Can one observe the bottleneckness of a space by the heat distribution? Текст научной статьи по специальности «Физика»

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DOI: https://doi.Org/10.15688/mpcm.jvolsu.2017.3.6

UDC 517 LBC 22.161

CAN ONE OBSERVE THE BOTTLENECKNESS OF A SPACE BY THE HEAT DISTRIBUTION?1

Satoshi Ishiwata

Professor, Department of Mathematical Sciences, Yamagata University [email protected] Yamagata 990-8560, Japan

Abstract. In this paper we discuss a bottleneck structure of a non-compact manifold appearing in the behavior of the heat kernel. This is regarded as an inverse problem of heat kernel estimates on manifolds with ends obtained in [10] and [8]. As a result, if a non-parabolic manifold is divided into two domains by a partition and we have suitable heat kernel estimates between different domains, we obtain an upper bound of the capacity growth of b-skin of the partition. By this estimate of the capacity, we obtain an upper bound of the first non-zero Neumann eigenvalue of Laplace — Beltrami operator on balls. Under the assumption of an isoperimetric inequality, an upper bound of the volume growth of the b-skin of the partition is also obtained.

Key words: heat kernel, manifold with ends, inverse problem.

1. Introduction

In the real world, waves, like as electromagnetic waves (light, radiation ray, infrared ray etc...) are good tools to observe a structure of given space (ex. location of obstacles, airplanes, planets etc...). However, in the following situation, heat distribution might be more useful to observe the structure of the space. Consider a space separated into two domains by a "dark" partition (difficult to see by the light). Here, assume that we know the heat distribution well on each separated domain and we have some data of the heat ^ distribution between two domains. Then what can one observe the dark partition? Can one S see how large it is? Can one see inside of the partition?

oor This problem is inspired by a recent progress of the study of the heat kernel on non-2 compact Riemannian manifolds. Let M be a geodesically complete Riemannian manifold. i? The heat kernel p(t,x,y) is the minimal positive fundamental solution of the heat equation

© dtu(x,t) = Au(x,t) (x,t) E M x (0, to),

where A is the Laplace-Beltrami operator on M. It is well-known that the heat kernel p(t,x,y) can be also regarded as the transition density of the Brownian motion ({Xt}r>o, {Px}xeM) generated by A on M, namely, for any Borel set A c M,

Px(Xt e A) = f p(t,x,y)dv(y),

J A

where fx is the Riemannian measure on M.

On Rra, the heat kernel is given by the Gaussian function:

^ N 1 ll^ll2 p(t,x,y) = ———e 4t . ' (4nt)n/2

On a general manifold, the heat kernel is sensitive to the underlying space and the dependence has been an important subject in probability theory, harmonic analysis and geometry. In particular, on a non-compact Riemannian manifold, long time behavior of the heat kernel has been investigated under various settings by many authors. For the detail, see [2; 5; 15] and references therein.

Recently, Grigor'yan and Saloff-Coste [10], Grigor'yan, and Saloff-Coste and the author [8] proved heat kernel estimates on manifolds with ends. Let us quickly review these results. Let Mi,..., Mk be non-compact manifolds and for 1 < I < k, Ki a compact set of Mi. A manifold M is called a connected sum of Mi,..., Mk with a central part K c M if

M\K = Q El = Q (Mi\Ki).

i=i i=i

Then the connected sum M is denoted by Mi#M2# ••• #Mk. To study the behavior of the heat kernel on connected sums, parabolicity/non-parabolicity of the manifold plays an important role. Here, a manifold is parabolic if the Brownian motion is recurrent and non-parabolic if the Brownian motion is transient. Various equivalent conditions to parabolicity/non-parabolicity are known. See [3] for the detail.

When the central part K is compact and the all ends are non-parabolic together with a suitable condition (Poincare inequality (PI), volume doubling property (VD) and relatively connected annuli condition (RCA)), behavior of the heat kernel is proved in [10]. For the detail of these conditions, see for instance [9; 11; 12; 14; 15] and references therein. Here we note only that Grigor'yan [6] and Saloff-Coste [14] proved that the condition (PI) together with (VD) is equivalent to the condition of the manifold that Li — Yau type heat kernel estimates (two-sided Gaussian estimates) holds:

jd2(x,y)

P(t,x,y) -^e ~, (1)

V (x,y/t)

where V(x, r) is the Riemannian measure of a geodesic ball B(x, r) and the symbol x means that both < and > hold by changing constants C > 0 and b > 0.

For x e Ei c Mi, denote by Vi(x,r) the Riemannian measure of a geodesic ball Bi(x, r) in Mi. Let

Vmin(r) = min Vi(r) = min Vi(ohr),

1< l<k Kl<k

where oi E Ki is a reference point on each Ml in Kl. For x E Ej, and t > 0, let |x| = = d(x,K) + 1 and we define a function H(x, t) as

H {x, () = ™{i,.gl + (£ J.

Then the following heat kernel estimates between different ends are obtained: Theorem 1.1 (Grigor'yan and Saloff-Coste [10, Theorems 4.9 and 5.10]). Let M be a connected sum of non-parabolic manifolds M1,M2,...,Mk with (PI), (VD) and (RCA). For x E Ei, y E Ej with i = j and t > 0,

P(t ,X,V) (H!*m0+hmi+Hi^) e-¿v. (2)

V VmUVt.) Vi(Vt) Vj (St))

In particular, for x E Ei, y E Ej with |x| & r and |y| « 1 (resp. |x| « 1 and |y| & r) (center-middle time regime), we obtain

p(r2,x, y) x (resp. ) .

Vi(r) \ Vj (r)J

Vi(r) \ Vj (r), For x E Ei and y E Ej with |x| « |y| & r (middle time regime), we have

(3)

r2 C

p(r2,x, y) x c << (4)

Vi (r)Vj (r) VmaX(r)

where Vmax(r) = max1<i<k Vi(r). These estimates motivate us to observe the bottleneckness of the space in terms of the heat kernel behavior. We note that, in these regimens (center-middle time regime, middle time regime), the effect of other ends (first term in (2)) to the heat kernel is dominated.

When the connected sum is non-parabolic but there exist some parabolic ends (mixed case), Grigor'yan and Saloff-Coste proved the heat kernel estimates by using Doob's transform. For 1 < I < k, set

hi(r) = 1 +

For x E Ei and r > 0, set also

С

Г2 ds \

i шщ) +

Щх, r) = Шх\) + h2(r))Vt(x, r)

and for t > 0,

H( ) = \x\^ 1 ( f" ds \

(x t] '= hj(\x\)Vt(\x\) + hi(\x\)hi(Vt) \J\x\2 ШЩ )

+

Then the following heat kernel estimates between different ends are obtained: Theorem 1.2 (Grigor'yan and Saloff-Coste [10, Theorem 6.5 and Remark 6.7]). Let M be a connected sum of manifolds with (PI), (VD) and (RCA) and at least one end is non-parabolic. Then, for x E Ei and y E Ej with i = j and t > 0,

P(t,x,y) x СЫШ, ы( H^■t] + + f^) e^

V Vmm(\t) Vj (Vt) Vi(Vt) J

Let us observe heat kernel behavior in some typical regimes. For simplicity, let us consider the case that Vi(r) ~ rai (see [10, Example 6.11 and Corollary 6.12]).

If 2 < Oi, a, the estimate is the same as the case that all ends are non-parabolic stated in (3), (4).

If 0 < Oi, a < 2, for x E Ei and y E Ej with | 'X ~ v, \ru\ ^ 1 (resp. | 1 and |y| &), we obtain

p(r2,x,y) x ^ << , | resp. ^ << , 1 . (5)

Fy ' ,yj rA-a Vi(r) V rA-a Vj(r)J w

For x E Ei and y E Ej with |x| & |y| & r, we obtain

C r2

P(r2,x,y) X << c-

r6-ai-aj Vi(r)Vj (r)'

In this situation, the estimate in (5) (center-middle time regime) shows that the heat kernel behavior on each one end is far different from the Li — Yau type bound in (1).

If a < 2 < a, for x E Ei and y E Ej with |x| & r and |y| & 1 (resp. |x| & 1 and M ~ r),

/ 2 C C ( c c \

p(r2,x,y) x —— << resp. — & . . . (6)

> r4-ni Vi(r) y F rnj v.(r)J

For x E Ei and y E Ej with |x| & |y| & r, we obtain

C f2

P(r2,*,y) X o+a-a- <<

r2+ai-aj Vi(r)Vj (r)'

In this situation, similar to the previous case, the estimate in (6) with |x| & r and |y| & 1 shows us the heat kernel behavior on Ei is different from the Li — Yau type bound in (1).

When the connected sum is parabolic (i.e., all ends are parabolic), the heat kernel estimates are obtained in [8]. To state the result, we prepare some notation. A parabolic manifold M is called critical if V(x,r) & r2 and subcritical if

h(r) = i< C

11 V (y/0)- V (r)'

We remark that R2 is critical and R is subcritical.

Then the following heat kernel estimates between different ends are obtained: Theorem 1.3 (Grigor'yan, Ishiwata, Saloff-Coste [8, Theorem 2.3]). Let M be a connected sum of parabolic manifolds with (PI), (VD) and (RCA). If all ends are subcritical, then for x E Ei and y E Ej,

P(t,x,y) —e ~.

'max( V t)

This shows that even if x E Ei and y E Ej are far enough from the center, the heat kernel is similar to the on-diagonal bound in the central part. Hence, in this situation, the bottleneckness of the space cannot be seen in the heat kernel estimates in the middle time regime.

When some ends are critical, we can see a bottleneck effect in the behavior of the heat kernel, but the effect is milder than that on non-parabolic connected sums. For example, consider M = R2#R2. Then, for x E E\ and y E E2 with |

p(r2,x,y) C

r log r

which is slightly smaller than the on-diagonal bound —.

2. Main results

In view of the above observations of the heat kernel behavior on connected sums, we consider an inverse problem, which asks a bottleneck structure of a manifold appearing in the heat kernel estimates.

The condition of the manifold we need to observe the bottleneck structure is as follows. First, from the heat kernel estimates on connected sums of parabolic manifolds in Theorem 1.3, the manifold is required to be non-parabolic. Take a non-parabolic manifold divided into two domains by a partition. From the heat kernel estimates in the center-middle time regime (3), (5), our second requirement of the space is that the heat kernel on each one domain (including near the partition) is well-known (i.e. Li — Yau type bound (1)). Under these settings of the underlying space, our interest is a bottleneck structure of the space appearing in a heat kernel behavior between different domains.

The bottleneckness of the space is characterized by the capacity of closed sets and the first non-zero Neumann eigenvalue on balls. For a non-empty closed set F on M, the capacity of the capacitor (F, M) is defined by

cap(F) := inf i |V/1>,

f SLip0(M), I f=lonF

where Lip0(M) is spaces of all compactly supported Lipschitz functions on M. For an open set Q C M, let Af(Q) be the first non-zero Neumann eigenvalue on Q, namely, by definition, Af(Q) is the minimum positive vaule so that there exist f E V := C2(Q) n C1 (Q) such that

Af = Af in Q, vf = 0 on dQ,

{

where v is the normal derivative on dQ. By max-min theorem (c.f. [1, p. 17]),

fn IV/

Af (Q) = inf

fL |/- /n|2^H'

where fa = -^y fa fdy.. We note that the (strong) Poincare inequality (PI) is rewritten by

Af (В(x,r)) > .

For a subset A C M and 6 > 0, let = B(A, b), that is, the 6-open neighborhood of A. We call A6\A the b-skin of A. Then we obtain the following.

Theorem 2.1. Let M be a geodesically complete non-parabolic manifold divided into two unbounded connected components and E2 by a closed set F c M. Fix a refernce point o G F and we assume that there exist curves xr G E\ and yr G E2 satisfying the following conditions:

(i) There exist constants a, b > 0 such that for all r > d(o, dF), r2/4 < s < r2 and for all z G Fb(o, r) := FS\F n B(o,r),

n

p(s, xr,z) >

p{s,yr,z) >

Vi(xr ,r)' a

У2(Уг,r),

where Vt(x, r) = |(B(x, r) H Et).

(ii) There exists a positive valued function D(r) such that

(See Fig. 1).

p(r2,xr,yr) <

D(r)

Vi(xr ,r)V2(yr ,r)'

El

F

Fs(o.r)

E,

Then we obtain

Fig. 1. Partition of M, xr, yr and F^(o, r)

cap(F5(o,r)) < —(f.

a2r2

(7)

This concludes also the upper bound of the first non-zero Neumann eigenvalue A^ on balls. Fix x G M and r > maxi=1,2 d(x, Ei). For all 0 < e < 1, set

E\(s) := {z G Ei : Pz(tf^) < e)}, E2(e) := {z G E2 : Pz(xFb{x,r) < e)}. (8)

Then we obtain

Af (B(x,r)) <

8D(d(x,o) + r)

a2(1 — t)2(d(x, o) + r)2 mini=l,2 y.(B(x, г) П Ei(t))

(9)

Corollary 2.2. Under the same setting of the above theorem, assume further that the isoperimetric inequality with isoperimetric function I holds:

-n-i(dQ) > I(-(Q)) VQ C M,

where -n-i(dQ) is the induced measure of the boundary of Q. Then by [4; 13], -(F6(o, r)) satisfies the following inequality:

l^(Fb(o,r))

In particular, if I satisfies

( dv \ \ (t?( 4d(t)

\J^(F6(o,r)) 12(V)J С2Г2

fOO

dv < (10)

J 12(v)

and D(r) = Cr2, we obtain

-(Fb(o,r)) < C'

for some constant C' > 0 and for all r > 0, namely,

-(Fb\F) <

As a more general situation, let us consider the case D(r) = Cra with some a > 2 and I(v) = vThen we obtain

-(Fs(o,r)) < C'r^(a-2). (11)

We note that the boundedness (10) implies the non-parabolicity of M (see [4, Section 3]). Moreover, we note also that the estimate in (11) may not be optimal. See Example 2.4 below.

Let us apply these results to some examples. Example 2.3 (Non-parabolic connected sum with compact central part). Let M is a

connected sum of manifolds M\,M2,... ,Mk with (PI), (VD) and (RCA) along a compact central part K. Assume that there exits at least one non-parabolic end. As a partition, take F = KUE3 • • •UEk and choose curves xr E E\ and yr E E2 so that d(xr, K) & d(yr, K) & r.

If Mi and M2 are non-parabolic, then the condition (i) in Theorem 2.1 holds by (3) and (ii) is also true with D(r) = Cr2 by (4). Then Theorem 2.1 asserts that

4C

cap(F&(o,r)) < —r a2

for any r > 0, namely,

4C

cap(Fb\F) < —.

a2

Moreover, for all x E M and for sufficiently large r > maxi=1,2 d(x,Ei), we obtain by (9)

fn fi'

Af (B(x,r)) < —- <<-2 Vmm(r) r2

for some constant C' > 0, which fails the Poincare inequality (PI). We note that in this situation, we cannot observe other ends.

If one of Mi, M2 is parabolic, then the condition (i) is failed by the estimates in (5) and (6).

Next, let us verify the effect of Theorem 2.1 on a connected sum with a non-compact central part.

Example 2.4 (Connected sum of two copies of 1 along a surface of revolution (c.f. [7])). For n > 3, let M be a connected sum of two copies of 1 along the surface of revolution A(m, a) (0 < m < n — 2, 0 < a < 1) defined by

A(m, a) = [x e Rra : h(x) < r(x)a]

where

r(x)

(1+ E , h(x)=( £ xA

\ 1 <i<<m / \m+1<i<n J

1/2

In the case m = 0, we always take a =0.

As a partition F of the space, take the joint part along the surface of A(m, a). We choose continuous curves xr in Ei and yr in E2) given by

yr = ((V-^O, 1 + r, 0,..., 0),

m

which is vertical to the subspace Rm С Rra (see Fig. 2).

]l(x) = /'(;!')Q

A(m, a)

Fig. 2. A(m, a) c 1" and xr Then, by using Theorem 1.1 in [7], for any z e Fs(o,r), we obtain

p(r2,xr,z) x p(r2, yr,z) X —

and

p(r2,xr ,yr)

с

y.n+(1-a)(n-m—2)

^^,n—(1 — a)(n—m—2)

whence the conditions (i) and (ii) in Theorem 2.1 are true with D(r) Then Theorem 2.1 asserts that

cap(F6(o,r)) < *lr™+a("-2), a2

which gives an optimal estimate of the capacity growth of the 6-skin of the surface of A(m, a). Moreover, for all x E M and sufficiently large r > maxi=1,2 d(x,Ei), we obtain by (9) that

fit rn

Af (B(x,r)) < 2Mi1 ,,-;- << ^

1 \ \ ) n — ^,2+(1 — a)(n—m—2) ^2

for some constant C' > 0, which fails the Poincare inequality (PI).

n — 1

Since the isoperimetric inequality with I(v) = cv~ is true (see [7, Section 3]), Corollary 2.2 asserts that

l(F6(o,r)) < C'ran+^(1-a)

while

l(F&(o,r)) « rm+a(n-m-1) << C'ran+ ^(1-a).

3. Proof of Theorem 2.1

The following lemma is a key to prove Theorem 2.1. Lemma 3.1 (c.f. [10; 11]). Let M be a non-parabolic manifold. For any compact set F C M and for all x,y G F and for all t > 0,

P(t,X,y) > -cap(F) mf p(s,x,z) mf p(s,z,y).

4 zEdF zEdF

t/4<s<t t/4<s<t

Proof. The proof is similar to Lemma 3.1 in [10]. Let tf be the first hitting time to F of a Brownian path u, namely

tf(u) = inf{t > 0 : Xt(u) G F}. We denote by (t,x) the hitting probability

P^Tf < t).

Let Ez be the expectation of the Brownian motion on M. For any Borel set A C M, we have

Px(Xt G A) = [ p(t,x,y)di(y) = Ex(bA(Xt)) =

J A

= Ex (( 1{tf<t/2} + 1{t/2<Tp<t} + 1{tf>t}) bA(Xt)) = = Ex (1{TF<t/2}bA(Xt)) + Ex (1{t/2<TF<t}bA(Xt)) +

+ Ex (1{TF>t}bA(Xt)) > > Ex (1{ tF<t/2}bA(Xt)) ,

where 6A is the characteristic function of A. By the strong Markov property of the Brownian motion on M, we obtain

E^ (1{tf<t/2}bA(Xt)) = Ez ^1{tf<t/2}ExTF (& A(Xt-TF))) = = Ex <t/2}PxTp (Xt-TF G A)} >

> l,x) inf P,(X, G A) >

V 2 J t/2<B<t

v ' zEdF

> Lbv tf p(8,z,y)dl(y).

A V ' zEdF

Since A C M is arbitrary, this concludes that

pit,x, y) > l,x) inf p(s,z, y).

\ 2 / t/2<s<t

v 7 zeeF

Since M is non-parabolic and F is compact, we obtain by Theorem 3.7 in [11] that for any

xGF,

ft \ f^2 t -,x) > cap(F) inf p(s,z,x)ds > -cap(F) inf p(s,z,x),

\2 ) In zedF 4 t/4<s<t/2

y 7 ,/U zEdF

which concludes the lemma.

Proof of Theorem 2.1. Substituting the assumptions (i) and (ii) of the heat kernel into the above lemma for F = F&(o, r) and t = r2, we obtain

D(r) r2 . ^ . a a

() > -rcap(F6(o, r)).

Vi(xr, r),V2(yr, r) 4 ' Vi(xr, r)V2(yr, r)

which concludes (7).

Next we prove the upper bound of the first non-zero Neumann eigenvalue (9). For x G M and r > maxi=1,2d(x, Ei), choose a test function f G C™(B(x, r)) as

( 1 - VFs(x,r)(x) ifx GB(x, r) HE1

f(x) = I 0 ifx G B(x, r) HK (12)

\ -c (1 - V Fs(x,r)(x)) ifx GB(x, r) HE2,

where V Fs(x,r)(x) = VFs(x,r)(<x,x) and c = c(r) is a positive constant so that fB(x,r) = 0. Since the hitting probability V F(x) is the equilibrium potential of the capacitor (F, M), that is,

cap( F) = |VФF |2 d^, jm

we obtain

A f (B(x, r)) <

Ntui u Jb{x,T) IV^2dV

<

!в(х,г) tf12

cap ( F{,(x, r)) + c2cap ( F^(x, r))

'¡B(x,r )r\Ei 11 - VF, (x,r )\2dl + C2JB(xr )nE2 11 - VFs(;c,r )|2 d!

By the definitions of the test function f in (12) and Ei(e) in (8), for i = 1, 2,

i \1 - VFb{x,r)\2di >/ \1 - VFb{x,r)\2di > (1 - e)2i(B(x, r) HEi(e)).

JB(x,r)rEi J B(x,r)rEi(e)

Then we obtain

An(B(xr))< _(1 + c2)cap( F&(x, r))__(13)

1 ( (, )) < (1 - e)2 h(B(x, r) HE1(e)) + c2i(B(x, r) H E2(e))]' K '

If с > 1, the estimate (13) implies that

Af (B(x,r)) < M™№(x,r))

(1 - e)2c2^(B(x,r) П E2(s)) 2cap(F,5(x, r))

(1 — e)2^(B(x,r) n E2(e)Y

If 0 < c < 1, we obtain

iNfnf w^ 2cap(F&(x,r))

Af (B(x,r)) <

(1 — i)2v(B(x,r) n Ei(e))' Since

F5(x, r) c F5 (o, d(x, o) + r), the estimate in (7) concludes that

Af (5(X,r)) < 8D (d(X,°) + r))

a2(1 — t)2(d(x, o) + r)2 mmi=1,2 ц(5(x, r) n Ei(t)) whence the proof of Theorem 2.1 is completed.

REMARK

1 This work was partially supported by JSPS, KAKENHI 21740034.

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МОЖНО ЛИ ПО РАСПРЕДЕЛЕНИЮ ТЕПЛА СУДИТЬ О НАЛИЧИИ «БУТЫЛОЧНОГО ГОРЛЫШКА» У ПРОСТРАНСТВА?

Сатоши Ишивата

Профессор, Факультет математических наук, Yamagata University [email protected] Yamagata 990-8560, Japan

Аннотация. В этой статье мы обсуждаем наличие узких мест в структуре некомпактного многообразия, проявляющееся в поведении ядра уравнения теплопроводности. Родственная обратная задача оценки ядра уравнения теплопроводности на многообразиях с концами изучалась в [8; 10]. В результате, если непараболическое многообразие делится на две области и имеются подходящие оценки ядра уравнения теплопроводности между разными областями, то мы получаем верхнюю оценку роста емкости 5-skin разбиения. По этой оценке емкости получаем верхнюю оценку первого ненулевого собственного числа Неймана оператора Лапласа — Бельтрами на шарах. В предположении изопериметрического неравенства также получен верхний предел роста объема 5-skin разбиения.

Ключевые слова: ядро уравнения теплопроводности, многообразие с концами, обратная задача.