LOCAL GRADIENT ESTIMATES FOR CONFORMAL QUOTIENT
EQUATIONS
PENGFEI GUAN, CHANG-SHOU LIN, AND GUOFANG WANG
Abstract. In this paper, we extend the local gradient estimates established in [8] and [10] to
the conformal quotient equations. An existence of solutions of the conformal quotient equations
follows from the local gradient estimates and the paper [14] of Gursky and Viaclovsky.
Dans ce travail, nous démontrons les estimations locales du gradient pour des équations
de quotient issues de la géométrie conforme, qui généralisent les résultats dans [8] et [10].
En utilisant ces estimations et les résultats dûs à Gursky et Viaclovsky [14], nous prouvons
l’existence des solutions pour ces équations de quotient.
1. Introduction
Let (M, g0 ) be a compact connected smooth Riemannian manifold of dimension n ≥ 3, and
let [g0 ] be the conformal class of g0 . The Schouten tensor of a metric g is defined as
µ
¶
Rg
1
Sg =
Ricg −
·g ,
n−2
2(n − 1)
where Ricg and Rg are the Ricci tensor and scalar curvature of g respectively. Following Viaclovsky [19], σk -scalar curvatures of g is defined as σk (g) := σk (g −1 · Sg ), where σk is the k-th
elementary symmetric function and g −1 · Sg is locally defined by (g −1 · Sg )ij = g ik (Sg )kj . We
are interested in certain local a priori estimates of the following conformally invariant quotient
equation for g ∈ [g0 ],
(1)
σk (g)
= 1,
σl (g)
0 ≤ l < k ≤ n.
If g = e−2u g0 , the Schouten tensor of g can be computed as
∇2 u + du ⊗ du −
|∇u|2
g0 + Sg0 .
2
We are lead to consider the following equation:
µ
¶
σk
|∇u|2
2
(2)
∇ u + du ⊗ du −
g0 + Sg0 = f e−2(k−l)u , 0 ≤ l < k ≤ n,
σl
2
where f is a nonnegative function. Here we denote σ0 = 1.
When k = 1 and l = 0, equation (1) is the Yamabe equation. We refer to the work of
Aubin [1] and Schoen [17]. Equation (1) was introduced by Viacolovsky in [19] for 2 ≤ k ≤ n,
1991 Mathematics Subject Classification. Primary 53C21; Secondary 35J60, 58E11 .
Key words and phrases. conformal metric, fully nonlinear equation, local gradient estimates, quotient equation.
Research of the first author was supported in part by an NSERC Discovery Grant.
1
2
PENGFEI GUAN, CHANG-SHOU LIN, AND GUOFANG WANG
l = 0. Since then, there has been a lot of work for equation (1) in this case. We refer to
[2, 3, 8, 9, 11, 13, 16] and [14]. The study of geometric inequalities in [10] leads us naturally to
consider the general form of equation (1). To solve the existence problem, one needs to establish
some a priori estimates for the solutions of these equations. Since equation (1) is conformally
invariant, in general there are no global a priori estimates for solutions. A localized estimate for
solutions will be the best that one can expect. As in the Yamabe problem, the blow-up analysis
is important to rule out the non-compact case. This local estimate is a crucial step to carry on
the blow-up analysis.
The main purpose of this paper is to prove a local gradient estimate for the conformal quotient
equation (2). Once the local gradient estimate is established, a local C 2 estimate can follow easily.
When l = 0, these local estimates were proved in our paper [8]. The local gradient estimates for
conformal quotient equation were established for equation (2) in the range (n−k +1)(n−l+1) >
2(n + 1), and were used in a crucial way in the proof of some new geometric inequalities in [10].
Here we will remove the restriction (n − k + 1)(n − l + 1) > 2(n + 1) and establish the local
gradient estimates for the full range of 0 ≤ l < k ≤ n.
We use the standard notation as in [8]. Let
n
Γ+
k = {Λ = (λ1 , λ2 , · · · , λn ) ∈ R | σj (Λ) > 0, ∀j ≤ k}.
A metric g is said to be k-admissible if g −1 · Sg ∈ Γ+
k for every point x ∈ M . We will simply
+
−2u
denote g ∈ Γk . If g = e g0 , we say u is k-admissible if g ∈ Γ+
k.
Theorem 1. Suppose that f is a positive function on M . Let u ∈ C 3 (Br ) be an admissible
solution of (2) in Br , a geodesic ball of radius r in a Riemannian manifold (M, g0 ). Then, there
exists a constant c1 > 0 depending only on r, kg0 kC 3 (Br ) and kf kC 1 (Br ) (independent of inf f ),
such that
(3)
sup {|∇u|2 } ≤ c1 (1 + e−2 inf Br u ).
Br/2
If, in addition, u ∈ C 4 (Br ), then
(4)
sup {|∇2 u|} ≤ c2 (1 + e−2 inf Br u ),
Br/2
1
for some constant c2 depending only on r, kg0 kC 4 (Br ) and kf k−l kC 2 (Br ) .
From Theorem 1, the “blow-up” analysis usually for semilinear equations, for example, harmonic map equation, Yang-Mills equation and the Yamabe equation, works for (2). It is an
interesting phenomenon, since typical fully nonlinear equations do not admit such blow-up analysis.
Corollary 1. There exists a constant ε0 > 0 such that for any sequence of solutions ui of (2)
in B1 with
Z
(5)
e−nu dvol(g0 ) ≤ ε0 ,
B1
either
(1) There is a subsequence uil uniformly converges to +∞ in any compact subset in B1 , or
LOCAL GRADIENT ESTIMATES FOR QUOTIENT EQUATIONS IN CONFORMAL GEOMETRY
3
1,α
(B1 ), ∀0 < α < 1. If f is smooth
(2) There is a subsequence uil converges strongly in Cloc
m (B ), ∀m.
and strictly positive in B1 , then uil converges strongly in Cloc
1
Gursky-Viaclovsky solved equation (1) in [11] for k > n2 , l = 0 under a condition on a conformal invariant. Recently, in [14] they removed the condition on conformal invariant and extended
to more general conformal invariant equations with certain structure conditions. In particular,
they proved the existence of a solution to (1) in the case k > n2 , l = 0 (Theorem 1.1 in [14]), and
the case k > n2 , (n − k + 1)(n − l + 1) > 2(n + 1) (Corollary 1.4 in [14]). An application of the
local gradient estimates in Theorem 1 and Theorem 1.3 in [14] is
Corollary 2. If k > n2 , 0 ≤ l < k and g0 ∈ Γ+
k , then equation (1) has a k-admissible solution.
The proof of Corollary 1 follows the same lines of proof of Corollary 1.2 in [8], this type
argument was first used by Schoen in [18] for the Yamabe equation.
The main result of this paper was obtained and circulated in 2002. It was included in an
unpublished lecture notes [5] of the first named author. Since the recent appearance of the paper
by Gursky-Viaclovsky [14], we are encouraged to publish our result which is a compliment to
their results. Corollary 2 was added after the paper [14]. We would like to thank Jeff Viaclovsky
for enlightening discussions on the subject. We would also like to call attention to a very recent
paper by S. Chen [4] where she found a proof of local C 2 estimates for a class of conformally
invariant differential equations, including equations treated in this paper.
2. Local gradient estimates
If the local gradient estimate (3) is established, local C 2 estimate (4) in Theorem 1 can be
1
deduced easily along the same lines as in the proof of Lemma 3.2 in [6] since F = ( σσkl ) k−l is
elliptic and concave in Γ+
k . Therefore we omit the proof here. The proof for quotient equation (2)
given here is a little more involved than the proof in [8] for σk . We will make use of some special
algebraic properties of the elementary symmetric functions. These properties of the elementary
symmetric functions were also used in crucial way in other contexts recently in [15, 7].
For Λ = (λ1 , . . . , λn ) ∈ Rn , the k-th elementary symmetric function is defined as
X
λi1 · · · λik .
σk (Λ) =
i1 <···<ik
Set σ0 = 1 and σq = 0 for q > n. σk can be extended as a function on real symmetric
+
n × n matrices. A real symmetric matrix A is said to lie in Γ+
k if its eigenvalues lie in Γk .
We denote Λi = (λ1 , · · · , λ̌i , · · · , λn ) = (λ1 , λ2 , · · · , λi−1 , λi+1 , · · · , λn ) and Λij = (λ1 , · · · , λ̌i ,
· · · , λˇj , · · · , λn ) for i 6= j. Therefore, σq (Λi ) (σq (Λij ) resp.) means the sum of the terms of σq (Λ)
not containing the factor λi (λi and λj resp.). It is easy to compute that, ∀i,
(6)
F ii = F ∗ σl−1 (Λi ){σl (Λ)
σk−1 (Λi )
− σk (Λ)},
σl−1 (Λi )
³
´ 1 −1
σk (Λ) k−l
1
1
where F ∗ = k−l
.
σl (Λ)
σl2 (Λ)
The following lemma will be used in our proof of the Claim.
4
PENGFEI GUAN, CHANG-SHOU LIN, AND GUOFANG WANG
Lemma 1. Let F (W ) =
∂F (W )
∂wij .
σk (W )
σl (W )
ij =
be defined on symmetric matrices with W ∈ Γ+
k , and let F
Suppose that W is diagonal, and denote wii by λi , ∀i = 1, · · · , n. Then
F ii ≤ F jj ,
if λi ≥ λj .
If, in addition, Λij ∈ Γ+
k−1 , then
F ii λ2i ≥ F jj λ2j ,
if λi ≥ λj ≥ 0.
Proof: The first statement follows from (6) and the monotonicity of σl−1 and
F ii λ2i
≥
check
m = 1, · · · , n,
(7)
F jj λ2j ,
under the condition that Λij ∈
Γ+
k−1 .
σk−1
σl−1 .
We now
It is easy to check that for any
σm (Λi ) = σm (Λij ) + λj σm−1 (Λij ),
σm (Λ) = σm (Λij ) + (λi + λj )σm−1 (Λij ) + λi λj σm−2 (Λij ).
By (7), we compute
F ii λ2i − F jj λ2j = (λ2i − λ2j )[σl (Λij )σk−1 (Λij ) − σk (Λij )σl−1 (Λij )]
+(λi − λj )λi λj [σl (Λij )σk−2 (Λij ) − σk (Λij )σl−2 (Λij )].
As Λij ∈ Γ+
k−1 , both terms in [· · · ] are positive by the Newton-MacLaurin inequality.
Proof of local gradient estimates in Theorem 1.
The first part of proof will follow from the
same lines in [8] to deduce it to Claim (17). The verification of Claim (17) will be carried out
in the next section.
We may assume r = 1. Let ρ be a test function ρ ∈ C0∞ (B1 ) such that
ρ ≥ 0,
(8)
in B1 ,
|∇ρ(x)| ≤ 200ρ1/2 (x),
ρ = 1,
in B1/2 ,
|∇2 ρ| ≤ 100,
in B1 .
Set H = ρ|∇u|2 , we estimate the maximum of H. Assume that H achieves its maximum
at x0 . After an appropriate choice of the normal frame at x0 , we may assume that W =
2
(uij + ui uj − |∇u|
2 δij + Sij ) is diagonal at the point, where ui and uij are the first order and
second order covariant derivatives respectively. Let wij be the entries of W , and let Sij be entries
of Sg0 . We have at x0 ,
(9)
wii = uii + u2i − 21 |∇u|2 + Sii ,
uij = −ui uj − Sij ,
By the choice of the test function ρ and Hi (x0 ) = 0, we have at x0
(10)
|
n
X
l=1
uil ul | ≤ 100ρ−1/2 |∇u|2 .
∀i 6= j.
LOCAL GRADIENT ESTIMATES FOR QUOTIENT EQUATIONS IN CONFORMAL GEOMETRY
5
1
2
We may assume that H(x0 ) ≥ 104 A20 , that is ρ−1/2 ≤ 100A
|∇u|, and |Sg0 | ≤ A−1
0 |∇u| for some
0
constant A0 to be chosen later, otherwise we are done. Thus, from (10) we have
n
X
|∇u|3
(11)
|
uil ul | ≤
(x0 ).
A0
l=1
We denote λi = wii and Λ = (λ1 , λ2 , · · · , λn ). In what follows, we denote C (which may vary
from line to line) as a constant depending only on kf kC 1 (B1 ) , k, n, and kg0 kC 3 (B1 ) . Since F is
elliptic in Γ+
k and Hij is non-positive definite at x0 , we have
½µ
¶
¾
ρi ρj
ij
ij
2
(12)
0 ≥ F Hij = F
−2
+ ρij |∇u| + 2ρulij ul + 2ρuil ujl .
ρ
P
The first term in (12) is bounded from below by 105 i≥1 F ii |∇u|2 . By interchanging covariant
derivatives, the second term in (12) can be estimated as follows,
X
X
X
F ij uijl ul ≥
F ij uijl ul − C|∇u|2
F ii
i,j,l
i
i,j,l
X
X
|∇u|2
{F ij (wij )l ul − F ij (ui uj −
δij )l ul } − C|∇u|2
F ii
2
i
i,j,l
X
X
X
X
F ii ukl uk ul − C|∇u|2
F ii
F ij uil uj ul +
Fl ul − 2
=
=
(13)
=
i,k,l
i,j,l
l
X
e−2u (fl ul − 2f |∇u|2 ) − 2
X
F ii uil ul ui +
X
≥ −C(1 + e−2u )|∇u|2 −
X
i
F ii
F ii uil ul ui − C|∇u|2
X
i,l
i,l
l
i
F ii
i
|∇u|4
.
A0
To obtain the local estimates, we need the following Lemma.
Lemma 2. There is constant A0 depending only on k, n, and kg0 kC 3 (B1 ) , such that,
X
X
−3
(14)
F ij uil ujl ≥ A0 4 |∇u|4
F ii .
i,j,l
i≥1
P
Assuming the lemma, local gradient estimate (3) can be proved as follows. As i F ii ≥ 1,
inequalities (12), (13) and (14) yield
µ
¶
X
X
(n + 2)2
− 34
5
2
jj
−2u
2
0 ≥ −10 |∇u|
F − Ce ρ|∇u| + −
+ A0
ρ|∇u|4
F jj
A0
j
j
(15)
½
µ
¶
¾
X
(n + 2)2
− 34
jj
5
2
−2 inf u
2
4
+ A0
≥
F
−10 n|∇u| − Ce
|∇u| + −
ρ|∇u| .
A0
j
Choosing A0 large enough so that A0 > 2((n + 2)2 )4 and multiplying (15) by ρ, we get
H 2 ≤ C(1 + e−2 inf u )H,
6
PENGFEI GUAN, CHANG-SHOU LIN, AND GUOFANG WANG
thus
|∇u(x)|2 ≤ C(1 + e−2 inf x∈B1 u ) for x ∈ B1/2 .
Therefore (3) of Theorem 1 is proved, assuming Lemma 2.
Proof of Lemma 2. Set ũij = uij + Sij , we estimate that,
X
X
1 X ii 2
1
(16)
F ij uil ujl ≥
F ũil − C 2 |∇u|4
F ii .
2
A0
i
i,j,l
i,l
Hence, to prove the Lemma we only need to check the following
Claim: There is a constant A0 depending only on k, n, and kg0 kC 3 (B1 ) , such that,
X
− 5 X ii
F |∇u|4 .
(17)
F ii ũ2il ≥ A0 8
i
i,l
From (9), the left hand side can be expressed as
P
P ii 2 P
ii 2
ii 2 2
i,l F ũil
i F ũii +
i6=l F ui ul
(18)
=
X
i
©
ª X ii 2
|∇u|4
).
F ii ũ2ii + u2i (|∇u|2 − u2i ) =
F (λi − 2u2i λi + λi |∇u|2 +
4
i
3. Local gradient estimates, continued
We verify the Claim in this section.
Proof of Claim. Set I = {1, 2, · · · , n}. Recall that at x0 , by (11), we have for any i ∈ I,
X
¡
¢ X
1
uil ul | ≤
Sil ul | = |
|∇u|3 .
|ui uii − (|∇u|2 − u2i ) −
A0
l
l
This implies
¡
¢
2
|∇u|3 .
|ui uii − (|∇u|2 − u2i ) | ≤
A0
(19)
−1/4
Set δ0 = A0 . Sometimes, for simplicity of notation, we denote Wii by λi . We divide I into
three subsets I1 , I2 and I3 by
I1 = {i ∈ I | u2i ≥ δ0 |∇u|2 },
I2 = {i ∈ I | u2i < δ0 |∇u|2 & ũ2ii ≥ δ02 |∇u|4 }
and
I3 = {i ∈ I | u2i < δ0 |∇u|2 & ũ2ii < δ02 |∇u|4 }
For any i ∈ I1 , by (19) we can deduce that
¯
¯
¯
|∇u|2 ¯¯
3
2
2
2
¯
(20)
¯λi − 2 ¯ < 2δ0 |∇u| < 2δ0 |∇u| .
For any j ∈ I3 , since λj = ũjj + u2j − |∇u|2 /2, we have
¯
¯
¯
|∇u|2 ¯¯
− 14
2
2
¯
(21)
¯λj + 2 ¯ < 2δ0 |∇u| = 2A0 |∇u| .
LOCAL GRADIENT ESTIMATES FOR QUOTIENT EQUATIONS IN CONFORMAL GEOMETRY
7
In particular, λi > 0 if i ∈ I1 and λj < 0 if j ∈ I3 , for large small δ0 .
We verify the Claim (17) by dividing into two cases.
Case 1. |I3 | = 0.
First we note that this case includes the case k = n. If ũ2ii + u2i (|∇u|2 − u2i ) ≥ δ02 |∇u|4 for all
i ∈ I, the Claim follows from (18) easily. Therefore we may assume that there is i0 such that
ũ2i0 i0 ≤ δ02 |∇u|4 . Recall that ũii = uii + Sii . Since I3 = 0, we have i0 ∈ I1 . Thus,
ũ2i0 i0 ≤ δ02 |∇u|4
(22)
and
u2i0 ≥ δ0 |∇u|2 .
Assume that i0 = 1. By (19) we have u21 ≥ (1 − 2δ0 )|∇u|2 and λ1 > 0. Now it is clear that
(|∇u|2 − u2j ) ≥ (1 − 2δ0 )|∇u|2 for all j > 1, and there is no other j ∈ I, j 6= 1 satisfying (22) if
A0 is large enough. Hence, for any j > 1, ũjj ≥ δ02 |∇u|4 , which implies
ũ2jj + u2j (|∇u|2 − u2j ) ≥ δ02 |∇u|4
(23)
for any j > 1.
If there is j0 ≥ 2 such that λj0 ≤ λ1 , by Lemma 1 we have F j0 j0 ≥ F 11 . By (23)
X
F ii ũ2il
≥
δ02 |∇u|4
n
X
i=2
i,l
n
X
1
F ≥ δ02 |∇u|4
F ii .
2
ii
i=1
Hence, we may assume that λj ≥ λ1 for any j ≥ 2. It follows that Λ = (λ1 , λ2 , · · · , λn ) ∈ Γ+
n.
By Lemma 1 we have F jj λ2j ≥ F 11 λ2i for any j ≥ 2. And we have |∇u|2 − 2u2j ≥ 0 for any j ≥ 2.
Note that λ1 ≥ ( 21 − 2δ02 )|∇u|2 by (20), altogether we have
X
i,l
F ii ũ2il ≥
n
X
F jj (λ2j +
j=2
n
n
n
j=2
j=1
j=1
X
X
X
λ2j
|∇u|2
1
)≥
F jj λ2j ≥
F jj
≥ ( − δ02 )2
F jj |∇u|2 .
4
2
4
Case 2. |I3 | 6= 0.
By (21), for j ∈ I3 we have
1
λ2j − 2uj λj ≥ ( − 2δ0 )|∇u|4 .
4
(24)
For j ∈ I2 , it is clear that λ2j − 2u2j λj = (λj − u2j )2 − u4j ≥ −δ02 |∇u|4 . Set F̃ 1 = maxi∈I1 F ii , we
have
X
X
(25)
F jj (λ2j − 2u2j λj ) ≥ −δ02 |∇u|4
F ii .
i
j∈I2
Observation: The Claim is true if
δ0 .
P
j∈I2 ∪I3
F jj ≥ (1 + c0 )F̃ 1 for some c0 > 0 independent of
The Observation follows from (19)–(25), since
8
PENGFEI GUAN, CHANG-SHOU LIN, AND GUOFANG WANG
X
F ii ũ2il =
X
i
i,l
≥ (
X
+
i∈I1
≥
(26)
X
X
+
i∈I2
µ
F ii
X
)F ii (λ2i − 2u2i λi ) +
i∈I3
|∇u|4
4
i∈I1
≥ F̃ 1
F ii (λ2i − 2u2i λi + λi |∇u|2 +
− 2u2i
|∇u|2
|∇u|4 X ii
F
4
i
¶
2
|∇u|4
)
4
X
+
F jj
|∇u|4
|∇u|4 X ii
+ (1 − 32δ02 )
F
4
4
i
j∈I3
X
|∇u|4
|∇u|4
|∇u|4 X ii
F ,
− F̃ 1 |∇u|4 +
F jj
+ (1 − 32δ02 )
4
4
4
i
j∈I3
≥ −F̃ 1
|∇u|4
2
+ (1 − 32δ02 )
|∇u|4
4
X
i
1
|∇u|4 X ii
F ii ≥ ( c0 − 32δ02 )
F .
2
4
i
We note that, if |I3 | ≥ 2, (20) and (21) imply that for any i ∈ I1 and j ∈ I3 we have λi > λj .
P
So F ii ≤ F jj by Lemma 1. Hence j∈I3 F jj ≥ |I3 |F̃ 1 ≥ 2F̃ 1 and the Claim follows from the
Observation. Therefore in the rest of proof, we may assume |I3 | = 1 and may take I3 = {n} .
We divide it into three subcases.
Subcase 2.1. |I3 | = 1, |I1 | ≥ 2.
P
Since F̃ 1 ≤ F nn , we may assume that F jj ≤ 12 F̃ 1 for any j ∈ I2 . Otherwise, j∈I2 ∪I3 F jj ≥
3 1
jj ≤ F̃ 1 implies
2 F̃ and the Claim is true by the Observation. From Lemma 1 and (20), F
that λj ≥ inf i∈I1 λi ≥ ( 21 − 2δ02 )|∇u|2 . It is clear to see that u2i ≤ (1 − δ0 )|∇u|2 , for |I1 | ≥ 2. By
the Observation we may assume F nn ≤ 2F̃ 1 . From these facts, together with (20) and (25),
we estimate
X
F ii ũ2il ≥
n−1
X
F ii (λ2i − 2u2i λi + λi |∇u|2 +
i=1
i,l
≥
X
F
ii
(λ2i
−
2u2i λi
n−1
|∇u|4 X ii
+ λi |∇u| ) +
F
4
2
i=1
i∈I1
≥ |∇u|4 {
n−1
3 X ii X ii
1 X ii 32δ02 X ii
F −
F (1 − δ0 ) +
F −
F }
4
4
4
i∈I1
≥
i=1
i∈I1
n−1
1 |∇u|4 X ii
|∇u|4 X ii
δ0
F − 32δ02
F
2
4
4
i=1
≥
|∇u|4
)
4
1
δ0
4
n−1
|∇u|4 X
4
i=1
i
1
F ii ≥ δ0
8
|∇u|4
4
n
X
i=1
F ii .
i
LOCAL GRADIENT ESTIMATES FOR QUOTIENT EQUATIONS IN CONFORMAL GEOMETRY
9
Subcase 2.2. |I3 | = 1, |I1 | = 1 and k ≤ n − 2.
In this subcase, I2 = {2, 3, · · · , n − 1}. As in Subcase 2.1, we may assume that λj ≥ λ1 for
any j ∈ I2 . First we assume that there is a j0 ∈ I2 such that Λ1j0 ∈ Γ+
k−1 . By Lemma 1, we
j
j
2
11
2
0
0
λj0 ≥ F λ1 .
have F
Using (20) and (21), we compute
X
F ii ũ2il ≥
X
F ii (λ2i − 2u2i λi + λi |∇u|2 +
i
i,l
n−1
≥ F
11
(λ21
n
X
1 X ii 2
|∇u|4
− 2|∇u| λ1 ) +
F λi + F nn λ2n +
F ii
+ F |∇u|2
2
4
2
j=2
i=1
3
1
|∇u|4
≥ − F 11 |∇u|4 + F j0 j0 λ2j0 + F nn
+
4
2
4
≥
|∇u|4
)
4
X
X
1
|∇u|4
F ii − 32δ02 |∇u|4
F ii .
8
i
n
X
i=1
F ii
X
|∇u|4
− 32δ02 |∇u|4
F ii
4
i
i
So the Claim will follow if we pick A0 large enough.
Hence, we may assume that for any j ∈ I2 , σk−1 (Λ1j ) ≤ 0. From this fact, we want to show
that
(27)
σk−1 (Λ1n ) ≤
n−2
(λ1 + |λn |)σk−2 (Λ1n ).
n−k−1
Assume that λ2 = minj∈I2 λj . From
0 ≥ σk−1 (Λ12 ) = σk−1 (Λ12n ) + λn σk−2 (Λ12n ),
we have σk−1 (Λ12n ) ≤ |λn |σk−2 (Λ12n ). (Recall that λn < 0.) As 0 < λ2 ≤ λj for any 3 ≤ j ≤
n − 1, by counting the terms, it’s easy to see that
σk−1 (Λ1n ) ≤
n−2
σk−1 (Λ12n ).
n−k−1
Altogether gives that
σk−1 (Λ1n ) ≤
≤
n−2
n−2
σk−1 (Λ1n ) ≤
|λn |σk−2 (Λ1n )
n−k−1
n−k−1
n−2
(λ1 + |λn |)σk−2 (Λ1n ).
n−k−1
We now want to make use of (27). By (7) we have
F 11 = F ∗ {[σl (Λ1n )σk−1 (Λ1n ) − σl−1 (Λ1n )σk (Λ1n )]
(28)
+λn [σl (Λ1n )σk−2 (Λ1n ) − σk (Λ1n )σl−2 (Λ1n )]
+λ2n [σl−1 (Λ1n )σk−2 (Λ1n ) − σk−1 (Λ1n )σl−2 (Λ1n )]}.
10
PENGFEI GUAN, CHANG-SHOU LIN, AND GUOFANG WANG
We have a similar expansion for F nn . Hence, we obtain
(29)
F nn − F 11 = F ∗ (λ1 − λn ){[σl (Λ1n )σk−2 (Λ1n ) − σk (Λ1n )σl−2 (Λ1n )]
+(λ1 + λn )[σl−1 (Λ1n )σk−2 (Λ1n ) − σk−1 (Λ1n )σl−2 (Λ1n )]}.
By the Newton-MacLaurin inequality, there is C1 > 0 depending only on n, k and l, such that
(30)
σl (Λ1n )σk−2 (Λ1n ) − σk (Λ1n )σl−2 (Λ1n ) ≥ C1 σl (Λ1n )σk−2 (Λ1n ),
σl−1 (Λ1n )σk−2 (Λ1n ) ≥ σk−1 (Λ1n )σl−2 (Λ1n ).
l−1
l−1
Since λ1 + λn ≤ 4δ0 |∇u|2 ≤ 2δ0 λ2 and σl−1 (Λ1n )λ2 ≤ Cn−2
σl (Λ1n ), where Cn−1
is the binomial
constant. Combining this fact with (30), if δ0 > 0 small enough, we have
(31)
(λ1 + λn )[σl−1 (Λ1n )σk−2 (Λ1n ) − σk−1 (Λ1n )σl−2 (Λ1n )] ≥ −
C1
σl (Λ1n )σk−2 (Λ1n ).
2
In view of (29) and (27), if δ0 > 0 small enough, we get
F nn − F 11 ≥
≥
C1 ∗
4 F (λ1
− λn )σl (Λ1n )σk−2 (Λ1n )
(n − k − 1)C1 ∗
F σl (Λ1n )σk−1 (Λ1n ) ≥ C2 F 11 ,
4(n − 2)
where the last inequality follows from the expansion (28) of F 11 , the fact that λn < 0 and
l−1
λ2n σl−1 (Λ1n ) ≤ 2λ22 σl−1 (Λ1n ) ≤ 2Cn−2
σl (Λ1n ). Hence, we have F nn ≥ (1 + C2 )F 11 and the
Claim follows from the Observation.
Subcase 2.3 |I3 | = 1, |I1 | = 1 and k = n − 1.
Again, we may assume that λj ≥ λ1 for any 2 ≤ j ≤ n − 1. Note that 2u2j ≤ |∇u|2 for any
2 ≤ j ≤ n − 1. Also as in Subcase 2.2, if δ0 > 0 is small enough,
(32)
(l + 1)σl+1 (Λ1n ) + (λ1 + λn )lσl (Λ1n ) ≥ 0.
LOCAL GRADIENT ESTIMATES FOR QUOTIENT EQUATIONS IN CONFORMAL GEOMETRY
11
It follows that
n−1
X
(σl (Λ)σk−1 (Λj ) − σk (Λ)λl−1 (Λj ))λ2j
j=2
=
n−1
X
λ2j {σl (Λ)(σn−2 (Λ1jn ) + (λ1 + λn )σn−3 (Λ1jn ) + λ1 λn σn−4 (Λ1jn ))
j=2
−σn−1 (Λ)(σl−1 (Λ1jn ) + (λ1 + λn )σl−2 (Λ1jn ) + λ1 λn σl−3 (Λ1jn ))}
= σl (Λ){[σk (Λ1n )σ1 (Λ1n ) − (k + 1)σk+1 (Λ1n )] + (λ1 + λn )[σk−1 (Λ1n )σ1 (Λ1n )
(33)
−kσk (Λ1n )] + λ1 λn [σk−2 (Λ1n )σ1 (Λ1n ) − (k − 1)σk−1 (Λ1n )]}
−σk (Λ){[σl (Λ1n )σ1 (Λ1n ) − (l + 1)σl+1 (Λ1n )] + (λ1 + λn )[σl−1 (Λ1n )σ1 (Λ1n )
−lσl (Λ1n )] + λ1 λn [σl−2 (Λ1n )σ1 (Λ1n ) − (l − 1)σl−1 (Λ1n )]}
= −(n − 2)λ1 λn σn−2 (Λ1n )σl (Λ)
+σn−1 (Λ)[(l + 1)σl+1 (Λ1n ) + (λ1 + λn )lσl (Λ1n ) + (l − 1)λ1 λn σl−1 (Λ1n )]
≥ λ1 |λn |{(n − 2)σl (Λ)σn−2 (Λ1n ) − (l − 1)σn−1 (Λ)σl−1 (Λ1n )}
≥ λ1 |λn |(n − l − 1)σl (Λ)σn−2 (Λ1n ).
From (33), we get
X
F ii ũ2il
F jj (λ2j − 2u2j λj + λj |∇u|2 +
j=1
i,l
≥
n
X
n−1
X
F jj (λ2j − 2u2j λj + λj |∇u|2 +
j=2
(34)
≥
n−1
X
F jj λ2j = F ∗
j=2
|∇u|4
)
4
|∇u|4
)
4
n−1
X
(σl (Λ)σk−1 (Λj ) − σk (Λ)λl−1 (Λj ))λ2j
j=2
≥ F ∗ λ1 |λn |(n − l − 1)σl (Λ)σn−2 (Λ1n )
1
≥ F ∗ ( − 2δ0 )|∇u|4 σl (Λ)σn−2 (Λ1n ).
4
1
Since λj ≥ λ1 for any j = 2, 3, · · · , n − 1, it is easy to see that σn−2 (Λ1n ) ≥ n−1
σn−2 (Λj ) for
P
1
∗
ii
any j = 1, 2, · · · , n. It follows that F σl (Λ)σn−2 (Λ1n ) ≥ (n−1)n i F . Hence, (34) implies
X
j,l
The proof is complete.
F ii ũ2il ≥
X
1
1
( − 2δ0 )|∇u|4
F ii .
(n − 1)n 4
i
12
PENGFEI GUAN, CHANG-SHOU LIN, AND GUOFANG WANG
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LOCAL GRADIENT ESTIMATES FOR QUOTIENT EQUATIONS IN CONFORMAL GEOMETRY
13
Department of Mathematics and Statistics, McGill University, Montreal, H3A 2K6 Canada.
E-mail address: [email protected]
Department of Mathematics, National Chung-Cheng University, Minghsiung, Chia-Yi, Taiwan
E-mail address: [email protected]
Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany
E-mail address: [email protected]