A nonlocal $\mathbf Q$-curvature flow on a class of closed manifolds of dimension $\mathbf{n \geq 5}$

0.25 MB·English

by  Xuezhang Chen

A nonlocal Q-curvature flow on a class of closed manifolds of
dimension n ≥ 5
5
1 Xuezhang Chen∗
0
2 ∗DepartmentofMathematics&IMS,NanjingUniversity,Nanjing210093, P.R.China

r
p
A

2

Abstract
]
G
D In this paper, weemploy a nonlocal Q-curvature flowinspired by Gursky-Malchiodi’s
. work[14]tosolvetheprescribed Q-curvature problem onaclassofclosedmanifolds: For
h
t n ≥ 5, let (Mn,g0) be a smooth closed manifold, which is not conformally diffeomor-
a
m phic to the standard sphere, satisfying either Gursky-Malchiodi’s semipositivity hypothe-
ses: scalar curvature R > 0 and Q ≥ 0 not identically zero or Hang-Yang’s: Yamabe
[ g0 g0
3 constantY(g0) > 0,Paneitz-Sobolevconstantq(g0)> 0andQg0 ≥ 0notidenticallyzero.
Letf beasmoothpositivefunction onMn andx besomemaximumpointoff. Suppose
v 0
8 either (a) n = 5,6,7 or (Mn,g ) is locally conformally flat; or (b) n ≥ 8, Weyl tensor at
0
1 x isnonzero. Inaddition,assumeallpartialderivativesoff vanishatx uptoordern−4,
6 0 0
thenthereexistsaconformalmetricg ofg withitsQ-curvatureQ equaltof. Thisresult
0 0 g
0 generalizes Escobar-Schoen’s work [Invent. Math. 1986] on prescribed scalar curvature
.
1 problem onanylocallyconformally flatmanifoldsofpositivescalarcurvature.
0 2010MSC:Primary53C44,58J05,53C21,45K05; Secondary35J30,35B40,35B50.
5
Keywords: Nonlocal Q-curvature flow, prescribed Q-curvature, locally conformally
1
: flat,asymptoticbehavior.
v
i
X
r
a Contents
1 Introduction 2
2 A non-local Q-curvature flow 4
2.1 Shorttimeexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Positivityofu(t,x)and energy estimates 7
3.1 Someelementary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Globalexistence 13
5 Asymptoticbehaviors 15
∗X.Chen:[email protected]
1
2
6 Construction ofinitialdata 17
6.1 (Mn,g0) : n ≥ 8 and notlocallyconformallyflat at x0 . . . . . . . . . . . . . 18
6.2 (Mn,g0) : n = 5,6,7orn ≥ 8 locallyconformallyflat . . . . . . . . . . . . 20
7 Sequential convergence ofthe nonlocal Q-curvature flow 24
1 Introduction
Let (Mn,g) be a smooth Riemannian manifold of dimension n ≥ 3, and R ,Ric be the
g g
scalar curvature, Ricci curvature of the metric g, respectively. The following conformally co-
variantoperatoroforderfourandQ-curvatureon(Mn,g)arefirstdiscoveredbyS.Paneitz[24]
in dimension four. Subsequently, in dimension n ≥ 3 and n 6= 4, they are recognized by T.
Branson [3]and Fefferman-Graham [13], concretely,
n−4
P = ∆2 −δ[(a R g +b Ric )d]+ Q , (1.1)
g g n g n g 2 g
(n−2)2 +4 4
with a = ,b = − ,
n 2(n−1)(n−2) n n−2
where δ is the divergent operator, d is the exterior derivative operator and the Q-curvature Q
g
ofmetricg isdefined by
1 n3 −4n2 +16n−16 2
Q = − ∆ R + R2 − |Ric |2.
g 2(n−1) g g 8(n−1)2(n−2)2 g (n−2)2 g
4
Underconformalchangeg˜ = un−4g forn 6= 4,thereholds
n+4
Pg(ϕu) = un−4Pg˜(ϕ) (1.2)
forall ϕ ∈ C∞(Mn).
Analogous to the Yamabe problem and the prescribed scalar curvature problem, the pre-
scribed Q-curvature problem on closed manifolds involving the fourth order conformally co-
variant operator can be proposed as follows: Give a smooth function f defined on a closed
manifold (Mn,g0) with n ≥ 5, can one find a conformal metric g = un−44g0,u > 0 such that
theQ-curvatureofthemetricg isequaltof? TheprescribedQ-curvatureproblemon(Mn,g )
0
withn ≥ 5 isreduced to thesolvabilityoftheequation
n−4
Pg0(u) = 2 funn−+44 and u > 0 on Mn, (1.3)
where P is fourth-order conformally covariant operator on (Mn,g ) defined by (1.1). Very
g0 0
recently, some remarkable progresses have been made in the constant Q-curvature problem
on closed manifolds after Chang-Yang’s pioneering work [7]. For the existence of conformal
metrictotheconstantQ-curvatureproblem,inwhichf isapositiveconstant,undertheassump-
tionsthat (a) Q ≥ 0 and positivesomewhere;(b) the scalarcurvature R ≥ 0, an affirmative
g0 g0
answer is recently provided by Gursky-Malchiodi in [14] adopting a nonlocal flow approach.
Especially,astrongmaximumprincipleforPaneitzoperatorwasfirstestablishedintheirpaper.
3
Later, Hang-Yang in their recent works [15, 16, 17] relaxed the positivity of scalar curvature
R in Gursky-Malchiodi’s assumptions to the Yamabe constant Y(g ) > 0 and the Paneitz-
g0 0
Sobolev constant q(g ) > 0, and used variational method to solve the constant Q-curvature
0
problemonaclassofclosedmanifoldsofdimensionsotherthanfour. Formorebackgroundon
theQ-curvatureproblem,onemay refer to[7, 10,14, 20]and thereferences therein.
For more than one decade, conformal geometric flows have played important roles in pre-
scribed curvature problems in conformal geometry and have become very powerful tools in
such problems comparing with the classical variational methods. In 2003, S. Brendle initiated
a negative gradient flow approach in [6] to deal with the prescribed Q-curvature problem with
criticalexponentialexponentonaclosedRiemmanianmanifold(Mn,g ). NotlongafterBren-
0
dle’swork,M.Struweadoptedthisapproachin[26]tostudytheNirenbergproblem,namelythe
prescribed Gauss curvature problem. Then Malchiodi and Struwe applied this approach to the
prescribed Q-curvature problem on S4 in [23], where they madesome further developmentsin
Morsetheorypart. Recently,theauthorandX.Xuin[9]adoptedthismethodtotheperturbation
result for the prescribed scalar curvature problem on Sn with n ≥ 3. However, to the author’s
knowledge, in the present stage, the presence of only one simple bubble in blow-up analysis
is crucial to all the above mentioned works. The closely related topics to the above conformal
geometricflowsaretheYamabeflow[4,5]andthefractionalYamabeflow[21]. Unfortunately,
there still exist sometechnical difficulties to show the long time existenceof positivesolutions
of such conformal geometric flows of higher order, even though such a strong maximum prin-
ciple(e.g. forPaneitzoperatorinsomespecialclass ofclosedmanifolds)has been established.
After Baird-Fardoun-Regbaoui’s work [2] and Gursky-Malchiodi’s[14] emerged, the nonlocal
flow comesintoplayand has shownitspowerinsuch prescribed curvatureproblems.
We arenowinpositionto stateourmaintheorem.
Theorem 1.1. Let (Mn,g ) be a smooth closed manifold of dimension n ≥ 5 and not
0
conformally diffeomorphic to the standard sphere Sn. Let f be a smooth positive function on
Mn andx beamaximumpointoff. Inthecaseofeither(a)n = 5,6,7or(Mn,g )islocally
0 0
conformally flat, or (b) n ≥ 8, Weyl tensor at x is nonzero, suppose either R > 0,Q ≥ 0
0 g0 g0
notidenticallyzeroortheYamabeconstantY(g ) > 0,thePaneitz-Sobolevconstantq(g ) > 0,
0 0
Q ≥ 0 notidenticallyzero. In addition,assume
g0
∇l f(x ) = 0 for 1 ≤ l ≤ n−4. (1.4)
g0 0
Then thereexistsa conformalmetricg of g withitsQ -curvatureequal tof.
0 g
Remark 1.1. Theorem 1.1 is a generalization of Theorem 2.1 in Escobar-Schoen’s work
[11] on prescribed scalar curvature problem for any locally conformally flat manifold of di-
mensionatleastthree, which isnotconformallydiffeomorphictothestandardsphere.
Remark 1.2. One case left open in Theorem 1.1 is that (Mn,g ) with n ≥ 8 is not locally
0
conformallyflat and Weyl tensor is zero at any maximum point of f, due to the lack of Positive
Mass Theorem for Paneitz operator, which is an obstruction in the construction of initial data
ofthenonlocalQ-curvatureflowinthiscase.
Thevanishingordercondition(1.4)onf atsomemaximumpointisusedtoconstructsome
positiveinitialdataofthenonlocalQ-curvatureflowsatisfyingeithersemi-positivityhypotheses
4
(2.1) or (2.2) below together with some restrictions on energy bounds. In dimension five, the
condition(1.4)isautomaticallysatisfied.
In section 2, we introduce a nonlocal Q-curvature flow which is a negative gradient flow
of E [u] in some suitable Hilbert space, and a detailed proof for short time existence of the
f
flow is available. In section 3, we show the positivity of u(t,·) for any time t ≥ 0, as well
as some elementary estimates involving E [u] and α(t) etc. In section 4, the global existence
f
of the nonlocal flow for some special class of initial data is presented. In section 5, we show
the asymptotic convergence of u P u dµ and the positivities of Q-curvature, as well as
Mn t g0 t g0
of the scalar curvature under hypotheses (2.1) for time t ≥ 0. Reminiscing about Aubin and
R
Schoen’s dichotomy on the Yamabe problem, in either (a) n = 5,6,7 or (Mn,g ) is locally
0
conformally flat, or (b) n ≥ 8, Weyl tensor at some maximum point of f is nonzero cases, in
section6,weconstructinitialdatasatisfyingeither(2.1)or(2.2),aswellassomerestrictionson
initialenergybounds. Finally,insection7,weestablishsequentialconvergenceofthenonlocal
flow meanwhilecompletingtheproofofTheorem 1.1.
Acknowledgments: Theauthoris supported throughNSFC (No.11201223),A Foundationfor
the Author of National Excellent Doctoral Dissertation of China (No. 201417) and Program
for New Century Excellent Talents in University (NCET-13-0271). He is grateful to Professor
XingwangXuforhisinvitationtotheInstituteforMathematicalSciencesatNationalUniversity
ofSingapore,wherepartsoftheworkwerecarriedout. HewouldliketothankProfessorYanyan
Li and Dr. JingangXiong for so many helpfuldiscussionsand so much precious adviceduring
thisproject,especiallythelatterpart ofthecurrent proofofLemma2.2suggestedbyProfessor
YanyanLi.
2 A non-local Q-curvature flow
Throughout this paper, suppose(Mn,g ) is a smoothclosed manifoldof dimension n ≥ 5,
0
satisfyingeitherGursky-Malchiodi’ssemipositivityhypotheses:
scalarcurvature R ≥ 0,Q ≥ 0 and Q > 0 somewhere; (2.1)
g0 g0 g0
orHang-Yang’s:
Yamabeconstant Y(g ) > 0,q(g ) > 0 and Q ≥ 0 isnotidenticallyzero. (2.2)
0 0 g0
Under the above assumption (2.1), a strong maximum principle for Paneitz operator P
g0
in [14] plays an important role in the existence of constant Q-curvature problem on closed
manifolds. Define aPaneitz-Sobolevconstantby
wP wdµ
q(g ) = q(Mn,g ) := inf Mn g0 g0 ;w ∈ H2(Mn,g )\{0}
0 0 (( RMnwn2−n4dµg0)n−n4 0 )
which is independent of the selection oRf the metric in the conformal class of g . The Paneitz-
0
Sobolevconstantenjoysan analogouspropertyofYamabeconstantprovedbyT.Aubin[1]:
Lemma2.1. OnaclosedRiemannianmanifold(Mn,g )ofdimensionn ≥ 8,supposethere
0
existsp ∈ Mn suchthattheWeyl tensorW (p) 6= 0,then q(g ) < q(Sn).
g0 0
5
Onemayrefertotheauthor’sunpublishednote[8]ortheprocessoftheproofin[12]Theo-
rem 1.1foradetailed proofofLemma2.1.
Later,HangandYang(cf. [16,17])relaxedtheabovehypotheses(2.1)toweakerones(2.2),
such a strong maximum principle still holds true. Both conditions (2.1) and (2.2) derive that
kerP = 0. However,uptonow,itisnotclearofthecoercivityofPaneitzoperatororq(g ) > 0
g0 0
can followfromY(g ) > 0,Q ≥ 0not identicallyzero whetherornot.
0 g0
Noticethat under theGursky-Malchiodi’shypotheses(2.1), it followsfrom Proposition2.3
in [14] that P is coercive and q(g ) is positive. Then, under either the hypotheses (2.1) or
g0 0
(2.2), weclaimthatthenorm
1
hv,P vi21 = vP vdµ 2, for v ∈ H2(Mn,g )
g0 g0 g0 g0 0
(cid:16)ZMn (cid:17)
and kvk are equivalent. Indeed, by the positivity of Paneitz-Sobolev constant q(g )
H2(Mn,g0) 0
and Ho¨lder’sinequality,thereholds
v2dµg0 ≤ vol(Mn,g0)n4 vn2−n4dµg0 n−n4 ≤ vol(Mq(gn,)g0)n4 vPg0vdµg0.
ZMn (cid:16)ZMn (cid:17) 0 ZMn
From theinterpolationSobolevinequality,given0 < ǫ < 1, onehas
|∆ v|2dµ = vP vdµ + |∆ v|2dµ − vP vdµ
g0 g0 g0 g0 g0 g0 g0 g0
ZMn ZMn (cid:16)ZMn ZMn (cid:17)
≤ vP vdµ +ǫ |∆ v|2dµ +C v2dµ .
g0 g0 g0 g0 ǫ g0
ZMn ZMn ZMn
Puttingtheabovefacts together, onehas
1
|∆ v|2dµ ≤ vP vdµ +C v2dµ
g0 g0 1−ǫ g0 g0 ǫ g0
ZMn ZMn ZMn
≤ C vP vdµ ,
g0 g0
ZMn
whichyields
kvk2 ≤ C vP vdµ .
H2(Mn,g0) g0 g0
ZMn
On theotherhand,it iseasy toverify theinverteddirectionoftheaboveinequality.
Letf beasmoothpositivefunctiondefinedonMn. MotivatedbyBaird-Fardoun-Regbaoui
[2]andGursky-Malchiodi[14],weextendtheirideastointroduceanonlocalQ-curvatureflow:
n+4
∂u = −u+ n−4P−1 αf|u|n−4 for (x,t) ∈ Mn ×[0,T);
∂t 2 g0 (2.3)
( u(0,x) = u ∈ C∞(Mn);
0
(cid:0) (cid:1)
coupledwiththeconstraintfunctionoftimet:
2 uP udµ
α(t) = Mn g0 g0. (2.4)
n−4 RMnfun2−n4dµg0
R
6
Remark2.1. From[14]Proposition2.4or[16]Lemma3.2,thereexistsaGreen’sfunction
G (p,·) of the Paneitz operator P with pole at p ∈ Mn, such that G (p,·) > 0 in Mn \
g0 g0 g0
{p}. Thus, given a function f ∈ C∞(Mn), the operator of P−1(f) can be interpreted as the
g0
convolutionbetween theGreen’s functionG (p,·)of P andthefunctionf, inother words,
g0 g0
P−1(f)(p) = G (p,x)f(x)dµ (x).
g0 g0 g0
ZMn
4
Remark2.2. Underconformalchangeofmetricsg = un−4g ,usingtheconformalcovari-
0
ance(1.2)ofthePaneitzoperatorP , wefind thatitsinverseP−1 isconformallycovariant:
g g
n+4
P−1 = u−1P−1 un−4 · ,
g g0
(cid:0) (cid:1)
duetothesimplefactthat∀ψ ∈ C∞(Mn),
n+4
u−1P−1 un−4P ψ = u−1P−1 P (uψ) = ψ.
g0 g g0 g0
(cid:0) (cid:1) (cid:0) (cid:1)
2.1 Short time existence
Thissubsectionisdevotedtothestudyoftheshorttimeexistencetotheflowproblem(2.3).
Lemma 2.2. There exists a unique solution in C0([0,T];C4,λ(Mn)) to the flow problem
(2.3)forany0 < λ < 1 and0 < T ≤ ∞.
Proof. It issufficientto showtheshorttimeexistenceofthefollowingmodifiedflowproblem
n+4
∂u˜ = −u˜+ n−4P−1 f|u˜|n−4 for (x,s) ∈ Mn ×[0,T);
∂s 2 g0 (2.5)
( u˜(s = 0,x) = u ∈ C∞(Mn).
0
(cid:0) (cid:1)
Indeed, the modified flow problem (2.5) differs from the original one (2.3) by scalings in time
and functions. Set
t
s(t) = µ(τ)dτ and u(t,x) = es(t)−tu˜(x,s(t))
Z0
where
2 u˜P u˜dµ
µ(t) = Mn g0 g0.
n−4 RMnfu˜n2−n4dµg0
By usingtheexpressionsofα(t)and µ(t) togeRt
8
α(t) = µ(t)e−n−4(s(t)−t),
we conclude that if the short time existence of the above modified flow problem (2.5) is estab-
lished,sois theflow problem(2.3).
7
Next we manage to establish the short time existence to (2.5) by the contraction mapping
principle. For simplicity we still use u instead of u˜ and time variable t instead of s. Let
X = C0([0,T];C4,λ(Mn)) forsomeλ ∈ (0,1). Foranyfixed δ > 0,define
X := {v ∈ X;v(0,x) = u ,kv −u k ≤ δ}
δ 0 0 X
whereu ∈ C∞(Mn), and putthedistanceon X by
0 δ
ρ(v,w) := kv −wk for v,w ∈ X .
X δ
Itisnothardtoverifythatthespace(X ,ρ)isacompletemetricspace. DefineamapL : X →
δ δ
X by
t n−4 t
L(u)(t,x) = u0(x)− u(τ,x)dτ + 2 Pg−01(f|u|nn−+44)(τ,x)dτ.
Z0 Z0
Recall that, fromtheSchauder estimatesofellipticequations,there holds
kP−1wk ≤ Ckwk ,
g0 C4,λ(Mn) Cλ(Mn)
whereC depends onlyonn,λ. Thenonehas
kPg−01(f|u|nn−+44)(t)kC4,λ(Mn) ≤ Ck(f|u|nn−+44)(t)kCλ(Mn)
8
≤ Cku(t)kn−4 ku(t)k ,
C0(Mn) Cλ(Mn)
and then
kL(u)(·,t)−u (·)k
0 C4,λ(Mn)
t
8
≤ Tku(t)k +C ku(τ)kn−4 ku(τ)k dτ
X C0(Mn) Cλ(Mn)
Z0
n+4
≤ CT(kuk +kukn−4)
X X
n+4
≤ CT(1+(δ +ku0kC4,λ(Mn))n−4).
Thuswehave
kL(u)−u k ≤ C(f,n,δ,λ)T.
0 X
By choosing T > 0 sufficiently small, it is not hard to verify that the map L is a contraction
on (X ,ρ). Then, by the contraction mapping theorem, there exists a unique fixed point of L.
δ
Thus,thelocalwell-possednessofthemodifiedflow problem(2.5)isestablished.
3 Positivity of u(t,x) and energy estimates
We first need to show that thepositivityof u is preserved along theflow. From now on, we
imposesomerestrictionsoninitialdata,that isu ∈ C∞,where
0 ∗
C∞ = {w ∈ C∞(Mn);w > 0,P w ≥ 0}.
∗ g0
It is easy to know C∞ 6= ∅ since 1 ∈ C∞ in view of the fact P 1 = n−4Q ≥ 0 due to
∗ ∗ g0 2 g0
Gursky-Malchiodi’ssemi-positivityhypotheses(2.1)orHang-Yang’s(2.2).
8
Lemma 3.1. Let u be a C4-solution to the nonlocal flow equation (2.3) with u(0,x) =
u (x) ∈ C∞, thenforall0 ≤ t ≤ T,therehold
0 ∗
u(x,t) > 0 and u ∈ C∞.
∗
Proof. A directcomputationyields
∂
P (u) = P (u )
∂t g0 g0 t
n−4
n+4
= −Pg0u+ 2 αf|u|n−4
≥ −P u.
g0
Then, wehave
P u(t,x) ≥ e−tP (u ) ≥ 0,
g0 g0 0
in view of u ∈ C∞. Thus, under either Gursky-Malchiodi’s (2.1) or Hang-Yang’s (2.2), the
0 ∗
strong maximum principle for P (cf. [14] Theorem 2.2 or [16] Proposition 3.1, respectively)
g0
gives
n−4
2 Qgunn−+44 = Pg0u(t,x) ≥ 0, u(t,x) > 0 forall (x,t) ∈ Mn ×[0,T].
Theabovefact alsoimpliesu ∈ C∞. Thiscompletestheproof.
∗
Consequently,theflow problem(2.3) turnsto
n+4
∂u = −u+ n−4P−1 αfun−4 ;
∂t 2 g0 (3.1)
( u(0,x) = u ∈ C∞;
0 ∗ (cid:0) (cid:1)
whereα(t) isgivenin(2.4). Forbrevity,let
ϕ = −u+ n−4P−1(αfunn−+44),
2 g0
and then
∂ n−4
n+4
∂tPg0u = −Pg0u+ 2 αfun−4 = Pg0ϕ. (3.2)
4 2n
From now on, denote by g(t) = u(t)n−2g0 the flow metric and dµg = u(t)n−4dµg0 the
volumeformoftheflowmetric, thenQ-curvatureequationgives
n−4
Pg0u = 2 Qunn−+44 on Mn, (3.3)
whereQ = Q istheQ-curvatureoftheflowmetricg(t). Definetheenergy functionals
g
n−4
E[u] = Q dµ = uP (u)dµ
2 g g g0 g0
ZMn ZMn
n−4
= (∆ u)2 +a R |∇u|2 +b Ric (∇u,∇u)+ Q u2 dµ
g0 n g0 g0 n g0 2 g0 g0
ZMn
h i
9
and
E[u]
E [u] = .
f 2n (n−4)/n
Mnfun−4dµg0
By (2.4)and (3.3),wehave
(cid:0)R (cid:1)
n−4
E[u(t)] = Qdµ
g
2
ZMn
and
2
0 = (αf −Q)dµ = uP ϕdµ . (3.4)
g n−4 g0 g0
ZMn ZMn
Alongthisflow,theenergy E[u(t)] ispreserved foralltimet ≥ 0.
Lemma 3.2. Along the nonlocal flow (3.1), the energy E[u(t)] is conserved for any time
t ≥ 0.
Proof. By theflowequation(3.1)and (2.4),we obtain
d
uP udµ = 2 uP (u )dµ
dt g0 g0 g0 t g0
ZMn ZMn
n+4
= 2 uP −u+ n−4P−1(αfun−4) dµ
g0 2 g0 g0
ZMn
(cid:16) (cid:17)
= −2 uPg0udµg0 − n−24α(t) fun2−n4dµg0
= 0, hZMn ZMn i
whichimpliesthedesired assertion.
From Remark 2.2, theflow equation(3.1)isequivalentto
n−4
u = P−1(αf −Q )u, (3.5)
t 2 g g
thatis,
∂
g = 2P−1(αf −Q )g.
∂t g g
SinceP is self-adjointand positive,wecan defineH2(Mn,g)innerproductby
g
hη,ζi = ηP ζdµ ,
g g g
ZMn
which induces the H2-norm k·k . In this sense, the nonlocal Q-curvature flow (3.1) is
H2(Mn,g)
a negative gradient flow of E [u] in the Hilbert space H2(Mn,g). Now we pause for a while
f
to give some explanations for the nonlocal Q-curvature flow. Up to a positiveconstant, regard
E [u]as afunctionalofthemetricg:
f
Q dµ
Q (g) = M g g
f n−4
R fdµ n
M g
(cid:16) (cid:17)
R
10
Set
g = φ g
ǫ ǫ
satisfying
dφ
φ = 1 and ǫ = φ ∈ C∞(M).
ǫ
dǫ
ǫ=0 ǫ=0
NoticethatP isinvertibleun(cid:12)derhypotheses(2.(cid:12)1)or(2.2), weobtain
g (cid:12) (cid:12)
(cid:12) (cid:12)
d Q dµ
Q′ (g)[φ] = M φǫg φǫg
f dǫ(cid:12)(cid:12)ǫ=0 RM fφǫn2dµg n−n4
(cid:12) (cid:16)R n−4(cid:17) n−4
2 d φ 4 P (φ 4 )dµ
= M ǫ g ǫ g
n−4dǫ(cid:12)(cid:12)ǫ=0R M fφǫn2dµg n−n4
φP ((cid:12)1)dµ(cid:16)R n−4 (cid:17) Q dµ
= M g g − M g g
n−4 2 n−4+1
R fdµ n fdµR n fφdµ
M g M g M g
=(cid:16)nR−4 (cid:17)fφn2dµ 4−nn (cid:16)R φ(Q (cid:17)−αf)dRµ
ǫ g g g
2
n−4(cid:16)ZM n (cid:17)4−nhZM i
= fφ2dµ n hφ,P−1(Q −αf)i .
2 ǫ g g g g
(cid:16)ZM (cid:17)
Lemma 3.3. Along the nonlocal flow (3.1), the energy E [u(t)] is non-increasing for all
f
timet ≥ 0.
Proof. By (3.5)and (2.4), wecompute
4−n
d 2n
E [u] = −(n−4) fun−4dµ n αfu−1u dµ
dt f g0 t g
(cid:16)ZMn (cid:17) ZMn
(n−4)2 2n 4−n
= − fun−4dµ n αfP−1(αf −Q)dµ .
2 g0 g g
(cid:16)ZMn (cid:17) ZMn
From (2.2),(3.3)and (2.4), as wellas from (3.4), wenoticethat
2
QP−1(αf −Q)dµ = P (u)uP−1(αf −Q)dµ
g g n−4 g0 g g0
ZMn ZMn
2 n+4
= P (u)P−1 (αf −Q)un−4 dµ
n−4 g0 g0 g0
ZMn (cid:16) (cid:17)
2
= (αf −Q)dµ = 0.
g
n−4
ZMn
Thus,weobtain
d (n−4)2 2n 4−n
E [u] = − fun−4dµ n (αf −Q)P−1(αf −Q)dµ
dt f 2 g0 g g
(cid:16)ZMn (cid:17) ZMn
(n−4)2 2n 4−n
= − fun−4dµ n kP−1(αf −Q)k2 ≤ 0.
2 g0 g H2(Mn,g)
(cid:16)ZMn (cid:17)
Thiscompletestheproof.