Poisson equation on complete manifolds

0.36 MB·

by  Ovidiu Munteanu

POISSON EQUATION ON COMPLETE MANIFOLDS
7 OVIDIUMUNTEANU,CHIUNG-JUEANNASUNG,ANDJIAPINGWANG
1
0
2 Abstract. We develop heat kernel and Green’s function estimates for man-
ifolds with positive bottom spectrum. The results are then used to establish
n
existence andsharpestimatesofthesolutiontothePoissonequationonsuch
a
manifolds with Ricci curvature bounded below. As an application, we show
J
thatthecurvatureofasteadygradientRiccisolitonmustdecayexponentially
1
ifitdecays fasterthanlinearandthepotential functionisboundedabove.
1

]
G
1. Introduction
D
. For a complete noncompact manifold (Mn,g) without boundary, consider the
h
Poissonequation
t
a
m
∆u= ϕ,
[ −
whereϕis agivensmoothfunctiononM. Inthispaper,weestablishexistence and

1 sharpestimates of the solution u and provide applications to steady gradientRicci
v solitons.
5
As well-known, the solvability of the Poisson equation is closely related to the
6
existence of the so-called Green’s function. In [16], Malgrange showed that M al-
8
2 waysadmitsaGreen’sfunctionG(x,y),namely,G(x,y)=G(y,x)and∆yG(x,y)=
0 δ (y). In particular, if ϕ C (M), then a solution u to the Poisson equation
− x ∈ 0∞
. exists and is given by
1
0
7
u(x)= G(x,y)ϕ(y)dy.
1
ZM
v: Malgrange’s proof is rather abstract. Later, Li and Tam [10] provided a more
i constructive proof. Among other things, the constructed Green’s function satisfies
X
r sup sup G(p,y) G(x,y) < .
a
| − | ∞
y M B(p,2R)x B(p,R)
∈ \ ∈
This turns out to be very useful in applications. For example, it was used to prove
an extension theorem for harmonic functions in [26].
Theorem 1.1. (Sung-Tam-Wang) For any harmonic function u defined on M Ω,
\
where Ω is a bounded subset of M, there exists a harmonic function v on M such
that u v is bounded on M Ω.
− \
Obviously, a good control of the Green’s function G(x,y) will enable one to
establishexistenceandestimatesofthesolutionutothePoissonequationformore
general ϕ. Recall that M is nonparabolic if M admits a positive Green’s function
ThefirstauthorwaspartiallysupportedbyNSFgrantDMS-1506220. Thesecondauthorwas
partially supported by MOST. The third author was partially supported by NSF grant DMS-
1606820.
1
2 OVIDIUMUNTEANU,CHIUNG-JUEANNASUNG,ANDJIAPINGWANG
and parabolic otherwise. It is well-known that M is nonparabolic if and only if
it admits a nonconstant bounded superharmonic function. Therefore, when M is
parabolic, the solution u to the Poisson equation must be unbounded if ϕ 0 but
≥
not identically 0. Since we are primarily concerned on the existence of bounded
solutions in this paper, we will restrict our attention to nonparabolic manifold M.
Inparticular,there exists a unique minimalpositive Green’s function onM. Inthe
following, it is understood that this is the Green’s function we refer to.
For manifolds with nonnegative Ricci curvature, a sharp pointwise estimate for
the Green’s function is available.
Theorem 1.2. (Li-Yau) Let Mn be a complete manifold with nonnegative Ricci
curvature. If 1∞V−1(p,√t)dt<∞ for some point p∈M, then M is nonparabolic
and its minimal positive Green’s function G(x,y) satisfies the estimate
R
C ∞ V 1(x,√t)dt G(x,y) C ∞ V 1(x,√t)dt
1 − 2 −
≤ ≤
Zr2(x,y) Zr2(x,y)
for some constants C and C depending only on the dimension n.
1 2
Hereandinthefollowing,V(p,r)denotesthevolumeofthegeodesicballB(p,r)
centered at point p with radius r, and r(x,y) the distance between points x and y
in M. The estimate follows from their famous upper and lower bounds of the heat
kernel [13] together with the fact that
∞
G(x,y)= H(x,y,t)dt.
Z0
Based on the estimates of the Green’s function, Ni, Shi and Tam [23] obtained
the following result concerning the Poissonequation.
Theorem1.3. (Ni-Shi-Tam)LetM beacompletemanifold withnonnegativeRicci
curvature. If M is nonparabolic, then for a locally Ho¨lder continuous function ϕ
with ϕ(x) cr k(x) for some k > 2, the Poisson equation ∆u = ϕ has a
−
| | ≤ −
solution u such that u(x) Cr k+2(x).
−
| | ≤
In fact, they haveprovedmore generalresults. For the existence, the decayrate
on ϕ is only assumed to be k >1. Moreover,the decay on ϕ is only required to be
true in the average sense over the geodesic balls centered at a fixed point. While
the solution u in general is no longer bounded, its growth is well controlled.
Theyappliedtheirresulttostudy,amongotherthings,thefollowinguniformiza-
tion conjecture of Yau.
Conjecture 1.4. (Yau) A complete noncompact K¨ahler manifold with positive
bisectional curvature is biholomorphic to the complex Euclidean space.
Indeed,byfirstsolvingthePoissonequation∆u=S onsuchmanifoldM,where
S is the scalar curvature, they demonstrated that under suitable assumptions u is
in fact a solution to the Poincar´e-Lelongequation
√ 1∂∂¯u=ρ,
−
where ρ is the Ricci form of M.
ThislineofideaswasinitiatedbyMok,SiuandYauin[17]. Whiletheconjecture
in its most general form is still open, there are various partial results. We refer
POISSON EQUATION ON COMPLETE MANIFOLDS 3
to the recent spectacular work of Liu [15] and the references therein for further
information.
Our focus will be on manifolds with positive spectrum. Denote by λ (∆) the
1
smallestspectrum of the Laplacianor the bottom spectrum of M. It is well-known
thatM isnonparabolicifλ (∆)>0.Recallthatλ (∆)canbecharacterizedasthe
1 1
best constant of the Poincar´einequality.
φ2dx
λ (∆)= inf M|∇ | .
1 φ∈C0∞ R Mφ2dx
As observed by Strichartz [25], if λ (∆)R> 0, then ∆ 1 is in fact a bounded
1 −
operatoronLp(M)for1<p< .Inparticular,thereexistsasolutionu Lp(M)
∞ ∈
to the Poisson equation for ϕ Lp(M).
∈
Ourachievementhereistoestablishanexistenceresultwithsharpcontrolofthe
solution u by only assuming a modest decay on the function ϕ, very much in the
spirit of the result alluded above by Ni, Shi and Tam for the case of nonnegative
Ricci curvature.
Theorem 1.5. Let M be a complete Riemannian manifold with bottom spectrum
λ (∆)>0 and Ricci curvature Ric (n 1)K for some constant K. Let ϕ be a
1
≥− −
smooth function such that
ϕ (x) c (1+r(x))−k
| | ≤
for some k >1, where r(x) is the distance function from x to a fixed point p M.
∈
Then the Poisson equation ∆u = ϕ admits a bounded solution u on M. If, in
−
addition, the volume of the ball B(x,1) satisfies V(x,1) c for all x M, then
≥ ∈
the solution u decays and
u (x) C (1+r(x))−k+1.
| | ≤
We point out that the existence of a solution u was previously proved by the
first author and Sesum [18]. However, their estimate on the solution u takes the
form
u (x) Cecr(x).
| | ≤
ItshouldalsobeemphasizedthattheassumptiononthevolumethatV(x,1) c
≥
is necessary to guarantee the solution u decays at infinity. Indeed, since u is a
bounded super-harmonic function when ϕ is positive, u can not possibly decay to
0 along a parabolic end of M.
The theoremis sharpasonecanseeasfollows. Onthe hyperbolicspaceHn,the
Green’s function is given by
∞ dt
G(x,y)= ,
A(t)
Zr(x,y)
whereA(t)istheareaofgeodesicsphereofradiustinHn.Forϕ(x)=(1+r(x))−k
with k >1, a direct calculation gives
4 OVIDIUMUNTEANU,CHIUNG-JUEANNASUNG,ANDJIAPINGWANG
u(x) = G(x,y)ϕ(y)dy
ZHn
c (1+r(x))−k+1.
≥
Our proof again relies on some sharp estimates of the Green’s function. Recall
the following result of the third author with Li [11], which is a sharp version of
Agmon’s work [1].
Theorem1.6. (Li-Wang)LetM beacompleteRiemannianmanifoldwithλ (∆)>
1
0. Let u be a nonnegative subharmonic function defined on M Ω, where Ω is a
\
compact domain. If u satisfies the growth condition
u2e−2√λ1(∆)r =o(R)
Z(M\Ω)∩B(p,R)
as R , then it must satisfy the decay estimate
→∞
u2 Ce−2√λ1(∆)R
≤
ZB(p,R+1)\B(p,R)
for some constant C >0 depending on u and λ (∆).
1
In particular, the theorem implies that the minimal positive Green’s function
satisfies
G2(p,y)dy Ce−2√λ1(∆)R.
≤
ZB(p,R+1)\B(p,R)
While this result provides a version of sharp estimate on the Green’s function,
toproveourtheorem,however,wealsoneedthefollowingdoubleintegralestimate.
G(x,y)dydx e√λ1(∆) V(A) V(B)(1+ r(A,B))e−√λ1(∆)r(A,B)
≤ λ (∆)
ZAZB 1
p p
for any bounded domains A and B of M.
For this purpose, we develop a parabolic versionof the aforementioned result of
Li and the third author.
Theorem1.7. LetM beacompleteRiemannianmanifoldwithλ (∆)>0.Suppose
1
that (∆ ∂ )u(x,t) 0 with u(x,t) 0,
− ∂t ≥ ≥
u2(x,0)e2√λ1(∆)r(x,A)dx<
∞
ZM
and
T
u2(x,t)e−2√λ1(∆)r(x,A)dxdt=o(R)
Z0 ZB(A,2R)\B(A,R)
for all T >0 as R . Then, for all R>0,
→∞
∞ u2(x,t)dxdt Ce−2√λ1(∆)R u2(x,0)e2√λ1(∆)r(x,A)dx.
≤
Z0 ZB(A,R+2)\B(A,R) ZM
POISSON EQUATION ON COMPLETE MANIFOLDS 5
Here, A is a bounded subset in M, r(x,A) the distance from x to A and
B(A,R)= x M r(x,A)<R . The constant C >0 depends only on λ (∆).
1
{ ∈ | }
By applying the theorem to the function u(x,t) = H(x,y,t)dy, one obtains
A
a sharp integral estimate of the heat kernel. This may be of independent interest.
R
The desired estimate of the Green’s function follows from the fact that
∞
G(x,y)= H(x,y,t)dt.
Z0
The following result is crucial to our proof of Theorem 1.5. It provides a sharp
integral control of the Green’s function.
Theorem 1.8. Let Mn be an n-dimensional complete manifold with λ (∆) > 0
1
and Ric (n 1)K. Then for any x M and r >0 we have
≥− − ∈
G(x,y)dy C (1+r)
≤
ZB(p,r)
for some constant C depending on n, K and λ (∆).
1
On top of the double integral estimate, the proof of the theorem utilizes an
idea originated in [12] and further illuminated in [18], where they used the co-area
formulatogetherwithsuitablychosencut-offfunctionstojustifythatforanyx M
∈
and 0<α<β,
β
G(x,y)dy c 1+ln .
≤ α
ZLx(α,β) (cid:18) (cid:19)
Here,
L (α,β):= y M :α<G(x,y)<β .
x
{ ∈ }
Partly motivated by applications to gradient Ricci solitons, we in fact consider
more generally the weighted Poisson equation on smooth metric measure space
(M,g,e fdx), that is,Riemannian manifold(M,g)together with aweightedmea-
−
sure e fdx, where f is a smoothfunction onM. The weightedPoissonequation is
−
given by
∆ u= ϕ,
f
−
where∆ u=∆u f, u istheweightedLaplacian. Anaturalcurvaturenotion
f
−h∇ ∇ i
correspondingtotheRiccicurvatureintheRiemanniansettingistheBakry-Emery
Ricci curvature, defined by
Ric =Ric+Hess(f).
f
Itisknown(see[20])thatresultssuchasvolumecomparison,gradientestimates
andmean value inequality are availableon the smoothmetric measurespaces with
the Bakry-Emery Ricci curvature bounded below together with suitable assump-
tions on the weight function f. With this in mind, we have a parallel version of
Theorem 1.5.
Theorem 1.9. Let Mn, g, e fdx be an n-dimensional smooth metric measure
−
space with Ric (n 1)K and the oscillation of f on any unit ball B(x,1)
f
≥ −(cid:0) − (cid:1)
bounded above by a fixed constant a. Assume that the bottom spectrum of the
weighted Laplacian λ (∆ ) is positive. Let ϕ be a smooth function such that
1 f
6 OVIDIUMUNTEANU,CHIUNG-JUEANNASUNG,ANDJIAPINGWANG
ϕ (x) c (1+r(x))−k
| | ≤
forsomek >1.Then ∆ u= ϕadmitsaboundedsolutionuonM.If, inaddition,
f
−
the weighted volume of the ball B(x,1) satisfies V (x,1) c for all x M, then
f
≥ ∈
the solution u decays and satisfies
u (x) C (1+r(x))−k+1.
| | ≤
In fact, we establish a slightly more general result (see Theorem 4.2). As an
immediate application,we obtainthe followingdecay estimate concerning the sub-
solutionstosemi-linearequations. Itwouldbeinterestingtoseeiftheestimatecan
be improved to exponential decay.
Theorem 1.10. Let Mn, g, e fdx be an n-dimensional smooth metric measure
−
space with Ric (n 1)K and the oscillation of f on any unit ball B(x,1)
f
≥ −(cid:0) − (cid:1)
boundedabovebyafixedconstanta.Assumethatthebottomspectrumλ (∆ )ofthe
1 f
weighted Laplacian is positive and the weighted volume has lower bound V (x,1)
f
≥
c>0 for all x M. Suppose ψ 0 satisfies
∈ ≥
∆ ψ cψq
f
≥−
for some q >1, and
1
lim ψ(x)rq−1 (x)=0.
x
→∞
Then there exist δ >0 and C >0 such that
ψ(x) Ce rδ(x).
−
≤
ThisresultmotivatedustostudythecurvaturebehaviorofsteadygradientRicci
solitons.
Definition 1.11. A steady gradient Ricci soliton is a complete manifold (M,g) on
which there exists a smooth potential function f such that
Ric+Hess(f)=0.
SteadygradientRiccisolitonsareself-similarsolutionstotheRicciflow. Indeed,
if we let g(t) = ψ(t) g, where ψ(t) is the diffeomorphism generated by the vector
∗
field f with ψ(0)=id , then g(t) is a solution to the Ricci flow
M
∇
∂
g(t)= 2Ric(g(t)).
∂t −
As such, they play important role in the study of the Ricci flows. Some prominent
examples of steady gradient Ricci solitons include the Euclidean space Rn with f
being a linear function, Hamilton’s cigar soliton (Σ, g ), where Σ=R2 and
Σ
dx2+dy2
g =
Σ 1+x2+y2
with the potential function f(x,y)= ln(1+x2+y2), and Bryantsoliton(Rn,g),
−
n 3,whereg isrotationallysymmetricandf =f(r)aswell. Thescalarcurvature
≥
of the cigar satisfies S = ef and decays exponentially S ce r(x) in the distance
−
≃
function. However, the curvature of the Bryant soliton decays linearly in distance.
For a steady gradient Ricci soliton, its Riemann curvature Rm satisfies
POISSON EQUATION ON COMPLETE MANIFOLDS 7
∆ Rm c Rm2
f
| |≥− | |
for some constant c > 0. Moreover, f is bounded and λ (∆ ) > 0 by [21]. So
1 f
|∇ |
Theorem1.10becomesapplicableoncethe weightedvolumeassumptionisverified.
This more or less follows from potential f being bounded above by a constant.
These considerations motivate the following theorem.
Theorem 1.12. Let (Mn,g,f) be a complete steady gradient Ricci soliton with
potential f bounded above by a constant. If its Riemann curvature satisfies
Rm (x)r(x)=o(1)
| |
as x , then
→∞
Rm (x) c (1+r(x))3(n+1) e r(x).
−
| | ≤
It is unclearat this point whether the assumptiononf is necessary. It is known
that the assumption automatically holds true when Ric>0. We also note that the
exponential decay rate in the theorem is sharp as seen from M =N Σ, where Σ
×
is the cigar soliton and N a compact Ricci flat manifold.
It view of our result, one may wonder whether there is a dichotomy for the cur-
vature decay rate of steady gradient Ricci solitons, namely, either exactly linear or
exponential. This dichotomy, if confirmed, should be very useful for the classifica-
tion of steady gradient Ricci solitons. In the three dimensional case, very recently,
Deng and Zhu [7] have shown that such a soliton must be the Bryant soliton if
its curvature decays exactly linearly. On the other hand, if the curvature decays
faster than linear,then it must be the product of the cigarsolitonand a circle (see
Corollary 5.5). We should also mention that Brendle [2] has confirmed Perelman’s
assertionin[24]thatanoncollapsedthreedimensionalsteadygradientRiccisoliton
must be the Bryant soliton.
Acknowledgement: We would like to dedicate this paper to Professor Peter
Li on the occasion of his sixty-fifth birthday. It can not be overstated how much
we have benefited from his teaching, encouragement and support over the years.
2. Heat kernel estimates
Inthissectionweextendthedecayestimateforsubharmonicfunctionsdeveloped
in[11]and[12]tothesubsolutionsoftheheatequation. Asaconsequence,weobtain
heat kernel estimate on complete manifolds with positive bottom spectrum. The
estimatewillbe appliedinnextsectiontoderiveintegralestimatesfortheminimal
Green’s function.
Wewillcastourresultinamoregeneralsettingofsmoothmetricmeasurespace
M,g,e fdx , where the following weighted Poincar´e inequality holds true for a
−
positive function ρ.
(cid:0) (cid:1)
(2.1) ρ(x)φ2(x)e−f dx φ2(x)e−fdx
≤ |∇ |
ZM ZM
for any compactly supported function φ C (M).
∈ 0∞
Let us define the ρ-metric by
ds2 =ρds2.
ρ
8 OVIDIUMUNTEANU,CHIUNG-JUEANNASUNG,ANDJIAPINGWANG
Using this metric, we consider the ρ-distance function defined to be
r (x,y)=infℓ (γ),
ρ ρ
γ
the infimum of the length of all smooth curves joining x and y with respect to
ds2. For a fixed point p M, one checks readily that r 2(p,x) = ρ(x). We say
ρ ∈ |∇ ρ|
thatmanifoldM hasproperty(P )ifthe ρ-metric iscomplete, andthis willbe our
ρ
standing assumption in this section.
Similarly, for a compact domain A M, we denote
⊂
r (x,A)= inf r (y,x)
ρ ρ
y A
∈
to be the ρ-distance to A and
B (A,R)= x M r (x,A)<R
ρ ρ
{ ∈ | }
to be the set of points in M that have ρ-distance less than R from set A.
Consider u(x,t) a nonnegative subsolution to the weighted heat equation
∂
(2.2) ∆ u 0.
f
− ∂t ≥
(cid:18) (cid:19)
We assume that u(x,t) satisfies the growth condition that
(2.3) u2(x,0)e2rρ(x,A)e−f(x)dx<
∞
ZM
and that for all T >0,
T
(2.4) ρ(x)u2(x,t)e−2rρ(x,A)e−f(x)dxdt=o(R)
Z0 ZBρ(A,2R)\Bρ(A,R)
as R .
→∞
Theorem 2.1. Let M,g,e fdx be a complete smooth metric measure space with
−
property (P ). Let u(x,t) satisfy (2.2), (2.3) and (2.4). Then for all R>0,
ρ
(cid:0) (cid:1)
∞ ρ(x) u2(x,t) e f(x)dxdt
−
Z0 ZBρ(A,R+2)\Bρ(A,R)
Ce−2R u2(x,0)e2rρ(x,A)e−f(x)dx
≤
ZM
for some absolute constant C >0.
Proof. Throughout the proof, we will denote by C an absolute constant which
may change from line to line. We also suppress the dependency of A and write
B (R) = B (A,R) and r (x) = r (x,A). The first step is to prove that for any
ρ ρ ρ ρ
0<δ <1, there exists a constant 0<C < such that
∞
∞ ρ(x)e2δrρ(x)u2(x,t)e−f(x)dxdt C u2(x,0)e2rρ(x)e−f(x)dx.
≤ 1 δ
Z0 ZM − ZM
POISSON EQUATION ON COMPLETE MANIFOLDS 9
Indeed, let φ(x) be a non-negative cut-off function onM. Then for any function
h(x) integration by parts yields
(2.5) (φueh)2e f = (φeh)2u2e f + (φeh)2 u2e f
− − −
|∇ | |∇ | |∇ |
ZM ZM ZM
+2 φeh u (φeh), u e f
−
h∇ ∇ i
ZM
(cid:0) (cid:1)
= (φeh)2u2e f + φ2 u2e2he f
− −
|∇ | |∇ |
ZM ZM
1
+ (φ2e2h), u2 e f
−
2 h∇ ∇ i
ZM
= (φeh)2u2e f + φ2 u2e2he f
− −
|∇ | |∇ |
ZM ZM
1
φ2∆f(u2)e2he−f
−2
ZM
= (φeh)2u2e−f φ2u(∆fu)e2he−f
|∇ | −
ZM ZM
φ2u2e2he f +2 φ φ, h u2e2he f
− −
≤ |∇ | h∇ ∇ i
ZM ZM
+ φ2 h2u2e2he f φ2uu e2he f,
− t −
|∇ | −
ZM ZM
where in the last line we have used (2.2). On the other hand, using the weighted
Poincar´einequality (2.1), we have
ρφ2u2e2he f (φueh)2e f.
− −
≤ |∇ |
ZM ZM
Hence (2.5) becomes
(2.6) ρφ2u2e2he f φ2u2e2he f +2 φ φ, h u2e2he f
− − −
≤ |∇ | h∇ ∇ i
ZM ZM ZM
1 d
+ φ2 h2u2e2he f φ2u2e2he f.
− −
|∇ | − 2dt
ZM ZM
Integrating with respect to t, we conclude
T 1
(2.7) ρφ2u2e2he fdxdt+ φ2u2(x,T)e2he fdx
− −
2
Z0 ZM ZM
T T
φ2u2e2he fdxdt+2 φu2 φ, h e2he fdxdt
− −
≤ |∇ | h∇ ∇ i
Z0 ZM Z0 ZM
T 1
+ φ2 h2u2e2he fdxdt+ φ2u2(x,0)e2he fdx.
− −
|∇ | 2
Z0 ZM ZM
Let us first choose
1 on B (R)
ρ
(2.8) φ(r (x))= R 1(2R r (x)) on B (2R) B (R)
ρ  − − ρ ρ \ ρ
0 on M B (2R)
 ρ
\

10 OVIDIUMUNTEANU,CHIUNG-JUEANNASUNG,ANDJIAPINGWANG
and
h(r (x))= δrρ(x) on Bρ (1+δ)−1K
ρ (cid:26) K−rρ(x) on M (cid:16)Bρ (1+δ)−(cid:17)1K
\
for some fixed K >1. Note that when R (1+δ)−1(cid:16)K, (cid:17)
≥
R 2ρ(x) on B (2R) B (R)
φ2(x)= − ρ \ ρ
|∇ | 0 on (M Bρ(2R)) Bρ(R)
(cid:26) \ ∪
and
R 1ρ(x) on B (2R) B (R)
φ, h (x)= − ρ \ ρ
h∇ ∇ i 0 on (M Bρ(2R)) Bρ(R),
(cid:26) \ ∪
whereas
h2(x)= δ2ρ on Bρ (1+δ)−1K
|∇ | (cid:26) ρ on M (cid:16)Bρ (1+δ)−(cid:17)1K .
\
Substituting all these into (2.7) implies (cid:16) (cid:17)
T 1
ρφ2u2e2he fdxdt+ φ2u2(x,T)e2he fdx
− −
2
Z0 ZM ZM
T
R 2 ρu2e2he fdxdt
− −
≤
Z0 ZBρ(2R)\Bρ(R)
T
+2R−1 ρu2e2he−fdxdt
Z0 ZBρ(2R)\Bρ(R)
T
+δ2 ρφ2u2e2he fdxdt
−
Z0 ZBρ((1+δ)−1K)
T
+ ρφ2u2e2he−fdxdt
Z0 ZBρ(2R)\Bρ((1+δ)−1K)
1
+ u2(x,0)e2he fdx.
−
2
ZM
This proves that
T
(1 δ2) ρu2e2he fdxdt
−
− Z0 ZBρ((1+δ)−1K)
1
+ u2(x,T)e2he fdx
−
2 ZBρ((1+δ)−1K)
T
R−2 ρu2e2he−fdxdt
≤
Z0 ZBρ(2R)\Bρ(R)
T
+2R 1 ρu2e2he fdxdt
− −
Z0 ZBρ(2R)\Bρ(R)
1
+ u2(x,0)e2rρe−fdx.
2
ZM
In view of the definition of h and (2.4), the first two terms on the right hand side
of this inequality tend to 0 as R . Therefore, we obtain the estimate
→∞