The 2-D stochastic Keller-Segel particle model : existence and uniqueness

0.2 MB·

by  Patrick Cattiaux

THE 2-D STOCHASTIC KELLER-SEGEL PARTICLE MODEL :
EXISTENCE AND UNIQUENESS.
6 PATRICK CATTIAUX ♠ AND LAURE PE´DE`CHES ♠
1
0
2
n ♠ Universit´e de Toulouse
a
J

9 Abstract. We introduce a stochastic system of interacting particles which is expected to
2 furnish as the number of particles goes to infinity a stochastic approach of the 2-D Keller-
Segelmodel. Inthisnote,weproveexistenceandsomeuniquenessforthestochasticmodel
]
fortheparabolic-ellipticKeller-Segelequation,forallregimesunderthecriticalmass. Prior
R
results for existence and weak uniqueness have been very recently obtained by N. Fournier
P
and B. Jourdain [6].
.
h
t
a Key words : Keller-Segel model, diffusion processes, Bessel processes.
m
[ MSC 2010 : 35Q92, 60J60, 60K35.

1
v
6 1. Introduction and main results.
2
0 The (Patlak) Keller-Segel system is a tentative model to describe chemo-taxis phenomenon,
8
an attractive chemical phenomenon between organisms. In two dimensions, the classical 2-D
0
. parabolic-elliptic Keller-Segel model reduces to a single non linear P.D.E.,
1
0 ∂ ρ (x) = ∆ ρ (x)+χ .((K ρ )ρ )(x) (1.1)
t t x t x t t
6 ∇ ∗
1 with some initial ρ0.
: Here ρ :R R2 R, χ > 0 and K : x R2 x R2 is the gradient of the harmonic
v +× → ∈ 7→ kxk2 ∈
i kernel, i.e. K(x)= log( x ).
X
∇ k k
It is not difficult to see that (1.1) preserves positivity and mass, so that we may assume that
r
a ρ is a density of probability, i.e. ρ 0 and ρ dx = ρ dx = 1. For an easy comparison
0 0 t 0
≥
with the P.D.E. literature, just remember that our parameter χ satisfies χ = m when m
R R 2π
denotes the total mass of ρ (the parameter 2π being usually included in the definition of K).
As usual, ρ is modeling a density of cells, and c = K ρ is (up to some constant) the
t t
∗
concentration of chemo-attractant.
A very interesting property of such an equation is a blow-up phenomenon
Theorem 1.2. Assume that ρ logρ L1(R2) and that (1+ x 2)ρ L1(R2). Then if
0 0 0
∈ k k ∈
χ > 4, the maximal time interval of existence of a classical solution of (1.1) is [0,T∗) with
1
T∗ x 2 ρ (x)dx.
0
≤ 2πχ(χ 4) k k
− Z
Date: February 1, 2016.
1
2 P.CATTIAUXANDL.PE´DE`CHES
If χ 4 then T∗ = + .
≤ ∞
For this result, a wonderful presentation of what Keller-Segel models are and an almost
up to date state of the art, we refer to the unpublished HDR document of Adrien Blanchet
(available onAdrien’swebpage[1]). We alsoapologize fornotfurnishingamorecomplete list
of references on the topic, where beautiful results were obtained by brilliant mathematicians.
But the present paper is intended to be a short note.
Actually (1.1) is nothingelse buta Mc Kean-Vlasov typeequation (non linear Fokker-Planck
equation if one prefers), involving a potential which is singular at 0. Hence one can expect
that the movement of a typical cell will be given by a non-linear diffusion process
dX = √2dB χ(K ρ )(X )dt, (1.3)
t t t t
− ∗
ρ (x)dx = (X ),
t t
L
where (X ) denotes the distribution of probability of X . The natural linearization of
t t
L
(1.3) is through the limit of a linear system of stochastic differential equations in mean field
interactions given for i= 1,...,N by
N i,N j,N
χ X X
dXi,N = √2dBi,N t − t dt, (1.4)
t t − N Xi,N Xj,N 2
Xj6=i k t − t k
for a well chosen initial distribution of the X.,N. Here the Bi,N are for each N independent
0 .
standard 2-D Brownian motions. Under some exchangeability assumptions, it is expected
that the distribution of any particle (say X1,N) converges to a solution of (1.3) as N ,
→ ∞
yielding a solution to (1.1). This strategy (including the celebrated propagation of chaos
phenomenon) has been well known for a long time. One can see [12] for bounded and
Lipschitz potentials, [11, 4] for unbounded potentials connected with the granular media
equation.
The goal of the present note is the study of existence, uniqueness and non explosion for the
system (1.4). That is, this is the very first step of the whole program we have described
previously. Moreover we will see how the N-particle system is feeling the blow-up property
of the Keller-Segel equation.
Forsuchsingularpotentialsveryfewisknown. Fournier,HaurayandMischler[5]havetackled
thecase ofthe2-D viscous vortex model, correspondingto K(x) = x⊥ forwhich noblow-up
kxk2
phenomenon occurs. In the same spirit the sub-critical Keller-Segel model corresponding to
K(x) = x for some ε > 0 is studied in [8]. The methods of both papers are close,
kxk2−ε
and mainly based on some entropic controls. These methods seem to fail for the classical
Keller-Segel model we are looking at. However, during the preparation of the manuscript,
we received the paper by N. Fournier and B. Jourdain [6], who prove existence and some
weak uniqueness by using approximations. Though some intermediate results are the same,
we shall here give a very different and much direct approach, at least for existence and some
uniqueness. However, we shall useone resultin [6] to prove a more general uniqueness result.
Also notice that when we replace the attractive potential K by a repulsive one (say K), we
−
find models connected with random matrix theory (like the Dyson Brownian motion).
Our main theorem in this paper is the following
STOCHASTIC KELLER-SEGEL 3
Theorem 1.5. Let M = there exists at most one pair i = j such that Xi = Xj . Then,
{ 6 }
for N 4 and χ < 4 1 1 , there exists a unique (in distribution) non explosive
• ≥ − N−1
solution of (1.4), star(cid:16)ting from(cid:17)any x M. Moreover, the process is strong Markov,
∈
lives in M and admits a symmetric σ-finite, invariant measure given by
µ(dX1,...,dXN)= Π1≤i<j≤N Xi Xj −Nχ dX1...dXN ,
k − k
for N 2, if χ > 4, the system (1.4) does not admit any global solution (i.e. defined
• on the≥whole time interval R+),
for N 2, if χ = 4, either the system (1.4) explodes or the N particles are glued in
• ≥
finite time.
Since later we will be interested in the limit N + , this theorem is in a sense optimal:
→ ∞
for χ < 4 we have no asymptotic explosion while for χ > 4 the system explodes. Also notice
that in the limiting case χ = 4, we have (at least) explosion for the density of the stochastic
system and not for the equation (1.1).
The proof of this theorem is partly “pathwise”, based on comparisons between one dimen-
sional diffusion processes and the behavior of squared Bessel processes, partly based on
Dirichlet forms theory and partly based on an uniqueness result for 2 dimensional skew
Bessel processes obtained in [6]. The latter is only used to get rid of a non allowed polar set
of starting positions which appears when using Dirichlet forms.
The remaining part of the whole program will be the aim of future works.
2. Study of the system (1.4).
Most of the proofs in this section will use comparison with squared Bessel processes. Let us
recall some basic results on these processes.
Definition 2.1. Let δ R. The unique strong solution (up to some explosion time τ) of the
∈
following one dimensional stochastic integral equation
t
Z = z+2 Z dB +δt,
t s s
Z0
p
is called the (generalized) squared Bessel process of dimension δ starting from z 0.
≥
In general squared Bessel processes are only defined for δ 0, that is why we used the word
≥
generalized in the previous definition. For these processes the following properties are known
Proposition 2.2. Let Z be a generalized squared Bessel process of dimension δ. Let τ the
0
first hitting time of the origin.
If δ < 0, then τ is almost surely finite and equal to the explosion time,
0
•
if δ = 0, then τ <+ and Z = 0 for all t τ almost surely,
0 t 0
• ∞ ≥
if 0 < δ < 2, then τ < + almost surely and the origin is instantaneously reflecting,
0
• ∞
if δ 2, then the origin is polar (hence τ = + almost surely).
0
• ≥ ∞
For all this see [13] chapter XI, proposition 1.5.
4 P.CATTIAUXANDL.PE´DE`CHES
Now come back to (1.4). For simplicity we skip the index N in the definition of the process.
Since all coefficients are locally Lipschitz outside the set
A = there exists (at least) a pair i= j such that Xi = Xj
6
(cid:8) (cid:9)
and bounded when the distance to A is bounded from below, the only problem is the one
of collisions between particles. As usual we denote by ξ the lifetime of the process. For
simplicity we also assume, for the moment, that the starting point does not belong to A, so
that the lifetime is almost surely positive.
For 2 k N we define K = 1,...,k and K¯2 = (i,j) K2 i = j . We shall say that
≤ ≤ { } { ∈ | 6 }
a k-collision occurs at (a random) time T if Xi = Xj for all (i,j) K¯2, Xl = Xi for all
T T ∈ T 6 T
l > k. Of course, there is no lack of generality when looking at the first k indices, and we
can also assume that at this peculiar time T, any other collision involves at most k other
particles.
In what follows we denote Di,j = Xi Xj, and
−
Zk = Di,j 2 .
k k
(i,jX)∈K¯2
Of course a k-collision occurs at time T if and only if T < ξ and Zk = 0.
T
Let us study the process Zk. Applying Ito’s formula we get on t < ξ
t χ
Zk = Zk +2√2 Di,j(dBi dBj)+4k(k 1) 2 t (2.3)
t 0 s s− s − − N
Z0 (i,jX)∈K¯2 (cid:16) (cid:17)
2χ t N Di,l Dl,j
Di,j s + s ds.
− N Z0 (i,jX)∈K¯2 s Xl=1 k Dsi,l k2 kDsl,j k2!
l6=i,j
We denote
dMk = Di,j(dBi dBj)
s s s − s
(i,jX)∈K¯2
N i,l l,j
D D
Ek = Di,j s + s
s (i,jX)∈K¯2 s Xl=1 kDsi,l k2 k Dsl,j k2!
l6=i,j
the martingale part and the non-constant drift part.
2.1. Investigation of the martingale part Mk. Letuscomputethemartingalebracket,
using the immediate Di,l = Dl,i and Di,l+Dl,j = Di,j.
−
STOCHASTIC KELLER-SEGEL 5
d< Mk > = < Di,j(dBi dBj),Dl,m(dBl dBm) >
s s s− s s s − s
(i,jX)∈K¯2
(l,m)∈K¯2
= Di,jDl,m(δ δ δ +δ )ds
s s il − im− jl jm
(i,jX)∈K¯2
(l,m)∈K¯2
= Di,j Di,m+ Di,l+ Dl,j + Dm,j ds
s  s s s s 
(i,jX)∈K¯2 mX∈K lX∈K lX∈K mX∈K
 m6=i l6=i l6=j m6=j 
 
= 2 Di,j Di,m+ Dm,j ds
s  s s 
(i,jX)∈K¯2 mX∈K mX∈K
m6=i m6=j 
 
= 2k Di,j 2
k s k
(i,jX)∈K¯2
i.e. finally
d< Mk > = 2kZKds. (2.4)
s s
According to Doob’s representation theorem (applied to 1I d < Mk > ), there exists (on
s<ξ s
an extension of the initial probability space) a one dimensional Brownian motion Wk such
that almost surely for t < ξ
t t
2√2 Di,j(dBi dBj) = 4√k ZkdWk. (2.5)
s s − s s s
Z0 (i,jX)∈K¯2 Z0 q
2.2. Reduction of the drift term. In order to study the drift term Ek we will divide
t
it into two sums: the first one, Ck taking into consideration the i and j in K, i.e. the
t
pair of particles which will be directly involved in the eventual k-collision; the other one Rk,
t
involving the remaining indices.
More precisely Ek = Ck +Rk with
t t t
i,l l,j
D D
Ck = Di,j t + t ,
t t Di,l 2 Dl,j 2!
(i,jX)∈K¯2 lX∈K k t k k t k
l6=i,j
N i,l l,j
D D
Rk = Di,j t + t .
t t Di,l 2 Dl,j 2!
(i,jX)∈K¯2 l=Xk+1 k t k k t k
6 P.CATTIAUXANDL.PE´DE`CHES
We will deal with Rk later. First we ought to simplify the expression of Ck. Indeed
t t
i,l l,j
D D
Ck = Di,j t + t
t t Di,l 2 Dl,j 2!
(i,jX)∈K¯2 lX∈K k t k k t k
l6=i,j
i,l l,j
D D
=  t Di,j+ t Di,j
Di,l 2 t  Dl,j 2 t 
(i,Xl)∈K¯2 k t k jjX6=∈iK,l  (j,Xl)∈K¯2 k t k iiX6=∈jK,l 
 
 
i,l l,i
D D
=  t Di,j + t Dl,j +
Di,l 2 t Dl,i 2 t
(i,Xl)∈K¯2 k t k jX∈K k t k jX∈K 
i<l  j6=i,l j6=i,l 
 
l,j j,l
D D
+  t Di,j + t Di,l
Dl,j 2 t Dj,l 2 t
(j,Xlj)<∈lK¯2 k t k iiX6=∈jK,l k t k jiX6=∈jK,l 
 
so that using again Dl,iDl,j = Di,lDj,l the latter is still equal to
i,l l,j
D D
= t  (Di,j +Dj,l)+ t (Di,j +Dl,i)
Di,l 2 t t Di,l 2  t t 
(i,Xli)<∈lK¯2 k t k jjX6=∈iK,l  (j,Xlj)<∈lK¯2 k t k iiX6=∈jK,l 
 
 
and using again Di,l+Dl,j = Di,j, we finally arrive at
Ck = 2(k 2) # (i,l) K2 i < l = k(k 1)(k 2). (2.6)
t − × { ∈ | } − −
2.3. Back to the process Zk. With the results obtained in (2.5) and (2.6) we may
simplify (2.3), writing (still on t < ξ)
t kχ 2χ t
Zk = Z0+4√k ZkdWk +2k(k 1) 4 t Rkds. (2.7)
t t s s − − N − N s
Z0 q (cid:18) (cid:19) Z0
Hence defining
1
Vk = Zk
t 4k t
the process Vk satisfies
kχ χ
dVk = 2 VkdWk +(k 1) 2 dt Rkdt, (2.8)
t t t − − 2N − 2kN t
q (cid:18) (cid:19)
i.e. can be viewed as a perturbation of a squared Bessel process of dimension
kχ
δ = (k 1) 2 ,
− − 2N
(cid:18) (cid:19)
we shall denote by Uk in the sequel.
STOCHASTIC KELLER-SEGEL 7
2.4. The case of an N-collision. If k = N, RN = 0 so that VN is exactly the squared
Bessel process of dimension N−1(4 χ). Hence, according to proposition 2.2
2 −
if χ > 4 there is explosion in finite time for the process VN (hence for X also),
•
if χ = 4, there is an almost sure N-collision in finite time, and then all the particles
•
are glued, provided no explosion occurred before for the process X,
if 4 1 1 < χ < 4 there is an almost sure N-collision in finite time, provided
• − N−1
no e(cid:16)xplosion o(cid:17)ccurred before for the process X,
if χ 4 1 1 there is almost surely no N-collision (before explosion).
• ≤ − N−1
(cid:16) (cid:17)
In particular we see that the particle system immediately feels the critical value χ = 4, in
particular explosion occurs in finite time as soon as χ > 4.
For4 1 1 < χ < 4weknowthatVN isinstantaneouslyreflectedonceithitstheorigin,
− N−1
but it(cid:16)does not(cid:17)indicate whether all or only some particles will separate (we only know that
they are not all glued). Notice that when N = 2 this condition reduces to 0 < χ < 4, and
then both particles are separated. Hence in this very specific case, there is no explosion
(for the distance between both particles) in finite time almost surely, but there are always
2-collisions.
2.5. Towards non explosion for χ 4 1 1 . As we said before, the lifetime ξ is
≤ − N−1
greater than or equal to the first multiple(cid:16)collision t(cid:17)ime T.
Since we shall consider Vk as a perturbation of Uk, what happens for the latter ?
for χ > 4N, Uk reaches 0 in finite time a.s. and then explosion occurs,
• k
for χ = 4N, Uk reaches 0 and is sticked,
• k
for 4N > χ > 4N 1 1 , Uk reaches 0 and is instantaneously reflected,
• k k − k−1
for χ 4N 1 (cid:16)1 , Uk(cid:17)does not hit 0 in finite time a.s.
• ≤ k − k−1
(cid:16) (cid:17)
Lemma 2.9. For all 3 k N, it holds
≤ ≤
4N 1 1
1 4 1 .
k − k 1 ≥ − N 1
(cid:18) − (cid:19) (cid:18) − (cid:19)
Proof. Introduce the function
4N 1
u g(u) = 1
7→ u − u 1
(cid:18) − (cid:19)
defined for u > 1. Then
4N 2 u 1
g′(u) = − +
u(u 1) u u 1
− (cid:18) − (cid:19)
is negative on [2 + √2,+ [, so that the lemma is proved for N k 4. For k = 3, it
amounts to N N−2 whi∞ch is true for all N 3 (with equality for≥N =≥3 and N = 4). (cid:3)
6 ≥ N−1 ≥
In particular, since χ 4 1 1 , Uk never hits 0 for 3 k N, while it reaches 0 but
≤ − N−1 ≤ ≤
is instantaneously reflected(cid:16)for k =(cid:17)2. What we expect is that the same occurs for Vk.
8 P.CATTIAUXANDL.PE´DE`CHES
In order to prove it, let T be the first multiple collision time. With our convention (changing
indices if necessary) there exists some 2 k N such that T is the first k-collision time Tk.
≤ ≤
Note that this does not prevent other k′-collisions (with k′ k) possibly at the same time
≤
T for the particles with indices larger than k +1. But as we will see this will not change
anything. The reasoning will be the same but the conclusion completely different for k = 2
and for k 3.
≥
2.5.1. No k-collisions for k 3. Introduce, for ε > 0, the random set
≥
Ak = T = Tk < + and inf inf Di,l 2ε .
ε { ∞ i∈K,l≥k+1 t≤T k t k≥ }
It holds
T = Tk < + = Ak.
{ ∞} ε
ε∈1/N
[
In particular if P(T = Tk < + )> 0 there exists some ε > 0 so that P(Ak) > 0.
∞ ε
We shall see that this is impossible when k 3.
≥
Indeed recall that
N i,l l,j
D D
Rk = Di,j t + t .
t t Di,l 2 Dl,j 2!
(i,jX)∈K¯2 l=Xk+1 k t k k t k
So on Ak, we have, for t T,
ε ≤
N
1 1
Rk Di,j +
| t| ≤ k t k Di,l 2 Dl,j 2!
(i,jX)∈K¯2 l=Xk+1 k t k k t k
N k
i,j
D −
≤ k t k ε
(i,jX)∈K¯2
N k
− k(k 1) Zk
≤ ε − t
q
the latter being a consequence of Cpauchy-Schwarz inequality. Thus on Ak, for t T
ε ≤
2
Rk (N k)k√k 1 Vk. (2.10)
| t| ≤ ε − − t
q
Hence, on Ak for t T the drift bk (which is not Markovian) of Vk satisfies
ε ≤ t
(k 1) Nχ 2
bk ˆbk(v) = − 4 (N k)k√k 1√v. (2.11)
≥ 2 − k − ε − −
(cid:18) (cid:19)
In particular for any θ > 0,
(k 1) Nχ
ˆbk(v) − 4 θ
≥ 2 − k −
(cid:18) (cid:19)
provided v is small enough. Thus the hitting time of the origin for the process with drift ˆbk
is larger than the one for the corresponding squared Bessel process (thanks to well known
comparison results between one dimensional Ito processes, see e.g. [9] Chap.6, Thm 1.1),
and since this holds for all θ, finally is larger than the one of Uk. But as we already saw,
Uk never hits the origin for 3 k. Using again the comparison theorem (this time with bk
≤
STOCHASTIC KELLER-SEGEL 9
and ˆbk(v)), Vk does not hit the origin in finite time on Ak which is in contradiction with
ε
P(Ak)> 0.
ε
2.5.2. About 2-collisions. Actually all we have done in the previous sub subsection is un-
changed for k = 2, except theconclusion. IndeedU2 reaches theorigin butis instantaneously
reflected. So V2 (on A2) can reach the origin too, but is also instantaneously reflected. Ac-
ε
tually using that
(k 1) Nχ 2
bk(v) ¯bk(v) = − 4 + (N k)k√k 1√v,
≤ 2 − k ε − −
(cid:18) (cid:19)
together with the Feller’s explosion test, it is easily seen that V2 will reach the origin with a
(strictly) positive probability (presumably equal to one, but this is not important for us).
ButthisinstantaneousreflectionisnotenoughforthenonexplosionoftheprocessX,because
Xi is R2 valued. Before going further in the construction, let us notice another important
fact: there are no multiple 2-collisions at the same time, i.e. starting from Ac the process
lives in M at least up to the explosion time ξ. Of course this is meaningful provided N 4.
≥
To prove the previous sentence, first look at
Y = D1,2 2 + D3,4 2,
t k t k k t k
assuming that Y = 0 and that no other 2-collision happens at time T. It is easily seen that
T
(just take care that we had an extra factor 2 in our definition of Zk)
t
t χ
Y = Y +2√2 D1,2(dB1 dB2)+D3,4(dB3 dB4) +8 2 t
t 0 s s − s s s − s − N
Z0 (cid:16) (cid:17)
(cid:0) (cid:1)
χ t N D1,l Dl,2 N D3,l Dl,4
D1,2 s + s + D3,4 s + s ds,
− N Z0  s Xl=1 kDs1,l k2 k Dsl,2 k2! s Xl=1 k Ds3,l k2 kDsl,4 k2!
 l=6 1,2 l6=3,4 
 
so that, defining V = Y /4 we get that
t t
χ
dV = 2 V dW +2 2 dt+ R dt
t t t t
− N
where R is a remaining term wpe can manag(cid:16)e just a(cid:17)s we did for Zk. Since for N 4,
2 2 χt 2, V , hence Y does not hit the origin. Notice that if wte consider k( 2≥) 2-
− N ≥ t t ≥
collisions, the same reasoning is still true, just replacing 4 by 2k, the final equation being
(cid:0) (cid:1)
unchanged except for R .
t
2.5.3. Nonexplosion. Accordingtoallwhatprecedeswhatweneedtoproveistheexistence
of the solution of (1.4) with an initial configuration satisfying X = x with x1 = x2, all other
0
coordinates beingdifferent and different from x1 = x2. Indeed, on ξ <+ , X δM the set
ξ
∞ ∈
of particles with exactly two glued particles, so that if we can prove that starting from any
point of δM, we can build a strong solution on an interval [0,S] for some strictly positive
stopping time S, it will show that ξ = + almost surely. However we will not be able to
∞
prove the existence of such a strong solution. Actually we think that it does not exist. We
will thus build some weak solution and show uniqueness in some specific sense.
This will be the goal of the next sections.
10 P.CATTIAUXANDL.PE´DE`CHES
3. Building a solution.
3.1. Existence of a weak solution. Writing
M = Xk = Xl
i<j k6=l;l6=i,j
∪ ∩ { 6 }
we see that M is an open subset of R2N.
Recall that x δM means that exactly two coordinates coincide (say x1 = x2), all other
∈
coordinates being distinct and distinct from x1. We may thus define
d = min i 3;i = j; j = 1,...,N ; xi xj > 0,
x
{ ≥ 6 k − k}
so that
Ω = ΠN B(xj,d /2) M, (3.1)
x j=1 x ⊂
and points y Ω δM will satisfy y1 = y2. If x / δM, we may similarly define d =
x x
∈ ∩ ∈
min i = j; j = 1,...,N ; xi xj and then Ω . In all cases the balls B(.,.) are the open
x
{ 6 k − k}
balls. Now if K is some compact subset of M we can cover K by a finite number of sets Ω ,
x
so that for any measure µ, a function g belongs to L1 (M,µ) if and only if g L1(Ω ,µ) for
loc ∈ x
all x in M.
The natural measure to be considered is
µ(dX1,...,dXN)= Π1≤i<j≤N Xi Xj −Nχ dX1...dXN , (3.2)
k − k
since it is, at least formally, the symmetric measure for the system (1.4).
It is clear that for x / δM, µ is a bounded measure on Ω . When x δM, say that x1 = x2
x
∈ ∈
and perform the change of variables
Y1 = X1 X2 , Y2 = X1+X2.
−
In restriction to Ω , µ can be written
x
µ(dX1,...,dXN)= C(N,x) Y1 −Nχ dY1dY2dX3...dXN ,
k k
hence is a bounded measure on Ω provided χ < 2N just looking at polar coordinates for
x
Y1. In this case it immediately follows that µ is a σ finite measure on M. Also remark that
if f is compactly supported by K and belongs to L2(dµ) then it belongs to L2(dX) and
f2dX sup( Y1 Nχ) f2dµ.
≤ k k
ZK K ZK
But we can say much more.
To this end consider the symmetric form
(f,g) = < f, g > dµ , f,g C∞(M). (3.3)
E ∇ ∇ ∈ 0
ZM
First we check that this form is closable in the sense of [7]. To this end it is enough to show
thatitisaclosableformwhenrestrictedtofunctionsf,g C∞(Ω )forallx M. Ifx / δM
∈ 0 x ∈ ∈
the form is equivalent to the usual scalar product on square Lebesgue integrable functions,
so that it is enough to look at x δM.
∈
Hence let f be a sequence of functions in C∞(Ω ), converging to 0 in L2(dµ) and such that
n 0 x
f converges to some vector valued function g in L2(dµ). What we need to prove is that g
n
∇
is equal to 0. To this end consider a vector valued function h which is smooth and compactly