We prove an a priori estimate of type sup*inf on Riemannian manifold of
dimension 3 (not necessarily compact).Comment: 9 page

Bahoura, Samy Skander

English

arXiv

9 pagesWe prove an a priori estimate of type sup*inf on Riemannian manifold of dimension 3 (not necessarily compact)

Hal-Diderot

Sup-Inf inequality on manifold of dimension 3
Samy Skander Bahoura
To cite this version:
Samy Skander Bahoura. Sup-Inf inequality on manifold of dimension 3. 9 pages. 2006. <hal-
00023382>
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SUP-INF INEQUALITY ON MANIFOLD OF DIMENSION 3
SAMY SKANDER BAHOURA
ABSTRACT. We give an estimate of type sup× inf on Riemannian manifold of Dimension 3 for
prescribed curvature equation.
INTRODUCTION AND RESULTS.
We are on Riemannian manifold (M, g) of dimension 3 not necessary compact. In this paper
we denote ∆ = −∇j(∇j) the geometric laplacian.
The scalar curvature equation is:
8∆u+Rgu = V u
N−1, u > 0. (E)
Where Rg is the scalar curvature and V is a function (prescribed scalar curvature).
We consider three positive real number a, bA and we suppose V lipschitzian:
0 < a ≤ V (x) ≤ b < +∞ and ||∇V ||L∞(M) ≤ A. (C)
The equation (E) was studied lot of when M = Ω ⊂ Rn or M = Sn see for example, [B],
[CL1], [L1]. In this case we have a sup× inf inequality.
The corresponding equation in two dimensions on open set Ω of R2, is:
∆u = V eu, (E′)
The equation (E′) was studed lot of and we can find very important result about a priori
estimates in [BM], [BLS], [CL2], [L2], and [S].
In the case V ≡ 1 and M compact, the equation (E) is Yamabe equation. It was studed lot
of, T.Aubin and R.Schoen have proved the existence of solution in this cas, see for example [Au]
and [L-P].
When M is a compact Riemannian manifold, it exist some compactness result for equation
(E) see [L-Zh]. Li and Zhu [L-Zh], proved that the energy is bounded and if we suppose M not
diffeormorfic to the three sphere, the solutions are uniformly bounded. To have this result they
use the positive mass theorem.
Now, if we suppose M Riemannian manifold (not necessarily compact) and V ≡ 1, Li and
Zhang [L-Z] proved that the product sup× inf is bounded.
Here, we give an equality of type sup× inf for the equation (E) with general conditions (C).
We have:
Theorem. For all compact set K of M and all positive numbers a,b,A, it exists a positive
constant c, which depends only on, K,a,b,A,M,g such that:
(sup
K
u)1/3 × inf
M
u ≤ c,
for all u solution of (E) with conditions (C).
Note that in our work, we have not assumption on energy or boundary condition if we suppose
the manifold M with boundary.
Next, in the proof of the previous theorem, we can replace the scalar curvature by any smooth
function f , but here we do the proof with Rg the scalar curvature.
1
Proof of the theorem.
Part I: The metric and the laplacian in polar coordinates.
Let (M, g) a Riemannian manifold. We note gx,ij the local expression of the metric g in the
exponential map centred in x.
We are concerning by the polar coordinates expression of the metric. By using Gauss lemma,
we can write:
g = ds2 = dt2 + gkij(r, θ)dθ
idθj = dt2 + r2g˜kij(r, θ)dθ
idθj = gx,ijdx
idxj ,
in a polar chart with origin x”, ]0, ǫ0[×Uk, with (Uk, ψ) a chart of Sn−1. We can write the
element volume:
dVg = r
n−1
√
|g˜k|drdθ1 . . . dθn−1 =
√
[det(gx,ij)]dx
1 . . . dxn,
then,
dVg = r
n−1
√
[det(gx,ij)][expx(rθ)]α
k(θ)drdθ1 . . . dθn−1,
where, αk is such that, dσSn−1 = αk(θ)dθ1 . . . dθn−1. (Riemannian volume element of the la
sphere in the chart (Uk, ψ) ).
Then, √
|g˜k| = αk(θ)
√
[det(gx,ij)],
Clearly, we have the following proposition:
Proposition 1: Let x0 ∈M , there exist ǫ1 > 0 and if we reduce Uk, we have:
|∂r g˜kij(x, r, θ)| + |∂r∂θm g˜kij(x, r, θ)| ≤ Cr, ∀ x ∈ B(x0, ǫ1) ∀ r ∈ [0, ǫ1], ∀ θ ∈ Uk.
and,
|∂r|g˜k|(x, r, θ)| + ∂r∂θm |g˜k|(x, r, θ) ≤ Cr, ∀ x ∈ B(x0, ǫ1) ∀ r ∈ [0, ǫ1], ∀ θ ∈ Uk.
Remark:
∂r[log
√
|g˜k|] is a local function of θ, and the restriction of the global function on the sphere
Sn−1, ∂r[log
√
det(gx,ij)]. We will note, J(x, r, θ) =
√
det(gx,ij).
Let’s write the laplacian in [0, ǫ1]× Uk,
−∆ = ∂rr + n− 1
r
∂r + ∂r[log
√
|g˜k|]∂r + 1
r2
√
|g˜k|∂θ
i(g˜θ
iθj
√
|g˜k|∂θj ).
We have,
−∆ = ∂rr + n− 1
r
∂r + ∂r log J(x, r, θ)∂r +
1
r2
√
|g˜k|∂θ
i(g˜θ
iθj
√
|g˜k|∂θj ).
We write the laplacian ( radial and angular decomposition),
−∆ = ∂rr + n− 1
r
∂r + ∂r[log J(x, r, θ)]∂r −∆Sr(x),
where ∆Sr(x) is the laplacian on the sphere Sr(x).
We set Lθ(x, r)(...) = r2∆Sr(x)(...)[expx(rθ)], clearly, this operator is a laplacian on Sn−1
for particular metric. We write,
Lθ(x, r) = ∆gx,r,Sn−1
,
and,
2
∆ = ∂rr +
n− 1
r
∂r + ∂r[J(x, r, θ)]∂r − 1
r2
Lθ(x, r).
If, u is function on M , then, u¯(r, θ) = uo expx(rθ) is the corresponding function in polar
coordinates centred in x. We have,
−∆u = ∂rru¯+ n− 1
r
∂ru¯+ ∂r[J(x, r, θ)]∂r u¯− 1
r2
Lθ(x, r)u¯.
Part II: ”Blow-up” and ”Moving-plane” methods
The ”blow-up” technic
Let, (ui)i a sequence of functions on M such that,
∆ui +Rgui = Viui
5, ui > 0, (E
′)
We argue by contradiction and we suppose that sup1/3× inf is not bounded.
We assume that:
∀ c, R > 0 ∃ uc,R solution of (E′) such that:
R[ sup
B(x0,R)
uc,R]
1/3 × inf
M
uc,R ≥ c, (H)
Proposition 2:
There exist a sequence of points (yi)i, yi → x0 and two sequences of positive real number
(li)i, (Li)i, li → 0, Li → +∞, such that if we consider vi(y) =
uio expyi(y)
ui(yi)
, we have:
0 < vi(y) ≤ βi ≤ 21/2, βi → 1.
vi(y)→
(
1
1 + |y|2
)1/2
, uniformly on every compact set of R3.
l
1/2
i [ui(yi)]
1/3 × inf
M
ui → +∞
Proof:
We use the hypothesis (H), we can take two sequences Ri > 0, Ri → 0 and ci → +∞, such
that,
Ri[ sup
B(x0,Ri)
]1/3ui × inf
M
ui ≥ ci → +∞,
Let, xi ∈ B(x0, Ri), such that supB(x0,Ri) ui = ui(xi) and si(x) = [Ri−d(x, xi)]1/2ui(x), x ∈
B(xi, Ri). Then, xi → x0.
We have,
max
B(xi,Ri)
si(x) = si(yi) ≥ si(xi) = Ri1/2ui(xi) ≥ √ci → +∞.
Set :
li = Ri − d(yi, xi), u¯i(y) = uio expyi(y), vi(z) =
uio expyi
(
z/[ui(yi)]
2
)
ui(yi)
.
Clearly, yi → x0. We obtain:
Li =
li
(ci)1/2
[ui(yi)]
2 =
[si(yi)]
2
c
1/2
i
≥ c
1
i
c
1/2
i
= c
1/2
i → +∞.
3
If |z| ≤ Li, then y = expyi [z/[ui(yi)]2] ∈ B(yi, δili) with δi =
1
(ci)1/2
and d(y, yi) <
Ri − d(yi, xi), thus, d(y, xi) < Ri and, si(y) ≤ si(yi), we can write,
ui(y)[Ri − d(y, yi)]1/2 ≤ ui(yi)(li)1/2.
But, d(y, yi) ≤ δili, Ri > li and Ri−d(y, yi) ≥ Ri− δili > li− δili = li(1− δi), we obtain,
0 < vi(z) =
ui(y)
ui(yi)
≤
[
li
li(1 − δi)
]1/2
≤ 21/2.
We set, βi =
(
1
1− δi
)1/2
, clearly βi → 1.
The function vi is solution of:
−gjk[expyi(y)]∂jkvi − ∂k
[
gjk
√
|g|
]
[expyi(y)]∂jvi +
Rg[expyi(y)]
[ui(yi)]4
vi = Vivi
5,
By elleptic estimates and Ascoli, Ladyzenskaya theorems, (vi)i converge uniformely on each
compact to the function v solution on R3 of,
∆v = V (x0)v
5, v(0) = 1, 0 ≤ v ≤ 1 ≤ 21/2,
Without loss of generality, we can suppose V (x0) = 3.
By using maximum principle, we have v > 0 on R3, the result of Caffarelli-Gidas-Spruck (
see [C-G-S]) give, v(y) =
(
1
1 + |y|2
)1/2
. We have the same propreties for vi in the previous
paper [B2].
Polar coordinates and ”moving-plane” method
Let,
wi(t, θ) = e
1/2u¯i(e
t, θ) = et/2uio expyi(e
tθ), et a(yi, t, θ) = log J(yi, e
t, θ).
Lemma 1:
The function wi is solution of:
−∂ttwi − ∂ta∂twi − Lθ(yi, et) + cwi = Viw5i ,
with,
c = c(yi, t, θ) =
(
1
2
)2
+
1
2
∂ta− λe2t,
Proof:
We write:
∂twi = e
3t/2∂ru¯i +
1
2
wi, ∂ttwi = e
5t/2
[
∂rru¯i +
2
et
∂ru¯i
]
+
(
1
2
)2
wi.
∂ta = e
t∂r log J(yi, e
t, θ), ∂ta∂twi = e
5t/2 [∂r log J∂ru¯i] +
1
2
∂tawi.
the lemma is proved.
Now we have, ∂ta =
∂tb1
b1
, b1(yi, t, θ) = J(yi, e
t, θ) > 0,
We can write,
− 1√
b1
∂tt(
√
b1wi)− Lθ(yi, et)wi + [c(t) + b−1/21 b2(t, θ)]wi = ViwiN−1,
4
where, b2(t, θ) = ∂tt(
√
b1) =
1
2
√
b1
∂ttb1 − 1
4(b1)3/2
(∂tb1)
2.
Let,
w˜i =
√
b1wi.
Lemma 2:
The function w˜i is solution of:
−∂ttw˜i +∆gyi,et,S2 (w˜i) + 2∇θ(w˜i).∇θ log(
√
b1) + (c+ b
−1/2
1 b2 − c2)w˜i =
= Vi
(
1
b1
)2
w˜5i ,
where, c2 = [
1√
b1
∆gyi,et,Sn−1
(
√
b1) + |∇θ log(
√
b1)|2].
Proof:
We have:
−∂ttw˜i −
√
b1∆gyi,et,S2
wi + (c+ b2)w˜i = Vi
(
1
b1
)2
w˜5i ,
But,
∆gyi,et,S2
(
√
b1wi) =
√
b1∆gyi,et,S2
wi − 2∇θwi.∇θ
√
b1 + wi∆gyi,et,S2
(
√
b1),
and,
∇θ(
√
b1wi) = wi∇θ
√
b1 +
√
b1∇θwi,
we deduce,
√
b1∆gyi,et,S2
wi = ∆gyi,et,S2
(w˜i) + 2∇θ(w˜i).∇θ log(
√
b1)− c2w˜i,
with c2 = [
1√
b1
∆gyi,et,S2
(
√
b1) + |∇θ log(
√
b1)|2]. The lemma is proved.
The ”moving-plane” method:
Let ξi a real number, and suppose ξi ≤ t, we set tξi = 2ξi − t and w˜ξii (t, θ) = w˜i(tξi , θ).
We have,
−∂ttw˜ξii +∆g
yi,e
tξi
S2
(w˜i)+2∇θ(w˜ξii ).∇θ log(
√
b1)w˜
ξi
i +[c(t
ξi)+b
−1/2
1 (t
ξi , .)b2(t
ξi)−cξi2 ]w˜ξii =
= V ξii
(
1
bξi1
)2
(w˜ξii )
5
.
By using the same arguments than in [B], we have:
Proposition 3:
We have:
1) w˜i(λi, θ)− w˜i(λi + 4, θ) ≥ k˜ > 0, ∀ θ ∈ S2.
For all β > 0, there exist cβ > 0 such than:
2)
1
cβ
et/2 ≤ w˜i(λi + t, θ) ≤ cβet/2, ∀ t ≤ β, ∀ θ ∈ S2.
We set,
5
Z¯i = −∂tt(...) + ∆gyi,et,S2 (...) + 2∇θ(...).∇θ log(
√
b1) + (c+ b
−1/2
1 b2 − c2)(...)
Remark: In the opertor Z¯i, by using the proposition 3, the coeficient c+ b−1/21 b2− c2 verify:
c+ b
−1/2
1 b2 − c2 ≥ k′ > 0, pour t << 0,
it is fundamental if we want to apply Hopf maximum principle.
Goal:
Like in [B], we have elliptic second order operator, here it’s Z¯i, the goal is to use the ”moving-
plane” method to have a contradiction. For this, we must have:
Z¯i(w˜
ξi
i − w˜i) ≤ 0, if w˜ξii − w˜i ≤ 0.
We write:
Z¯i(w˜
ξi
i − w˜i) = (∆g
yi,e
tξi ,S2
−∆gyi,et,S2 )(w˜
ξi
i )+
+2(∇
θ,et
ξi −∇θ,et)(wξii ).∇θ,etξi log(
√
bξi1 ) + 2∇θ,et(w˜ξii ).∇θ,etξi [log(
√
bξi1 )− log
√
b1]+
+2∇θ,etwξii .(∇θ,etξi −∇θ,et) log
√
b1 − [(c+ b−1/21 b2 − c2)ξi − (c+ b−1/21 b2 − c2)]w˜ξii +
+V ξii
(
1
bξi1
)2
(w˜ξii )
5 − Vi
(
1
b1
)2
w˜5i . (∗ ∗ ∗1)
Clearly, we have:
Lemma 3 :
b1(yi, t, θ) = 1− 1
3
Ricciyi(θ, θ)e
2t + . . . ,
Rg(e
tθ) = Rg(yi)+ < ∇Rg(yi)|θ > et + . . . .
According to proposition 1 and lemma 3,
Propostion 4 :
Z¯i(w˜
ξi
i − w˜i) ≤ Vib1(−2[(w˜ξii )5 − w˜5i ] + 2(w˜ξii )5|V ξii − Vi|+
+C|e2t−e2tξi |
[
|∇θw˜ξii |+ |∇2θ(w˜ξii )|+ |Ricciyi|[w˜ξii + (w˜ξii )5] + |Rg(yi)|w˜ξii
]
+C′|e3tξi−e3t|.
Proof:
We use proposition 1, we have:
a(yi, t, θ) = log J(yi, e
t, θ) = log b1, |∂tb1(t)|+ |∂ttb1(t)|+ |∂tta(t)| ≤ Ce2t,
and,
|∂θjb1|+ |∂θj,θkb1|+ ∂t,θjb1|+ |∂t,θj,θkb1| ≤ Ce2t,
then,
|∂tb1(tξi)− ∂tb1(t)| ≤ C′|e2t − e2t
ξi |, sur ]−∞, log ǫ1]× S2, ∀ x ∈ B(x0, ǫ1)
Locally,
∆gyi,et,S2
= Lθ(yi, e
t) = − 1√|g˜k(et, θ)|∂θl [g˜θ
lθj (et, θ)
√
|g˜k(et, θ)|∂θj ].
6
Thus, in [0, ǫ1]× Uk, we have,
Ai =


[
1√
|g˜k|∂θl(g˜
θlθj
√
|g˜k|∂θj )
]ξi
− 1√|g˜k|∂θl(g˜θ
lθj
√
|g˜k|∂θj )

 (w˜ξii )
then, Ai = Bi +Di with,
Bi =
[
g˜θ
lθj(et
ξi
, θ)− g˜θlθj(et, θ)
]
∂θlθj w˜
ξi
i (t, θ),
and,
Di =
[
1√
|g˜k|(etξi , θ)∂θl [g˜
θlθj (et
ξi
, θ)
√
|g˜k|(etξi , θ)]− 1√|g˜k|(et, θ)∂θl [g˜θ
lθj (et, θ)
√
|g˜k|(et, θ)]
]
∂θj w˜
ξi
i (t, θ),
we deduce,
Ai ≤ Ck|e2t − e2t
ξi |
[
|∇θw˜ξii |+ |∇2θ(w˜ξii )|
]
,
If we take C = max{Ci, 1 ≤ i ≤ q} and if w use (∗ ∗ ∗1), we obtain proposition 4.
We have,
c(yi, t, θ) =
(
n− 2
2
)2
+
n− 2
2
∂ta+Rge
2t, (α1)
b2(t, θ) = ∂tt(
√
b1) =
1
2
√
b1
∂ttb1 − 1
4(b1)3/2
(∂tb1)
2, (α2)
c2 = [
1√
b1
∆gyi,et,Sn−1
(
√
b1) + |∇θ log(
√
b1)|2], (α3)
Then,
∂tc(yi, t, θ) =
1
2
∂tta+ 2e
2tRg(e
tθ) + e3t < ∇Rg(etθ)|θ >,
by proposition 1,
|∂tc2|+ |∂tb1|+ |∂tb2|+ |∂tc| ≤ K1e2t,
Now, we consider the function, w¯i(t, θ) = w˜i(t, θ)− [ui(yi)]
1/3 ×minM ui
2
et, and λ > 2 >
0.
For t ≤ ti = −(2/3) logui(yi), we have:
w¯i(t, θ) = e
t
[
b1(t, θ)e
−t/2uio expyi(e
tθ)− [ui(yi)]
1/3 ×minM ui
2
]
≥
≥ et [ui(yi)]
1/3 ×minM ui
2
> 0,
We set, µi =
[ui(yi)]
1/3 ×minM ui
2
.
We use proposition 3 and the same arguments than in [B], we obtain:
Lemma 4:
There exists ν < 0 such that for µ ≤ ν :
w¯µi (t, θ) − w¯i(t, θ) ≤ 0, ∀ (t, θ) ∈ [µ, ti]× S2,
We set, λi = −2 logui(yi), then,
Lemma 5:
w¯i(λi, θ)− w¯i(λi + 4, θ) > 0.
7
Proof of lemma 5:
Clearly:
w¯i(λi, θ)− w¯i(λi + 4, θ) = w˜i(λi, θ)− w˜i(λi + 4, θ) + µi(e4 − 1),
we deduce lemma 6 from proposition 3.
Let, ξi = sup{µ ≤ λi + 2, w¯ξii (t, θ)− w¯i(t, θ) ≤ 0, ∀ (t, θ) ∈ [ξi, ti]× S2}.
The real ξi exists (see [B]), if we use (∗ ∗ ∗2), we have:
w˜ξii (t, θ) + |∇θw˜ξii (t, θ)|+ |∇2θw˜ξii (t, θ)| ≤ C(R), ∀ (t, θ) ∈]−∞, logR]× S2,
We can write:
Z¯i(w¯
ξi
i − w¯i) = Z¯i(w˜ξii − w˜i)− µiZ¯i(e2t
ξi − e2t),
−Z¯i(e2t
ξi − e2t) = [1− 1
4
− 3
2
∂ta−Rge2t + b−1/21 b2 − c2](e2t
ξi − e2t) ≤ c1(et
ξi − et),
with c1 > 0, car |∂ta| + |∂tb1| + |∂ttb1| + |∂t,θjb1| + |∂t,θj ,θkb1| ≤ C′e2t < 1, for t very
small.
We use proposition 4, to obtain on, [ξi, ti]× S2,
Z¯i(w¯
ξi
i − w¯i) ≤ c2[Vi(w˜ξii )5 − w˜5i ] + |V ξii − Vi|(wξii )5 + [µic1 − C′(R)](et
ξi − et) ≤ 0,
Like in [B], after using Hopf maximum principle, we have,
sup
θ∈S2
[w¯ξii (ti, θ)− w¯i(ti, θ)] = 0.
We have:
w¯ξii (ti, θi)− w¯i(ti, θi) = 0, ∀ i.
We deduce,
[u(yi)]
1/3 × inf
M
ui ≤ c, ∀ i.
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[C-G-S], L. Caffarelli, B. Gidas, J. Spruck. Asymptotic symmetry and local behavior of semi-
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DEPARTMENT OF MATHEMATICS, PATRAS UNIVERSITY, 26500 PATRAS , GREECE
E-mail address: samybahoura@yahoo.fr, bahoura@ccr.jussieu.fr
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Sup-Inf inequality on manifold of dimension 3

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