A dual Moser-Onofri inequality and its extensions to higher dimensional spheres

0.21 MB·English

by  Martial Agueh

A dual Moser-Onofri inequality and its extensions to
higher dimensional spheres
Martial Agueh ∗ Shirin Boroushaki † and Nassif Ghoussoub ‡
5 DedicatedtoDominiqueBakryontheoccasionofhis60thbirthday
1
0
2
Abstract
n
a
We use optimal mass transport to provide a new proof and a dual formula to the
J
Moser-OnofriinequalityonS2 inthesamespiritastheapproachofCordero-Erausquin,
6
Nazaret and Villani [4] to the Sobolev inequality and of Agueh-Ghoussoub-Kang[1] to

moregeneralsettings. Therearehowevermanyhurdlestoovercomeonceastereographic
]
P projection onR2 isperformed: Functionsarenotnecessarily ofcompactsupport,hence
A boundary terms need to be evaluated. Moreover, the corresponding dual free energy
of the reference probability density µ (x)= 1 is not finite on the whole space,
. 2 π(1+|x|2)2
h whichrequirestheintroductionofarenormalizedfreeenergyintothedualformula. We
t
a also extend this duality to higher dimensions and establish an extension of the Onofri
m inequality to spheres Sn with n ≥ 2. What is remarkable is that the corresponding
[ free energy is again given by F(ρ) = −nρ1−n1, which means that both the prescribed
scalar curvature problemandtheprescribed Gaussian curvature problemleadessentially

1 tothesame dualproblem whose extremals are stationary solutions of thefast diffusion
v
equations.
7
6 Keywords: Onofri inequality, duality, mass transport, prescribed Gaussian curvature,
2 fast diffusion.
1
0
Mathematics subject classification: 39B62,35J20, 34A34, 34A30.
.
1
0
5 1 Introduction
1
:
v One of the equivalent forms of Moser’s inequality [10] on the 2-dimensional sphere S2 states
i that the functional
X
r 1
a J(u):= u2dω udω log eudω (1.1)
4ZS2|∇ | −ZS2 − (cid:18)ZS2 (cid:19)
is bounded below on H1(S2), where dω is the Lebesgue measure on S2, normalized so that
dω = 1. Later, Onofri [11] showed that the infimum is actually zero, and that modulo
S2
conformaltransformations,u=0istheoptimalfunction. Notethatthisinequalityisrelated
tRo the “prescribed Gaussian curvature” problem on S2,
∆u+K(x)e2u =1 on S2, (1.2)
∗Department of Mathematics and Statistics University of Victoria, Victoria, BC, V8W 3R4, CANADA,
[email protected].
†DepartmentofMathematics,1984MathematicsRoad,UniversityofBritishColumbia,BC,V6T1Z2,CANADA
[email protected].
‡DepartmentofMathematics,1984MathematicsRoad,UniversityofBritishColumbia,BC,V6T1Z2,CANADA
[email protected]
1
whereK(x)istheGaussiancurvatureassociatedtothemetricg =e2ug onS2,and∆=∆
0 g0
is the Laplace-Beltrami operator corresponding to the standard metric g . Finding g for a
0
given K leads to solving (1.2). Variationally, this reduces to minimizing the functional
dV dV dV
(u)= u2 0 +2 u 0 log K(x)e2u 0 onH1(S2), (1.3)
F ZS2|∇ | 4π ZS2 4π − (cid:18)ZS2 4π (cid:19)
wherethevolumeformissuchthat dV =4π. Onofri’sresultsaysthat,moduloconformal
S2 0
transformations, u 0 is the only solution of the “prescribed Gaussian curvature” problem
(1.2) for K =1, i.e.≡, 1∆u+eu =1Ron S2, which after rescaling, u 2u, gives
2 7→
∆u+e2u =1 on S2. (1.4)
The proof given by Onofri in [11] makes use of an “exponential” Sobolev inequality due to
Aubin [2] combined with the invariance of the functional (1.1) under conformal transforma-
tions. Other proofs were given by Osgood-Philips-Sarnak [12] and by Hong [8]. See also
Ghoussoub-Moradifam[7].
In this paper we shall give a proof based on mass transport. While this approachhas by
nowbecomestandard,therearemanyreasonswhyithasn’tbeensofarspelledout. Thefirst
originates in the fact that, unlike the case of Rn, optimal mass transport on the sphere is
harder to work with. To avoidthis difficulty, we use an equivalent formulation of the Onofri
inequality (1.1), that we call the Euclidean Onofri inequality on R2, namely
1
u2dx+ udµ log eudµ 0 u H1(R2), (1.5)
2 2
16π ZR2|∇ | ZR2 − (cid:18)ZR2 (cid:19)≥ ∀ ∈
which is obtained by projecting (1.1) on R2 via the stereographic projection with respect to
the North pole N =(0,0,1),i.e., Π:S2 R2, Π(x):= x1 , x2 where x=(x ,x ,x ).
Here µ is the probability density on R2→defined by µ (x(cid:16))1−=x3 1−1x3(cid:17) , and dµ =1µ (2x)d3x.
2 2 π(1+x2)2 2 2
| |
One can then try to apply the Cordero-Nazaret-Villani [4] approach as generalized by
Agueh-Ghoussoub-Kang [1] and write the Energy-Entropy production duality for functions
that are of compact support in Ω,
1
sup (F(ρ)+ x2ρ)dx; ρdx=1 (1.6)
− 2| |
(cid:26) ZΩ ZΩ (cid:27)
=inf α u2 G(ψ u) dx; ψ(u)dx=1 ,
|∇ | − ◦
(cid:26)ZΩ ZΩ (cid:27)
where G(x)=(1 n)F(x)+nxF (x) and where ψ and α are also computable from F.
′
Recall that by−choosing F(x) = nx1 1/n and ψ(t) = t2∗ where 2 = 2n and n > 2,
− − | | ∗ n 2
one obtains the following duality formula for the Sobolev inequality −
1
sup n ρ1 1/ndx x2ρdx; ρdx=1
−
n ZRn − 2ZRn| | ZRn o
2
=inf 2 n−1 u2dx; u2∗dx=1 , (1.7)
n (cid:18)n−2(cid:19) ZRn|∇ | ZRn| | o
where u and ρ have compact support in Rn. The extremal u and ρ are then obtained as
∞ ∞
solutions of
|x|2 n−1 =0, ρ =u2∗ (Rn). (1.8)
∇ 2 − ρ1/n ! ∞ ∞ ∈P
∞
The bestconstants arethen obtainedby computing ρ from(1.8)andinserting it into (1.7)
∞
in such a way that
2
inf 2 n−1 u2dx: u2∗dx=1 =n ρ1−1/ndx 1 x2ρ dx.
n (cid:18)n−2(cid:19) ZRn|∇ | ZRn| | o ZRn ∞ − 2ZRn| | ∞
2
Note that this duality leads to a correspondencebetween a solutionto the Yamabe equation
∆u= u2∗−2u on Rn (1.9)
− | |
and stationary solution to the rescaled fast diffusion equation
∂tρ=∆ρ1−n1 +div(xρ) on Rn. (1.10)
The abovescheme doesnothoweverapply to inequality(1.5). Forone, the functions euµ =
2
eu(x) do not have compact support, and if one restricts them to bounded domains, we
π(1+x2)2
then|n|eed to take into consideration various boundary terms. What is remarkable is that a
similar program can be carried out provided the dual formula involving the free energy
J (ρ)= (F(ρ)+ x2ρ)dx
Ω
− | |
ZΩ
is renormalized by substituting it with J (ρ) J (µ ).
Ω Ω 2
−
Another remarkable fact is that the corresponding free energy turned out to be F(ρ) =
2ρ21, which is the same as the one associated to the critical case of the Sobolev inequality
−F(ρ) = nρ1−n1 when n 3. In other words, the Moser-Onofri inequality and the Sobolev
− ≥
inequality“dualize”inthesameway,andboththeYamabeproblem(1.9)andtheprescribed
Gaussian curvature problem (1.4) reduce to the study of the fast diffusion equation (1.10),
with the caveatthat in dimension n=2, the above equation needs to be considered only on
bounded domains, with Neumann boundary conditions.
More precisely, we shall show that, when restricted to balls B of radius R in R2, there
R
is a duality between the “Onofri functional”
I (u)= 1 u2dx+ udµ on X := u H1(B ); eudµ = dµ
R 16π BR|∇ | BR 2 R { ∈ 0 R BR 2 BR 2}
and the free eRnergy R R R
JR(ρ)= √2π BR√ρdx− BR|x|2ρdx on YR :={ρ∈P(R2); BRρdx= BR dµ2}.
R R R R
Once the latter free energy is re-normalized, we then get
sup J (ρ) J (µ );ρ Y =0=inf I (u); u X . (1.11)
R R 2 R R R
{ − ∈ } { ∈ }
Note that when R + , the right hand side yields the Onofri inequality
→ ∞
1
inf u2dx+ udµ ; u H1(R2), eudµ =1 =0,
(cid:26)16π ZR2|∇ | ZR2 2 ∈ 0 ZR2 2 (cid:27)
while the left-hand side becomes undefined since J (µ ) + as R .
R 2
→ ∞ →∞
We also show that our approach generalizes to higher dimensions. More precisely, if B
R
is a ball of radius R in Rn where n 2, and if one considers the probability density µ on
n
Rn defined by ≥
n
µ (x)=
n ωn(1+ x n−n1)n
| |
(ω is the area of the unit sphere in Rn), and the operator H (u,µ ) on W1,n(Rn) by
n n n
H (u,µ ):= u+ (logµ )n (logµ ))n n (logµ ))n 2 (logµ ) u,
n n n n n − n
|∇ ∇ | −|∇ | − |∇ | ∇ ·∇
there is then a duality between the functional
1
I (u)= H (u,µ )dx+ u(x)dµ on X := u W1,n(B ); eudµ = dµ
R β(n) n n n R { ∈ 0 R BR n BR n}
ZBR ZBR
R R
3
and the free energy – renormalized by again subtracting J (µ ) –
R n
JR(ρ)=α(n) ρ(x)n−n1 dx− |x|nn−1ρ(x)dx on YN :={ρ∈P(Rn); BRρdx= BR dµn}.
ZBR ZBR
R R
where α(n)= n ( n )1/n and β(n)=ω ( n )n 1nn. We then deduce the following higher
dimensional venr−si1onωonf the Euclidean Onnofrni−in1eq−uality in Rn, n 2:
≥
1
H (u,µ )dx+ udµ log eudµ 0 for all u W1,n(Rn), (1.12)
p n n n
β(n)ZRn ZRn − (cid:18)ZRn (cid:19)≥ ∈
and u 0 is the extremal.
≡
We finish this introduction by mentioning that the Euclidean Onofri inequality (1.5) in
theradialcasewasrecentlyobtainedin[6]usingmasstransporttechniques. However,toour
knowledge, our duality result, the extensions of Onofri’s inequality to higher dimensions, as
well as the mass transport proof of the general (non-radial) Onofri inequality are new.
The paper is organized as follows. In section 2, we recall the mass transport approach
to sharp Sobolev inequalities and some consequences. In section 3, we establish the mass
transportdualityprincipleleadingtotheEuclideanOnofriinequality(1.5). Finally,insection
4, we extend the 2-dimensional analysis of section 3 to higher dimensions n 2, and we
≥
producean-dimensionalversionoftheEuclideanOnofriinequality,aswellasotherinteresting
inequalities.
2 Preliminaries
We start by briefly describing the mass transport approach to sharp Sobolev inequalities as
proposed by [4]. We will follow here the framework of [1] as it clearly shows the correspon-
dence between the Yamabe equation (1.9) and the rescaled fast diffision equation (1.10).
Let ρ ,ρ (Rn). If T is the optimal map pushing ρ forward to ρ (i.e. T ρ = ρ )
0 1 0 1 # 0 1
∈ P
in the mass transport problem for the quadratic cost c(x y)= |x−y|2 (see [14] for details),
then [0,1] t ρ = (T ) ρ is the “geodesic” joinin−g ρ and2ρ in ( (Rn),d ); here
t t # 0 0 1 2
∋ 7→ P
T :=(1 t)id+tT and d denotes the quadratic Wasserstein distance (see [14]). Moreover,
t 2
given a f−unction F :[0, ) R such that F(0)=0 and x xnF(x n) is convex and non-
−
∞ → 7→
increasing, the functional HF(ρ):= F (ρ(x)) dx is displacement convex [9], in the sense
Rn
that[0,1] t HF(ρ ) R is convex(in the usualsense), for allpairs(ρ ,ρ ) in (Rn). A
t 0 1
∋ 7→ ∈ R P
direct consequence is the following convexity inequality, known as “energy inequality”:
d
HF(ρ1) HF(ρ0) HF(ρt) = ρ0 (F′(ρ0)) (T id) dx,
− ≥(cid:20)dt (cid:21)t=0 ZRn ∇ · −
which, after integration by parts of the right hand side term, reads as
HF(ρ1) HF+nPF(ρ0) ρ0 (F′(ρ0)) T(x)dx, (2.1)
− ≤− −ZRn ∇ ·
where P (x) = xF (x) F(x); here id denotes the identity function on Rn. By the Young
F ′
−
inequality
F (ρ )p T(x)q 1 1
′ 0
(F′(ρ0)) T(x) |∇ | + | | p,q >1 such that + =1, (2.2)
−∇ · ≤ p q ∀ p q
(2.1) gives
1 1
HF(ρ1) HF+nPF(ρ0)+ ρ0 F′(ρ0)pdx+ ρ0(x)T(x)qdx,
− ≤− pZRn |∇ | q ZRn | |
4
i.e.,
1 1
HF(ρ1) y qρ1(y)dy HF+nPF(ρ0)+ ρ0 F′(ρ0)pdx, (2.3)
− − q ZRn| | ≤− pZRn |∇ |
where we use that T ρ = ρ . Furthermore, if ρ = ρ , then T = id and equality holds in
# 0 1 0 1
(2.1). Thenequalityholdsin(2.3)ifitholdsintheYounginequality(2.2). Thisoccurswhen
ρ0 = ρ1 satisfies ∇ F′(ρ0(x))+ |xq|q = 0. Therefore, we have established the following
duality: (cid:16) (cid:17)
1
sup HF(ρ ) y qρ (y)dy : ρ (Rn)
1 1 1
n− − q ZRn| | ∈P o
1
=inf HF+nPF(ρ0)+ ρ0 F′(ρ0)pdx: ρ0 (Rn) , (2.4)
n− pZRn |∇ | ∈P o
and an optimal function in both problems is ρ =ρ :=ρ solution of
0 1
∞
xq
F′(ρ (x))+ | | =0. (2.5)
∇ ∞ q
(cid:18) (cid:19)
In particular, choosing F(x) = nx1 1/n and ρ = u2∗ where 2 = 2n and n > 2, then
− − 0 ∗ n 2
HF+nPF =0, and (2.4)-(2.5) gives the duality formula for the Sobolev i−nequality (1.7).
Our goal now is to extend this mass transport proof of the Sobolev inequality to the
EuclideanOnofri inequality (1.5). As already mentioned in the introduction, a first attempt
on this issue was recently made by [6], but the result produced was only restricted to the
radial case. Here we show in full generality (without restricting to radial functions u) that
the Euclidean Onofri inequality (1.5) can be proved by mass transport techniques. More
precisely, we establish an analogue of the duality (1.7) for Euclidean Onofri inequality (see
Theorem 3.1), from which we deduce the Onofri inequality (1.5) and its optimal function
(see Theorem 3.3). Furthermore, we obtain as for the Sobolev inequality, a correspondence
between the prescribed Gaussian curvature problem (1.4) and the rescaled fast diffusion
equation (1.10). Finally, we extend our analysis to higher dimensions, and then produce a
new version of the Onofri inequality in dimensions n 2 (see Theorem 4.3).
≥
The following general lemma from mass transport theory will be needed in the sequel.
Lemma 2.1. Let ρ ,ρ (B ), where (B ) denotes the set of probability densities on
0 1 R R
the ball B Rn. Let T∈bePthe optimal mPap pushing ρ forward to ρ (i.e. T ρ = ρ ) in
R 0 1 # 0 1
⊂
the mass transport problem for the quadratic cost. Then
1
ρ1(y)1−n1 dy ρ0(x)1−n1div(T(x))dx. (2.6)
≤ n
ZBR ZBR
Proof. By [3], the optimal map T : B B is such that T = ϕ where ϕ : B R is
R R R
→ ∇ →
convex. Since T ρ =ρ , then it satisfies the Monge-Amp`ere equation
# 0 1
ρ (x)=ρ (T(x))det T(x) (2.7)
0 1
∇
or equivalently
ρ (T(x))=ρ (x)[det T(x)] 1. (2.8)
1 0 −
∇
By the arithmetic-geometric-meaninequality
1
[det T(x)]n1 div(T(x)),
∇ ≤ n
5
(2.8) gives
1
ρ1(T(x))−n1 ρ0(x)−n1div(T(x)). (2.9)
≤ n
Now using the change of variable y =T(x), we have
ρ1(y)1−n1 dy = ρ1(T(x))1−n1 det( T(x))dx,
∇
ZBR ZBR
which implies by (2.7) and (2.9),
1 1
ρ1(y)1−n1 dy ρ0(x)−n1div(T(x))ρ0(x)dx= ρ0(x)1−n1div(T(x))dx.
≤ n n
ZBR ZBR ZBR
3 Euclidean 2-dimensional Onofri inequality: a duality
formula
Consider the probability density
1
µ (y)= y R2,
2 π(1+ y 2)2 ∀ ∈
| |
and set
R2
θ := µ (y)dy = , (3.1)
R 2 1+R2
ZBR
where B R2. We now establish the following duality.
R
⊂
Theorem 3.1. (Duality for the 2-dimensional Euclidean Onofri inequality)
Consider the functionals
1
I (u)= u(x)2dx+ u(x)dµ (x) on H1(B ),
R 16π |∇ | 2 0 R
ZBR ZBR
and
2
J (ρ)= ρ(y)dy y 2ρ(y)dy on (R2).
R
√π − | | P
ZBR ZBR
p
Then, the following holds:
sup J (ρ) J (µ ); ρ (R2), ρdy =θ
R R 2 R
− ∈P
(cid:26) ZBR (cid:27)
=inf I (u); u H1(B ), eudµ =θ . (3.2)
R ∈ 0 R 2 R
(cid:26) ZBR (cid:27)
Moreover, the maximum on the l.h.s. is attained at ρ = µ , and the minimum on the
max 2
r.h.s. is attained at u = 0. Furthermore, the maximum and minimum value of both
min
functionals in (3.2) is 0.
Proof. Applying lemma 2.1 in dimension n=2, and integrating by parts, we get
2 ρ (y) dy ρ (x) div(T(x))dx
1 0
≤
ZBR ZBR (3.3)
p p
= (√ρ ) T(x)dx+ ρ (x) T(x) νdS
0 0
− ∇ · ·
ZBR Z∂BR
p
6
where ν = x/x denotes the outward unit normal to the boundary ∂B of B , and dS is
R R
| |
the surface element. By the basic identity
1
(√ρ )= √ρ (logρ ),
0 0 0
∇ 2 ∇
we can rewrite (3.3) as
4 ρ (y)dy √ρ (logρ ) T(x)dx+2 ρ (x) T(x) νdS. (3.4)
1 0 0 0
≤− ∇ · ·
ZBR ZBR Z∂BR
p p
Now introduce the probability density
eu(x)µ (x)
2
ρ (x)= x B ,
0 R
θ ∀ ∈
R
where u H1(B ) satisfies eudµ =θ . Then (3.4) becomes
∈ 0 R BR 2 R
R
4 θ ρ dy √euµ (log(euµ )) T(x)dx+2 √euµ T(x) νdS. (3.5)
R 1 2 2 2
≤− ∇ · ·
ZBR ZBR Z∂BR
p
We use Young inequality
1 ε
√euµ (log(euµ )) T(x) (log(euµ )) 2+ euµ T(x)2 with ε=4√π,
2 2 2 2
− ∇ · ≤ 2ε|∇ | 2 | |
and the fact that T ρ =ρ , to rewrite (3.5) as
# 0 1
32 πθ ρ dy 16π y 2θ ρ (y)dy 16 √πeuµ T(x) νdS
R 1 R 1 2
− | | − ·
ZBR ZBR Z∂BR
p
u+ (logµ )2dx. (3.6)
2
≤ |∇ ∇ |
ZBR
It is easy to check that
u+ (logµ ))2dx= u2dx+ (logµ )2dx 2 u∆(logµ )dx,
2 2 2
|∇ ∇ | |∇ | |∇ | −
ZBR ZBR ZBR ZBR
and
∆(logµ )= 8πµ .
2 2
−
Inserting these identities into (3.6) and using that u =0, we have
|∂BR
32 πθ ρ dy 16π y 2θ ρ (y)dy 16 √πµ T(x) νdS
R 1 R 1 2
− | | − ·
ZBR ZBR Z∂BR
p
u2dx+16π udµ + (logµ )2dx. (3.7)
2 2
≤ |∇ | |∇ |
ZBR ZBR ZBR
We now estimate the boundary term. Since T : B B , then T(x) R for all x B ,
R R R
→ | |≤ ∈
so that
T(x) xdS T(x) x dS R2 dS =2πR3.
· ≤ | || | ≤
Z∂BR Z∂BR Z∂BR
Then using that ν =x/x, we have
| |
1 2πR2
√πµ T(x) νdS = T(x) xdS = 2πθ . (3.8)
− 2 · −R(1+R2) · ≥−1+R2 − R
Z∂BR Z∂BR
7
We insert (3.8) into (3.7) to have
32 πθ ρ dy 16π y 2θ ρ (y)dy 32πθ
R 1 R 1 R
− | | −
ZBR ZBR
p
u2dx+16π udµ + (logµ )2dx. (3.9)
2 2
≤ |∇ | |∇ |
ZBR ZBR ZBR
By a direct computation, we have
(logµ )2dx=16 √πµ dy 16πθ .
2 2 R
|∇ | −
ZBR ZBR
This implies that
32 πθ ρ dy 16π y 2θ ρ (y)dy 16 √πµ dy 16πθ
R 1 R 1 2 R
− | | − −
ZBR ZBR ZBR
p
u2dx+16π udµ ,
2
≤ |∇ |
ZBR ZBR
that is, after dividing by 16,
2 πθ ρ dy π y 2θ ρ dy √πµ dy+πθ
R 1 R 1 2 R
− | | −
ZBR ZBR (cid:18)ZBR (cid:19)
p 1
u2dx+π udµ . (3.10)
2
≤ 16 |∇ |
ZBR ZBR
We easily check that
√πµ dy+πθ =2 √πµ dy π y 2µ dy.
2 R 2 2
− | |
ZBR ZBR ZBR
Inserting this identity into (3.10) yields
2 πθ ρ dy y 2πθ ρ dy 2 √πµ dy y 2πµ dy
R 1 R 1 2 2
− | | − − | |
ZBR ZBR (cid:18) ZBR ZBR (cid:19)
p
1
u2dx+π udµ . (3.11)
2
≤ 16 |∇ |
ZBR ZBR
Now, we simplify (3.11) by π and set ρ=θ ρ , to have that
R 1
J (ρ) J (µ ) I (u) (3.12)
R R 2 R
− ≤
forallfunctions uandρ suchthatu H1(B )and eudµ =θ = ρ(y)dy. Observe
∈ 0 R BR 2 R BR
that if ρ=µ , then the l.h.s of (3.12) is 0, and so is the r.h.s if u=0. Therefore
2 R R
0 sup J (ρ) J (µ ): ρ(y)dy =θ inf I (u): eudµ =θ 0.
R R 2 R R 2 R
≤ ρ (cid:26) − ZBR (cid:27)≤u∈H01(BR)(cid:26) ZBR (cid:27)≤
This completes the proof of the theorem.
Remark 3.2. Note that the duality formula (3.2) for Onofri’s inequality is restricted to balls
B , while for the Sobolev inequality, it is defined on the whole space Rn, n 3. The reason
R
≥
is simply because the free energy
2
J (ρ)= ρ(y)dy y 2ρ(y)dy
R
√π − | |
ZBR ZBR
p
8
for(3.2)isnotalwaysfinite onthewholedomainR2,thatiswhenR . Thiscanbe seen
→∞
when we choose for example ρ=µ . We then have
2
R2
J (µ )=log(1+R2)+
R 2 1+R2
so that lim J (µ )= .
R R 2
→∞ ∞
From Theorem 3.1, we can deduce a certain number of results. One of them is the
Euclidean Onofri inequality.
Theorem 3.3. (2-dimensional Euclidean Onofri inequality)
The following holds:
1
log eudµ udµ u2dx u H1(R2). (3.13)
2 2
(cid:18)ZR2 (cid:19)−ZR2 ≤ 16π ZR2|∇ | ∀ ∈
Moreover u=0 is an optimal function.
Proof. By Theorem 3.1, we have for all u H1(B ) such that eudµ =θ = R2 ,
∈ 0 R BR 2 R 1+R2
R
1
I (u)= u(x)2dx+ u(x)dµ (x) 0, (3.14)
R 2
16π |∇ | ≥
ZBR ZBR
with equality if u=0. Letting R , we have θ 1; then (3.14) becomes
R
→∞ →
1
u(x)2dx+ u(x)dµ (x) 0=log eudµ ,
2 2
16π ZR2|∇ | ZR2 ≥ (cid:18)ZR2 (cid:19)
for all u H1(R2), with equality if u=0. This completes the proof of the theorem.
∈
Another consequence of Theorem 3.1 is the correspondence between solutions to the
prescribed Gaussian curvature problem and stationary solutions to a rescaled fast diffusion
equation. Indeed, it is known from the theory of mass transport (see [14] for example) that
a maximizer ρ to the supremum problem in (3.2) is a solution to the pde:
1
∆(√ρ)+div(xρ)=0 in B ,
R
2√π
(with Neumann boundary condition) that is the stationary solution of the rescaled fast dif-
fusion equation
1
∂ ρ= ∆(√ρ)+div(xρ) in B . (3.15)
t R
2√π
On the other hand, it can be shown that a minimizer u to the infimum problem in (3.2)
solves the semilinear pde:
1
∆u+λµ eu =µ in B , (3.16)
2 2 R
8π
where λ is the Lagrange multiplier for the constraint eudµ = θ . The duality formula
BR 2 R
(3.2) establishes then a correspondence between stationary solutions to the rescaled fast
R
diffusion equation (3.15) and solutions to the semilinear equation (3.16).
9
4 Extension to higher dimensions
Here, we extend the 2-dimensional analysis of section 3 to higher dimensions n 2, thereby
≥
producing a n-dimensional version of the Euclidean Onofri inequality (3.13). We start by
proving a n-dimensional version of the duality formula (3.2). Hereafter, we denote by
n
µ (y)=
n ωn(1+ y n−n1)n
| |
a probability density on Rn, where ω is the area of the unit sphere in Rn. Set
n
Rn
θ := µ (y)dy = .
R ZBR n (1+Rnn−1)n−1
Theorem 4.1. (Duality for n-dimensional Euclidean Onofri inequality)
Consider the functionals
n−1 n
JR(ρ)=α(n) ρ(y) n dy y n−1ρ(y)dy
− | |
ZBR ZBR
and
1
I (u)= H (u,µ )dx+ u(x)dµ u W1,n(B )
R β(n) n n n ∀ ∈ 0 R
ZBR ZBR
where
1/n n 1
α(n)= n n , β(n)= n − nnω
n
n 1 ω n 1
− (cid:18) n(cid:19) (cid:18) − (cid:19)
and
Hn(u,µn):= u+ (logµn)n (logµn)n n (logµn)n−2 (logµn) u.
|∇ ∇ | −|∇ | − |∇ | ∇ ·∇
Then the following duality holds:
sup J (ρ) J (µ );ρ (Rn), ρdy =θ =inf I (u); u W1,n(B ), eudµ =θ .
R − R n ∈P R R ∈ 0 R n R
(cid:26) ZBR (cid:27) (cid:26) ZBR (cid:27)
(4.1)
Moreover, the maximum on the l.h.s. is attained at ρ = µ , and the minimum on the
max n
r.h.s. is attained at u = 0. Furthermore, the maximum and minimum value of both
min
functionals in (4.1) is 0.
Remark 4.2. Beforeprovingthetheorem,wemakeafewremarksonthe operatorH (u,µ ).
n n
1. Consider the function c:Rn R, c(z)= z n, n 2. Clearly c is strictly convex, and
→ | | ≥
c(z)=nz n 2z. So we have the convexity inequality
−
∇ | |
c(z ) c(z ) c(z ) (z z ) 0 z ,z Rn. (4.2)
1 0 0 1 0 0 1
− −∇ · − ≥ ∀ ∈
Setting z = (logµ ) and z = u+ (logµ ), we see that H (u,µ ) is nothing but
0 n 1 n n n
∇ ∇ ∇
the l.h.s of (4.2); we then deduce that
H (u,µ ) 0 u,µ .
n n n
≥ ∀
2. Forallu W1,n(B ),theintegralofH (u,µ )overB involvesawell-knownoperator,
∈ 0 R n n R
the n-Laplacian ∆ , defined by
n
∆nv :=div( v n−2 v). (4.3)
|∇ | ∇
Indeed, this can be seen after performing an integration by parts in the last term of
H (u,µ ),
n n
H (u,µ )dy = u+ logµ ndy (logµ ))ndy+n u∆ (logµ )dy.
n n n n n n
|∇ ∇ | − |∇ |
ZBR ZBR ZBR ZBR
10