function u class C∞, that is, u :U →R, such that a metric
(local) ˜g =e2ug is flat in U.
1.2
The Musical Isomorphism and Divergence of Tensors
Consider x∈M. # is the musical isomorphism between TxM and (TxM)∗, defined as:
# :TxM −→ (TxM)∗
X 7−→
(
(TxM) −→ R
Y 7−→ hX, Yig(x),
This isomorphism is the identification of a Euclidean space with its dual. We denote:
X# the image of X via # and η# the image of η ∈ (T
xM)∗ via the inverse of #. This definition extends naturally vector field (that is, (0,1)-tensors) and to (1,0)-tensors. IfX
is vector field and η is a (1,0)−tensor, the coordinates of their images in a chart are:
Xi := (X#)i =gijXj
ηi := (η#)i =gijη j,
that is an expression that is independent of the chart. Clearly, we have (X#)# = X e
(η#)# =η.
In what follows, in this work, A is defined as:
A (X)#,(X)#= k
X
i=1
Ai (Xi)#,(Xi)#
, (1.2)
whereAi, i = 1, ..., k, are positive and symmetric tensores, that is,Ais a sum of continuous symmetric (2,0)−tensor. We have X = (X1, . . . , Xk) is such that each Xi, i = 1, . . . , k
is a (1,0)-tensor. In what follows, for simplicity, we say that A is a smooth symmetric (2,0)-tensor when we refer the sum (1.2) above. Here Al(Xl)# is a (1,0)-tensor whose
local coordinates are
A(Xl)#i =Aij (Xl)#
j
=Aijgik(Xl)k
As the manifold M is compact and A is continuous, there exists a constant C > 0 such that:
Z
M
A (X)#,(X)# dvg
≤C
Z
M|
X|2gdvg,
Z
M
A (X)#,(X)# dvg = k
X
i=1
Z
M
Ai (Xi)#,(Xi)#
dvg.
We have that the manifoldM is compact andAis a symmetric and positive (2,0)−tensor, then there are positive constants cand C, such that:
c|X|2g ≤Ai (X)#,(X)#
≤C|X|2g, (1.3)
for X a (1,0)-tensor.
LetX be a smooth vector field inM. Thedivergence ofX is a smooth function inM
given by:
divgX :M → R
p 7→ (divgX)(p) = tr{v 7→(∇vX)(p)} ,
where v ∈ TpM and tr is the trace of the linear operator. To define on charts, consider
X a (1,0)-tensor inM. The divergence is defined as
divg(X) = gij(∇X)ij =gij ∂iXj−ΓkijXk
.
This expression is independent of the chart.
Throughout this work we use sometimes the following useful theorem.
Theorem 4(Divergence theorem). Let(M, g)be a compact Riemannian manifold without boundary. Let η be a smooth (1,0)-tensor. Then we have:
Z
M
divg(η)dvg = 0.
in particular, given u, v ∈ C∞(M), we have
Z
M
u∆gv dvg =
Z
Mh∇
u,∇vig dvg
Z
M
(∆gu)v dvg.
1.3
Homogeneous Functions
Consider G : M × Rk → R a continuous function and 2-homogeneous in the second variable class C1 in the second variable. For example:
G(x, t) = k
X
i,j=1
Aij(x)titj,
where t= (t1, . . . , tk) ∈Rk and A= (Aij)M →Mks(R) is continuous, such that (Aij(x)) is positive defined, for all x ∈ M, and Ms
k(R) is the space of real symmetric matrices
k×k.
LetF :Rk→Rbe a positive function classC1and 2#-homogeneous1, where 2#= 2n n−4.
For example:
F(t) = k
X
i=1 |ti|2
#
,
where t= (t1, . . . , tk). In this case,
∂iF(t) =|ti|2
#−2
ti.
Now let F : Rk → R be a continuous function and q-homogeneous. Consider the unitary sphere in the norm p, that is:
∂Bp[0,1] =
t∈Rk; |t|p = 1 , where |t|p = (|t1|p+· · ·+|tk|p)
1
p being t = (t
1, . . . , tk). As the set ∂Bp[0,1] is compact, then there are constants mF,p >0 e MF,p >0 such that:
mF,p ≤F(t)≤MF,p ∀t∈∂Bp[0,1] .
Now consider any t∈Rk\ {0}. Then, by the q-homogeneity of F, we have
F
t
|t|p
= 1
|t|qp
F(t),
for all t ∈Rk\ {0}. Thus,
mF,p|t|qp ≤F(t)≤MF,p|t|qp ∀t∈R k
. (1.4)
As the space Rk has finite dimension, then all norms are equivalent. Therefore, there exists a constant c >0 such that
1
F(λt) =λ2#
|t|p ≤c|t|q ∀t∈Rk.
then
F(t)≤MF,p′ |t|q
q ∀t∈R k,
whereM′
F,p =cqMF,p is a positive constant.
In this work, in most situations, we use that F is a 2#-homogeneous function.
There-fore, this case will simplify the notation where F is a 2#-homogeneous function and the
norm is Euclidean: mF,2 =mF e MF,2 =MF. Then
mF|t|2
#
2 ≤F(t)≤MF|t|2
#
2 ∀t∈Rk.
Similarly, if G: M×Rk →R is a q-homogeneous function in the second variable, we have:
mG,p|t|qp ≤G(x, t)≤MG,p|t|qp ∀t∈R k
.
In this work, we use, unless the contrary that G is a 2-homogeneous function on the second variable. Therefore, this case will simplify the notation where G is a 2-homogeneous function on the second variable and the norm is Euclidean: mG,2 =mG and
MG,2 =MG. Thereby, we have:
mG|t|22 ≤G(x, t)≤MG|t|22 ∀t ∈Rk. (1.5)
Let F :Rk → R be a C1 function q-homogeneous. In this work, we use the following
Euler identity:
k
X
i=1
∂iF(t)ti =q F(t).
Part of the vector theory of best constants does not follow directly from scalar theory. One of the differences is in relation to the nature of vector functions satisfying the condi-tions of homogeneity. In the case (k ≥2), there are examples of homogeneous functions they are just continuous. Note the following example. Consider
F(t) = |t|2µ# e G(x, t) = β(x)|t|2µ, where | · |µ is a µ-norm defined by |t|µ = P
k i=1|ti|µ
1
µ
for 1 ≤ µ < ∞ e |t|∞ = max{|t|i; i= 1, . . . , k}.
1.4
Sobolev Spaces of Vector Valued Maps of Second Order
We consider (M, g) a compact Riemannian manifold and dvg = dv(g) the Riemannian measure associated with the metric g. Given a function u: M →R class C∞(M) and k a integer, we denote ∇ku the k-nth covariant derivative ofu and|∇ku
|the norm of ∇ku, defined by
|∇ku|=gi1j1· · ·gikjk ∇ku
i1···ik ∇
k
uj
1···jk ,
where ∇ku
i1···ik denotes the components of∇u in a chart. However this definition does
not depend on the choice of the chart.
If (M, g) is a compact Riemannian manifold, the Sobolev spaceH2,2(M) is, by
defini-tion, the completion of C∞(M) em L2(M) by the norm
kuk′H2,2(M) =
2
X
j=0
Z
M|∇ j
u|2g dvg
1 2
,
wheredvg =dv(g) is Riemannian measure associated with the metric g. The space H2,2(M) is a Hilbert space, with norm:
kuk2H2,2(M) =
Z
M|
u|2dvg+
Z
M|∇
u|2dvg +
Z
M
(∆gu)2 dvg
Note that the normsk·k′
H2,2(M)andk·kH2,2(M)are equivalent. Indeed, by the
Bochner-Lichnerowitz-Weitzenb¨ock formula, we have that
Z
M
(∆gu)2 dvg =
Z
M|∇
2u
|2gdvg+
Z
M
Ricg (∇u)#,(∇u)#
dvg
for all u∈ H2,2(M). then, we have
k∇2uk22+k∇uk22+kuk22
= k∆guk22−
Z
M
Ricg (∇u)#,(∇u)#
dvg+k∇uk22+kuk22 ≤ |∆gu|22+Ck∇uk22+kuk22
for allu∈ H2,2(M). But, hence we have that there exists a positive constantC such that k · k′
H2,2(M)≤Ck · kH2,2(M). On the other hand, note that, for all function u∈ C∞(M):
(∆gu)2 ≤ |∇2u|2
The associate scalar product is defined by:
hu, vi =
Z
Mh
u, vig dvg+
Z
Mh∇
u,∇vigdvg+
Z
Mh∇
2u,
∇2vigdvg
=
2
X
j=0
Z
Mh∇ ju,
∇jv
igdvg.
As M is compact, the spaceH2,2(M) does not depend on a metric Riemannian. If M
is a compact manifold with two metrics g and ˜g, then exist a real number C > 1 such that, in any point of M:
1
Cg ≤ ˜g ≤ Cg .
These two inequalities we reagrd as inequalities between bilinear forms.
A linear operator T : E → F between two Banach spaces (generally, we use E =
H2,2(M) and F = Rk, with k
∈ N) is compact if, for any sequences (un)n∈N ∈ E
uniformly bounded in the norm of E, then exists u ∈ F and a subsequence (unk) such
that:
lim
n→∞T(unk) =u , strongly in F.
The space H2,2(M) is reflexive. Thus, all bounded sequence in H2,2(M) have weakly
convergent subsequences. And, if T : H2,2(M) → Rk is compact, then T takes limited sequences inH2,2(M) in sequences that have convergent subsequences inRk, for allk
∈N. The space H2,2(M) it is also a separable space. Therefore, every bouded sequence in
(H2,2(M))∗ has convergent subsequences (in the weak∗ topology).
Consider (Tn)n∈N ∈(H2
,2(M))∗ andT ∈ H2,2(M). We say that (Tn) converges weakly
for T if
lim
n→∞Tn(u) =T(u) for all u∈ H
2,2(M)∗ ,
or
Tn ⇀ T em H2,2(M)
∗ ,
where n→ ∞. (H2,2(M))∗ is the space of continuous linear forms of H2,2(M).
If M is a Riemannian manifold of dimension n≥5, then the immersion
H2,2(M)֒→Lq(M),
is compact for q∈ 1, 2n n−4
H2,2(M)⇀ Ln2−n4(M),
is continuous, but not compact. That is, there exists a constant C >0 such that
kuk
Ln2−n4(M) ≤ CkukH2,2(M).
We denote by Hk2,2(M) = H2,2 M,Rk the vector Sobolev space H2,2(M)× · · · ×
H2,2(M), that is
Hk2,2(M) = U = (u1,· · ·, uk);ui ∈ H2,2(M) for all i= 1,· · · , k . This space has the norm:
||U||Hk2,2(M)=
Z
M
(∆gU)2dvg+
Z
M |∇
gU|2+|U|2
dvg
1 2
,
whereU = (u1, . . . , uk) and,
Z
M|U|
2
2dvg = k
X
i=1
Z
M |
ui|2dvg,
Z
M |∇
gU|22dvg = k
X
i=1
Z
M|∇
gui|2dvg
Z
M
(∆gU)2dvg = k
X
i=1
Z
M
(∆gui)2dvg.
Also:
|U|p = (|u1|p+· · ·+|uk|p)
1
p .
To simplify, we use that:
|U|=|U|2.
The vector-valued Sobolev spaceHk2,2(M) has different properties due to the properties of the spaceH2,2(M). The space H2,2
k (M) is a Hilbert space and has the following scalar product:
hU,ViH2,2
k (M) =
k
X
i=1
hui, vii,
where h·,·i is the usual scalar product in H2,2(M) and U = (u
1,· · · , uk) and V =
(v1,· · · , vk).
The spaceHk2,2(M) is separable and reflexive. Thus, the unit ball ofHk2,2(M) is weakly compact. In other words, for any sequences (Un)n∈N ∈ H
2,2
kUnkH2,2
k (M) ≤C for all n∈
N,
there exists a subsequence (Uni)n∈N ∈ H
2,2
k (M) and exists U ∈H
2,2
k (M) such that
Uni ⇀ U , (1.6)
weakly in Hk2,2(M) when n→ ∞.
Let us return to the same bounded sequences (Un)n∈NinH
2,2
k (M). LetT :H
2,2
k (M)→
F be a compact operator andF a Banach space. Then the sequences (T(Un))n∈Nconverge
to T(U) (note the weak convergence of (1.6)). Furthermore, if Tn → T in Hk2,2(M)
∗
, then
Tn(Un)→T(U).
We define the spaces Lqk(M) = Lq(M,Rk), for each q with 1
≤ q < ∞, as the space
Lq(M)
× · · · × Lq(M). That is,
Lqk(M) = {U = (u1,· · · , uk); ui ∈ Lq(M), i= 1,· · · , k} , with norm
kUkLqk(M) =
k
X
i=1 kuikq
!1
q
,
where U = (u1, . . . , uk) and k · kp is the norm of Lp(M), defined by
kukp =
Z
M
|u|pdv g
1
p
.
To simplify, when there is no ambiguity,
kUkp
to indicate the norm of U = (u1, . . . , uk) in Lpk(M)
The Sobolev immersionHk2,2(M)֒→Lqk(M) is compact for 1≤q < n2−n4 and continuous ifq = 2n
n−4. So we say that 2∗ = 2n
n−4 is the critical exponent with respect to the immersion
1.5
Coercivity
Consider U = (u1, ..., uk) ∈ Hk2,2(M). We define the functional Φ : H
2,2
k (M) → R and Ψ = ΨG:Hk2,2(M)→R, respectively, by:
Φ(U) =
Z
M
F(U)dvg (1.7)
and
Ψ(U) = ΨG(U) =
Z
M
(∆gU)2 dvg+
Z
M
A (∇U)#,(∇U)# dvg +
Z
M
G(x, U)dvg
Z
M
F(U)dvg
2 2#
,
where A is as defined at the beginning (a sum of smooth (2,0)-tensors). We also define ˆ
Ψ :Hk2,2(M)→Rby ˆ
Ψ(U) =
Z
M
(∆gU)2 dvg+
Z
M
A (∇U)#,(∇U)# dvg+
Z
M
G(x, U)dvg. Thus,
Ψ(U) = ΨG(U) = Z Ψ(ˆ U) M
F(U)dvg
2 2#
. (1.8)
We denote by L2
k(M) the Sobolev space ofL2(M)×L2(M)× · · · ×L2(M) with norm:
kUk2L2
k(M) =
Z
M|
U|2dvg = k
X
i=1
Z
M|
ui|2dvg,
whereU = (u1,· · · , uk). Similarly, we define the space L2
#
k (M). Using the Sobolev immersions Hk2,2(M) ֒→ L2
k(M) and H
2,2
k (M) ֒→ L2
#
k (M) the functional ΨG and Φ are well defined.
Definition 1 (coercivity). We say that ˆΨ is coercive if exists α >0 such that:
Z
M
(∆gU)2+A (∇U)#,(∇U)#+G(x, U) dvg ≥ α
Z
M|
U|2dvg,
for all U ∈Hk2,2(M), where A and G:M ×Rk →R are as defined before.
We have the following proposition that show equivalent definitions of coercivity. Proposition 5. The definitions below are equivalent:
(ii) Exists α >0 such that:
Z
M
(∆gU)2+A (∇U)#,(∇U)#+G(x, U) dvg ≥ αkUk22#
≥ α M−
2 2#
F
Z
M
F(U)dvg
2 2#
,
for all U ∈ Hk2,2(M). (iii) Exists α >0 such that:
Z
M
(∆gU)2+A (∇U)#,(∇U)#+G(x, U) dvg ≥ αkUk2H2,2
k (M),
for all U ∈ Hk2,2(M). Proof. (iii)⇒(ii)
Follows from the immersion:
Hk2,2(M)֒→L2
#
k (M). (ii)⇒(i)
We obtain using the immersion:
L2k#(M)֒→L2k(M). (i)⇒(iii)
With (i) we have:
Z
M
(∆gU)2+A (∇U)#,(∇U)#+G(x, U) dvg ≥ αkUk2L2
k(M),
for some α > 0. Consider 0 < ε < 1, such that ε ≤ αα+k, where k > 0 is a constant presented below. We have:
L(U) :=
Z
M
(∆gU)2+A (∇U)#,(∇U)#+G(x, U) dv
g, (1.9)
is equal to
L(U) =εL(U) + (1−ε)L(U).
Hence,
L(U)≥ε
Z
M
(∆gU)2+A (∇U)#,(∇U)#+G(x, U) dvg+ (1−ε)α
Z
M |
Using the homogeneity of Gand that A is bounded, we have:
L(U)≥ε
Z
M
(∆gU)2+c|∇U|2+mG|U|2
dvg+ (1−ε)α
Z
M|
U|2dvg.
Consider ε0 = min{ε, εc, εmG}, that is, ε0 =rε, wherer = 1, c, or mG. Then
L(U)≥ε0
Z
M
(∆gU)2+|∇U|2+|U|2 dvg+ (1−ε)α
Z
M |
U|2dvg,
take (1−ε)α ≥ε0. Hence:
Z
M
(∆gU)2+A (∇U)#,(∇U)#+G(x, U) dvg ≥ ε0kUk2H2,2
k (M)
1.6
The Scalar AB Program
In Rn, there exists a constant A >0 such thatkuk2L2#(Rn) ≤A
Z
Rn
(∆ξu)2 dx , (1.10)
for all u∈ C∞
c (Rn) (the set of smooth functions with compact support in Rn) and where
ξ is the Euclidean metric ofRn.
Let A0 =A0(n) be the sharp constant in the Sobolev inequality (1.10). That is, A0
is the smallest constant satisfying (1.10). We define:
1
A2 0
= inf
u∈D22(Rn)\{0}
Z
Rn
(∆ξu)2 dx
Z
Rn|
u|2# dx
2 2#
, (1.11)
where D2
2(Rn) is is the completion of the Cc∞(Rn) com a norma kukD22(Rn) :=k∆ξuk2.
Follows from the Sobolev immersion theorem that the constant A0 >0 is well defined.
This constant was calculated by Lieb [25], Lions [26], Edmunds-Fortunato-Jannelli [14] and Swason [37]. We have:
1
A2 0
= n(n
2−4)(n−4)wn4
n
16 ,
where wn is the volume of the canonical unit sphereSn in Rn+1.
Moreover, the extremal of the sharp inequality, that is, functions in D2
2(Rn) that
achieve the infimum in (1.11) are known and they are of the form:
uλ,µ,x0(x) = µ
λ λ2 +|x−x
0|2
n−4 2
,
where µ6= 0, λ >0 andx0 ∈Rn are arbitrary.
The following Sobolev immersion is continuous, but not compact. If (M, g) is a Rie-mannian manifold of dimension n ≥ 5 then H2,2(M) ֒→ L2#
(M) continuously, where 2#= 2n
n−4. That is, exists A >0 such that:
kukL2#(M) ≤ AkukH2,2(M).
Further, from the continuous immersionH2,2(M)֒
→ L2#(M), then there are constants
A, B >0 such that:
kuk2
2# ≤Ak∆guk22+Bkuk2H1,2(M) , ∀u∈ H2,2(M), (1.12)
kuk2H1,2(M) =k∇uk22+kuk22.
We are interested in the sharp constantA, Bfrom the inequality above. More precisely, we are interested in taking A minimized.
Easily see that A≥ A2
0. Moreover, it holds true, for all ε >0 exists Bε >0 such that, for any u∈ H2,2(M) we have:
Z
M|
u|2#dvg
2 2#
≤(A20+ε)
Z
M
(∆gu)2 dvg
+Bε
Z
M |∇
gu|2+|u|2
dvg. (1.13)
In particular, we define
Kn= inf
A ∈R; exists B such that the above inequality (1.12) is valid ∀u∈ H2,2(M) , (1.14) then Kn=A20 for all manifold (M, g). A question that naturally comes is the following:
• The infimum Kn is achieved?
Or, equivalently, we can take ε = 0 in (1.13)?A positive response was given in 2000 by Djadli-Hebey-Ledoux [10] with the restriction that the metric g is conformally flat. Then in 2003, Emmanuel Hebey proved the result for any Riemannian manifold [16]. Specifically:
Theorem 6 (E. Hebey, 2003). Let (M, g) be a compact manifold of dimension n ≥ 5. Then exists B > 0, which depends on the manifold and the metric g, such that, for all u∈ H2,2(M)
Z
M|
u|2# dvg
2 2#
≤ A20
Z
M
(∆gu)2 dvg+B
Z
M |∇
gu|2+|u|2
dvg.
In particular, the infimum is achieved in (1.12).
Let (M, g) be a compact Riemannian manifold of dimension n ≥ 5. We denote
H2,2(M) as the completion of the space C∞(M) in the norm:
kuk2H2,2(M) =
Z
M
(∆gu)2 dvg+
Z
M |∇
gu|2 +|u|2
dvg.
As 2# = 2n
n−4, consider two positive functions f ∈ C∞(M) and h ∈ C
0(M), exist two
positive constantesA, B such that, for any u∈ H2,2(M):
Z
M |
u|2#dvg
2 2#
≤A
Z
M
(∆gu)2 dvg +B
Z
M
f(x)|∇gu|2+h(x)|u|2
The first best Sobolev constant associated to the (1.15) is defined by:
A0(f, h, g) = inf{A∈ R; exists B ∈ R such that (1.15) is valid} .
The first sharp Riemannian Sobolev inequality asserts that, for any u∈ H2,2(M),
Z
M|
u|2#dvg
2 2#
≤ A0(f, h, g)
Z
M
(∆gu)2 dvg +B
Z
M
f(x)|∇gu|2+h(x)|u|2
dvg,
(1.16) for some constant B ∈R.
The second sharp Sobolev constant associated to (1.15) is defined by:
B0(f, h, g) = inf{B ∈ R; (1.16) is valid} .
The second Sobolev Riemannian sharp inequality sates that, for any u∈ H2,2(M) we
have:
Z
M|
u|2# dvg
2 2#
≤ A0(f, h, g)
Z
M
(∆gu)2 dvg+ + B0(f, h, g)
Z
M
f(x)|∇gu|2+h(x)|u|2
dvg. (1.17)
A nonzero function u0 ∈ H2,2(M) is called extremal for the inequality (1.17) if
Z
M |
u|2#dvg
2 2#
=A0(f, h, g)
Z
M
(∆gu)2dvg+B0(f, h, g)
Z
M
f(x)|∇gu|2+h(x)|u|2
dvg.
The AB scalar program consists of several questions of interest involving the sharp constant A0(f, h, g) e B0(f, h, g), and the inequalities ´otimas (1.17) and (1.16). In what
follows, we will divide this program into parts: A program A and B program.
The A program consists of some problems involving A0(f, h, g) and the inequality
(1.16).
• question 1A: What is the exact value, or estimates ofA0(f, h, g)?
• question 2A: The inequality (1.16) is valid?
• question 3A: The validity of (1.16) implies some geometric obstruction?
The B program consists of some questions involving B0(f, h, g) and the inequality
(1.17):
• question 1B: What is the exact value, or estimates ofB0(f, h, g)?
• question 2B: B0(f, h, g) depends continuously on f and h in some topology? • question 3B: B0(f, h, g) depends continuously on the metric g in some topology? • question 4B: The inequality (1.17) has extremal function?
• question 5B: The set of the extremal functionsE(f, h, g) of L2#
-norm is compact in theC0 topology?
• question 6B: What is the role of geometry on these questions?
1.6.1 Partial Answers
As shown at the beginning of the previous section, we have answers only to the questions 1A and 2A.
There are no reponses for the questions of the scalar case in the B program. That is, no author has worked on these questions.
1.7
AB Vector Program
Let n ≥ 5 and k ≥ 1 be integer. We denote by Dk2,2(Rn) the Euclidean Sobolev vector space D2,2(Rn)
× · · · × D2,2(Rn) with norm
k∆UkD2,2
k (Rn) =
Z
Rn
(∆U)2 dx
1 2
,
where
U = (u1, . . . , uk)
and
Z
Rn
(∆U)2 dx= k
X
i=1
Z
Rn
(∆ui)2 dx .
Let F :Rk →R be a positive continuous function and 2#-homogeneous. In this case,
follows directly from (1.10) the existence of a constant A >0 such that
Z
Rn
F(U)dx
2 2#
≤ A
Z
Rn
(∆U)2 dx , (1.18)
for all U ∈ D2k,2(Rn).
The sharp Euclidean Sobolev constant A associated with the inequality (1.18) is:
A0(F, n) = inf{A ∈R; (1.18) is valid} .
The sharp Euclidean Sobolev vector inequality states that:
Z
Rn
F(U)dx
2 2#
≤ A0(F, n)
Z
Rn
(∆U)2 dx , (1.19)
for all U ∈ D2k,2(Rn).
A nonzero map U0 ∈ D2k,2(Rn) is called extremal of (1.19), if
Z
Rn
F(U)dx
2 2#
=A0(F, n)
Z
Rn
(∆U)2 dx . (1.20)
two basic questions related to (1.19) are:
(a) What is the exact value of A0(F, n)?
(b) (1.19) has extremal map?
Proposition 7. We have that
A0(F, n) =M
2 2#
F A0(n),
whereMF =maxSk−1
2 F andS
k−1
2 =
n
t∈Rk;Pk
i=1|ti|2 = 1
o
andA0(n)is given by (1.11).
Furthermore,U0 ∈ Dk2,2(R
n)is a extremal map of (1.19)if and only if,U
0 =t0u0 for some
t0 ∈ S2k−1 such that MF =F(t0) and some extremal function u0 ∈ D2,2(Rn) of (1.10).
Proof. From the 2#− homogeneity ofF:
F(t)≤ MF k
X
i=1 |ti|2
!2# 2
, ∀t∈Rk.
Thus, using the Minkowski inequality and by (1.10), we have for any U ∈ Dk2,2(Rn):
Z
M
F(U)dx
2 2# ≤ M 2 2# F Z M k X i=1 |ui|2
!2# 2 dvg 2 2# ≤ M 2 2# F k X i=1 Z M|
ui|2
# dvg 2 2# ≤ M 2 2#
F A0(n)
Z
Rn
k
X
i=1
(∆ui)2 dx=M
2 2#
F A0(n)
Z
Rn
(∆U)2 dx . (1.21)
Hence, we obtain that:
A0(F, n)≤M
2 2#
F A0(n).
now, by choosing U0 = t0u0 where t0 ∈ S2k−1 is such that MF = F(t0) and u0 ∈ D2,2(Rn) is a extremal function of (1.10), we have2
Z
Rn
F(U0)dx
2 2# = M 2 2# F Z
Rn|
u0|2#
dx 2 2# =M 2 2#
F A0(n)
Z
Rn
(∆u0)2 dx
= M
2 2#
F A0(n)
Z
Rn
k
X
i=1
∆ ti0u0
!2
dx
= M
2 2#
F A0(n)
Z
Rn
(∆U0)2 dx . (1.22)
A0(F, n)≥M
2 2#
F A0(n).
2
Note that, in this case:
(∆U0) 2
=
k
X
i=1
∆ ti0u0
2
Therefore,
A0(F, n) = M
2 2#
F A0(n).
We conclude also that the k-mapsU0 =t0u0, as constructed above, are extremals. We
affirm that all extremals (1.19) has this form. In fact, let U ∈ Dk2,2(Rn) be a extremal of (1.19). In this case, U satisfies (1.21) with equality instead of the three inequalities. But note that the second equality corresponds to the Minkowski inequality . This implies that exists t ∈ Rk in |t|2 = 1 and u ∈ D2,2(Rn) such that U = tu. And, concluding, by the first equality it follows that F(t) = MF and, from the third equality, u is extremal function of (1.10).
Let F : Rk → R be a positive continuous and 2#-homogeneous function, and G :
M ×Rk → R a positive continuous and 2-homogeneous function on the second variable. Follows from the continuous immersion H2,2(M) ֒→ L2#
(M) that exist two positive constants A and B such that
Z
M
F(U)dvg
2 2#
≤ A
Z
M
(∆gU)2dvg +
+ B
Z
M
A (∇gU)#,(∇gU)#
+G(x, U) dvg. (1.23) The first Sobolev best constant associated to (1.23) is defined as:
A0 =A0(A, F, G, g) = inf{A ∈R; exists B ∈R such that (1.23) is valid} .
The first sharp Riemannian Sobolev vector inequality states that, for anyU ∈Hk2,2(M), we have:
Z
M
F(U)dvg
2 2#
≤ A0
Z
M
(∆gU)2dvg+
+ B
Z
M
A (∇gU)#,(∇gU)#
+G(x, U) dvg, (1.24) for some constante B ∈R.
The second Sobolev best constant associated to (1.24) is defined as:
B0 =B0(A, F, G, g) = inf{B ∈R; (1.24) is valid } .
Z
M
F(U)dvg
2 2#
≤ A0
Z
M
(∆gU)2dvg + + B0
Z
M
(A (∇gU)#,(∇gU)#
+G(x, U) dvg. (1.25) This inequality is sharp in relation to the first and the second sharp Sobolev constant, in that where neither can be reduced.
A natural question arises: It is possible to achieve equality in (1.25)? A nonzero k-function U0 ∈Hk2,2(M) is called extremal of (1.25), if
Z
M
F(U0)dvg
2 2#
= A0
Z
M
(∆gU0)2dvg+ + B0
Z
M
(A (∇gU0)#,(∇gU)#
+G(x, U0)
dvg.
The AB vector program consists of several questions of interest involving the best constantsA0(A, F, G, g) and B0(A, F, G, g) and the sharp inequalities (1.24) AND (1.25).
This program is separated into two parts: the A program and the B program.
The A program is composed of the following questions involving A0(A, F, G, g) and
(1.24):
• question 1A: What is the exact value or estimates ofA0(A, F, G, g)?
• question 2A: The inequality (1.24) is valid?
• question 3A: The validity of (1.24) implies in some geometric obstruction?
• question 4A: A0(A, F, G, g) depends continuously ofF and Gin some topology? • question 5A: A0(A, F, G, g) depends continuously of g and A in some topology? • question 6A: What is the role of geometry in these questions?
The B program consists of the following questions involvingB0(A, F, G, g) and (1.25): • question 1B: What is the exact value or estimates ofB0(A, F, G, g)?
• question 2B: B0(A, F, G, g) depends continuously of F and G in some topology? • question 3B: B0(A, F, G, g) depends continuously on the metric g and A in some
topology?
