OZAWA, SHIN

ROPPONGI, SUSUMU

Ozawa, Shin

Roppongi, Susumu

Osaka University Knowledge Archive

Heat kernel and singular variation of domains

Osaka City University Repository

Ozawa, S. and Roppongi, S.
Osaka J. Math.
32 (1995), 941-957
HEAT KERNEL AND SINGULAR VARIATION OF
DOMAINS
SHIN OZAWA and SUSUMU ROPPONGI
(Received March 9, 1994)
1. Introduction
We consider a bounded region M in Rn (n = 2 or 3) whose boundary is
smooth. Let w be a fixed point in M. By B(ε w) we denote a ball of radius ε
with the center w. We put ME = M\B(ε w).
Let U(x,y,t) (U(ε\x,y,i)\ respectively) be the heat kernel in M (ME respectively)
with the Dirichlet condition on its boundary dM (δM
ε
; respectively). That is, it
satisfies
(1.1) xedM,
Urn U(x,y,t) = δ(x —y) x,y e M
ί->0
lim U(£\xy,i) = δ(x-y) x,y e M
ε
ί-»0
(δt - ΔJ U(ε\x,y,t) = 0 x,y e Mε, ? > 0
(1-1) J
We put
(1.2)
and
(1.3)
JKx)= ί i/(xj,
JM
?β)/K^)= ί UV(x,y9
JME
Then, (7f/ and U^f satisfy the following.
f (dt-Δx)(Utf)(x)=Q xεM, t>0
942 S. OZAWA AND S. ROPPONGI
Q xedM, ί>0
xeMt, ί>0
E, 00
We want to construct an approximate kernel of U(e\x,y,t) by using U(x,y,ί). We
put
(1.4) y(ε\x,y,ί)= U(X,y,t)-Lβ(ε) f U(x,w,τ)U(w,y,t-τ)dτ,
Jo
where
( -2π(logε)-1 (ifn = 2)
Ll
n\^)~ J
I 4πε (ifΛ = 3).
and we put
' - f 1JM feL'(λf).
Let Γ and Γ
ε
 be operators on M and M& respectively. Then, ||7Ί|p, || Γε||pε
denotes the operator norm on LP(M\ LP(ME), respectively. Let/ and ft be functions
on M and M
ε9 respectively. Then, ||/||p, \\fε\\p,ε denotes the norm on LP(M\
Lp(M
ε
), respectively.
Let χ
ε
 denote the characteristic function of M
ε
. Then, we have the following
Theorems 1 and 2.
Theorem 1. Assume that n = 2. Then, there exists a constant C,, which
may depend on t but which is independent of ε such that
if pe (2,oo)
\ί p = 2
hold. Here je(0,l) is an arbitrary fixed constant.
Theorem 2. Assume that n = 3. Then, there exists a constant C, independent
HEAT KERNEL AND VARIATION OF DOMAINS 943
of ε such that
|| jy(ε)_y 0ε)y II
II ut %εyt Λε l lp ,ε
C,ε1+(2/p) if pe (3,oo)
Q(5-
s)/3 if ^ 6 [3/2, 3]
C(ε3-<2/P) if ;>e (1,3/2)
hold. Here ίe(0,l) is an arbitrary fixed constant.
REMARK. Thus, by Theorems 1 and 2, we know that
U(x,w,τ)U(w,y,t-r)dτ
gives a main asymptotic term of the difference between U(ε\x,y,t) and U(x,y,t).
)χJy) Γ
Jo
The Hadamard variation of the heat kernel was discussed in [2]. And we
have various papers on singular variation of domain. See, for example,
[3], [4], [5], [6].
We give the proof of Theorems 1 and 2 in section 2 and section 3,
respectively. In Appendix we give some properties of U(x,y,t) and U(ε\x,y9t).
Following an usual custom, we use the same letter C in inegualities which are
independent of ε.
2. Proof of Theorem 1
Throughout this section we assume that n = 2. We put
(2.1) U(x,y,t)=
where
(2.2) W(x^t) = (4πO~ w/2exp( - \x -y\2/ 4t).
We write B(ε',w) = B
ε
. Without loss of generality we may assume that κ> = 0.
We take arbitrary /e Lp(M
ε
). Let /be an extension of /to M which is 0 on
B
ε
. At first we want to estimate |(Fjβ)/Xx)||je6aBβ. By (2.2), we have
W(x9w9τ
Jo
(2.3) )dτ\ xedB
ε
= Γ(4πτ)-1exp(-e2/4τ)£/τ
Jo
fJε2/4ί
944 S. OZAWA AND S. ROPPONGI
S= 00 4-
Γ
Jε2J /4ί
= (4π)-1(-21ogε+/?(ε,/)),
where
) = 2(l-exp(-ε2/40)logε+(exp(-ε2/40)log(40
Jε2/4t
Let y be the Euler constant. Then,
y = — e~s(\ogs)ds.
Jo
Thus, we have
(2.4) Λ(ε,0 = 2(1 - exp( - ε2 / 40)log ε
+ (exp(-ε2/40)log(40
Λε
2/4t
Jo
Λε
2/4ί
= — y -h log(4ί) -f e ~ slog(ε2 / (4ts))ds.
Jo
Since 0<log(ε2/(4/j))<2(ε2/(4^))1/2 = ε(^)~1/2 hold for any ,se(0, ε2/4/), we have
Λε
2/4ί
(2.5) I e-logO"
Jo
fε2/4t
<εΓl'2\ s-
Jo
It is easy to see that |logί|<2(f + r 1/2) holds for any fe(0,oo). Thus, by (2.3),
(2.4) and (2.5), we get
(2.6)
where
Γ m^
Jo
HEAT KERNEL AND VARIATION OF DOMAINS 945
hold for any sufficiently small ε>0.
On the other hand, since
)- Γ-
J o^
s)ds
= - —U(w,y,t-s)ds
hold for τe(0,f) and yeM
ε
, we see that
W(x
Jo
Thus, we have
ί (Γ W(x9w,τ)U(w,y,t-τ)dτ)f(y)dy
JME Jo
= ( f W(x,w,τ)dτ) I U(w,y,ί)f(y)dy
Jo J M
ε
- Γ W(x,w,τl Γ( ί 1-U(w,y,t-s)f(y)dy)dS)dτ
Jo Jo JMft
= ( Γ W(x,w,τ)dτ)(Utf)(w)- Γ W^τX f ^Ut_
Jo Jo Jo^ί
for xeM. Combining this equality with (1.2), (1.4) and (2.1), we can easily see
(2.7) (K,(ε»/Xjc)
= U(x,y,t)f(y)dy
-L
n
(ε) ί ( Γ W(x
JMC Jo
946 S. OZAWA AND S. ROPPONGI
( Γ S(x,W,τ)U(w,y,t - τ]dτ)f(y)dy
M, Jo
)-LM Γ ^ x,w,τ
Jo
Γ W(x,w,τ
Jo
for
We recall that L
n
(ε) = - 2π(log ε) ~ 1 for n = 2. Thus, by (2.6) and (2.7), we have
(2.8)
where
72(ε,0= -2π(logε)-1 Γ ^(x,w,
Jo
73(ε,0 = 2π(log ε)' ^SfaHUJKw) + Γ S^τXt/, _
Jo
for xεdB
ε
.
Notice that S(x,w,τ) is uniformly bounded for xεM and τe[0,/]. Thus, by
(2.6) and Lemma A.3 in Appendix,
(2.9)
< C\R(ε9t)\r ίlp\\f\\
and
(2.10)
HEAT KERNEL AND VARIATION OF DOMAINS 947
hold for p>\ and xeM. Therefore, by (2.8), (2.9) and (2.10), we have
(2.H) l/3(e,OI^Cllogεr1/-1^(/ + /-1/2
for /?>!. The same calculation as above yields
(2.12)
_*)-»-<!*><&
o
for /?>!, τe(0,0 and
(2.13)
for /?>!, where 0e(0,l) denotes some constant. Furhermore, by (2.2), (2.8)
and (2.12),
(2.14) |/2(ε,OI
Γτ^Λ-^-
Jo
<qiogε|- lri
hold for p>\.
Summing up (2.8), (2.11), (2.13) and (2.14), we get the following.
Proposition 2.1. Fixp > 1 and t>0. Then, there exists a constant C independent
of ε, t such that
holds for any /e L"(M
e
).
948 S. OZAWA AND S. ROPPONGI
Now we are in a position to prove Theorem 1. We put v(x,t)=(U^f)(x)
-(F?e)/Xjc). Then v(x,ή satisfies the following.
(2.15)
t>0
f-»0
xεdM,
a.a. xeME.
ί
 t>0
By the maximum principle we have
sup
xeδM
ε
xeQB
ε
Thus, by Proposition 2.1,
(2.16)
Ml/11,..
hold for /J>1.
On the other hand, by (!.!)„ and (2.15), φc,ί) is explicitly represented as follows.
(2.17)
φc,0 =
θ J6B
ε
Here δ/δv
z
 denotes the derivative along the exterior normal direction with respect
to M
ε
. Thus, by (2.17), Proposition 2.1 and Lemma A. 5 in Appendix, we have
(2.18) llW-χ§*r/llι..
ar/(ε)Λ
(sup|(^>
Jθ^B«
< Cε|log ε\ ~ l \\f\\p,εt(l/2)-(l/p)(t + r 1/2 -f 1)
for /? > 2.
We fix /?>2 and />0. Then, by (2.16), (2.18) and the interpolation
inequality, we see
(2.19)
HEAT KERNEL AND VARIATION OF DOMAINS 949
< II W-χ. Wll i/f || W-JE. Wll ^ (1/P)
£ Colleger1 11/11 „.
Therefore we get the following.
Proposition 2.2. Fix p>2 and t>Q. Then, there exists a constant Ct
independent of ε such that
holds.
From (1.4) and Lemma A.2 in Appendix, we can see that (F/ε))*= Kf(ε) and
(C/f(β))*=l/t(ί). Thus, by the duality argument,
n ?rn II Γ/(ε)— v F(ε)v II — II Γ/(ε) — v F(ε)v II\L.Z\)) \\Ut Λ ε * ί Aεllp',ε~ II ut Λ ε κ ί %ε\\p,ε
holds for any /?e(2,oo), where p' = (l — I//?)"1. Furthermore, by Proposition 2.2,
(2.20) and the Riesz-Thorin interpolation theorem,
II Γ/(ε) — v F(ε)v IL < II f/(ε) — v F<ε)y IIII ut %εyt Λεl l2,ε^ II ^ ί Λ« κ t Λεl lp,ε
holds for any /?e(2,oo).
From Proposition 2.2, (2.20) and (2.21), we can easily get Theorem 1.
3. Proof of Theorem 2
Throughout this section we assume that n = 3. We recall (2.2). Then
W[xwW*\xedB.
Jo
(4πτ)~3/2exp(-ε2/4τ)ί/τ
Jo
Since
ΓJo
and
p2/4f Λ>2
Jo Jo
950 S. OZAWA AND S. ROPPONGI
ε /4f
hold, we have
(3.1)
where
We fix an arbitrary /e Lp(M
ε
). We recall (2.7) and L
n
(ε)=4πε for n = 3. Thus,
by (2.7) and (3.1), we have
(3.2) (WX*W =
where
= 4πε Γ W[*,w,τX f^ϋ1,- JX»f)ώ)A
Jo Jodf
+ f S(x,w,τ\Ut
Jo
/5(ε,ί)
76(ε,ί)= -
for xedB
ε
. By (3.1) and Lemma A.3 in Appendix,
(3.3) |/6(ε,OI
< Cε(r 1/2|(ί/,/)(H>)| + Γ KC/.-JXivίl
Jo
ll/ll, f(ί-τ)-3/2Ά)
Jo
(p>3/2X
(3.4) |/4(ε,ί)| = ε| Vx( t/,/Kw + θ(x -
<CεΓ™2»-W>\\f\\f,c (p>\)
and
(3.5)
HEAT KERNEL AND VARIATION OF DOMAINS 951
hold for τe(0,ί), where 0e(0,l) denotes some constant.
Next we want to estimate 75(ε,ί). By (2.2), (3.2) and (3.5), we see
(3.6)
where
77(ε,f) = f τ-3/2exp(-ε2/4τ)( \(t-sΓl
Jo Jo
Since
77(ε,f) = ίί τ~3 / 2 exp(-e2/4τ)(t-sΓl~(3l2p}d5<h
J J θ < s < τ < ί
= Ut-sΓ1'13'2^ Γτ-3/2exp(-ε2/4τ)ί/τ)(ώ
Jθ Js
and
Γτ-
3/2
exp(-ε2/4τ)rfτ
Js
V4s
:
2/4ί
:
2/4s
ε
2/4ί
hold for 56(0,ί), we have
τ"
1 / 2(l— τ)~(1/2)"(3/2p)dτ
Jo
for ? > 3. Combining this inequality with (3.6), we get
(3.7) l
for p>3.
Summing up (3.2), (3.3), (3.4) and (3.7), we can get the following.
952 S. OZAWA AND S. ROPPONGI
Proposition 3.1 . Fix p>3 and t > 0. Then there exists a constant C independent
of ε, t such that
holds for any /e L"(M
ε
).
By Proposition 3.1, Lemma A.5 in Appendix and the same argument as in
section 2, we have
(3-8)
and
(3.9)
» Γ Λ w
τ)\dX)dσz)dτ
f [/
( ( \—(X,Z,t-
JβB.JM. CVZ
for p > 3.
From (3.8), (3.9) and the interpolation inequality (see (2.19)), we get
the following.
Proposition 3.2. Fixp > 3 and t > 0. Then there exists a constant Ct independent
of s such that
holds.
Furthermore, by the duality argument and the Riesz-Thorin interpolation
theorem, we have (2.20) for any /?e(3,oo) and
Π 1 (ft II Γ/(ε) — v 0ε) v II < II 77(ε) — v F(ε) v IIV 3--1^ II ut Λε κ ί Λεl l r .ε— If ut %εy t %ε\\ p,ε
for any /7e(3,oo) and re[3\2, 3].
HEAT KERNEL AND VARIATION OF DOMAINS 953
From Proposition 3.2, (2.20) and (3.10), we can easily get Theorem 2.
4. Appendix
Let M, M
ε
, U(x,yJ\ U(ε\x,y,t) be as in Introduction. See Friendman [1] for
the fundamental properties of the heat kernel. We have the following.
Lemma 1.1. There exists a constant C independent of x,yj such that
(A.1) 0< U(x,y,t)<CΓn/2exp(-\x-y\2/Ct)
(A.2) &WOI < Cr (M+ Wexp(-\x-y\2 / Ct) (\<i<ri)
(A.3) wOI £ CΓ <"+2>/2exp(- \x -y\2 / Ct)
ct
hold for x, j>eM,
Lemma A.2. We have
U(xy,t) = C/0,x,0 Λ:, y e M, ί > 0
jc, yeM
ε
, ί>0.
Let Ut be as in (1.2). Then we have the following.
Lemma A.3. Fix /?e(l,oo). Then there exists a constant C independent of t
such that
(A.4)
xeM
(A.5)
(A.6) sup(t/ f /)W| < CΓ ("/2'>-l \\f\\,
xeM ^^
hold for feLp(M) and t>0.
Proof. We take an arabitrary xeM. Then, by (1.2), (A.I) and using the
transformation of co-ordinates : y = x + (Ct)ί/2z, we have
954 S. OZAWA AND S. ROPPONGI
<a
π/2
 I exp(- \x-y\2 / Ct)\f(y)\dy
JM
f exp(-|*-;
JM
nW)\\f\\p( I exp(-|z|
jRn
-
(
"
m+(
where (!//>) + (!///)=!.
Therefore we get (A.4). By the same argument as above, we get (A.5) and
(A.6) from (A.2) and (A.3), respectively.
q.e.d
By B(r\w) we denote a ball of radius r>0 with the center w. And we
write B
r
 = B(r'9w) as before.
Lemma A.4. There exists a constant C independent of ε, x, t such that
dU(ε}(A.7) 0< -- (x,z,0<Cr(w+1)/2exp(-|x-z|2/CO
dvz
hold for zedB
ε9 xeME and t>0.
Here d /Svz denotes the derivative along the exterior normal direction with respect
to M
ε
.
Proof. Let F(r\x,y,t) be the fundamental solution of the heat equation in
Rn\B
r
 under the Dirichlet condition on dB
r
. Then we have the following identity.
i
x9ε-
ly,ε-2t)ε-n x, yeRn\B
ε
,
Thus,
holds for x, yeRn\B
ε
,
It is well known that there exists a constant C such that
Γ
 w/2exp( - \χ -y\2 / Ct)
dF(i}
HEAT KERNEL AND VARIATION OF DOMAINS 955
hold for jc, yeRH\Bl9 t>0. Thus, we get
dF(ε)(A.8)
for x, yeRn\B
εί
 t>Q.
It should be remarked that the constant C in (A.8) does not depend on ε.
Let H(ε\x,y,t)=F(ε\x,y,ή-U(ε\x,y,t). Then, H(ε\x,y,t) satisfies the following.
(dt - Δ J//<'W,0 = 0 x, y e Mε9 t > 0
H(ε\xyfy = F(ε\x,y,t) > 0 x e dλf, y e M
ε
, / > 0
H(ε\x,y,t) = 0 x e δ5
ε
, j e M
 ε
, ί > 0
By the maximum principle, H(ε\x9y9t)^0 holds for x,yeλϊε, t>Q. Therefore,
(A.9) 0<U(ε\x9y>t)<F(ε\x,y9t)
hold for x,yeM, t>Q.
We fix an arbitrary zedB
ε
. Then, vz= — (z — w)/|z — w| denotes the exterior
normal unit vector at zedB
ε
 with respect to M
ε
. We recall that
(A. 10) C/(ε)(jc,z,0 = F(ε)(x,z,0 = 0
for xeM
ε
, t>0. Therefore, by (A.9) and (A. 10),
(A. 1 1) 0 < ( U(ε\x,z - hvz,t) - C/(ε)(jc,z,0) / h
hold for x 6 M
ε
, t > 0 and any sufficiently small h > 0. Letting h 1 0 in (A. 1 1), we have
0 <
for xeM
ε
, t>0.
Combining this inequality with (A.8),
8U(ε}
<-^xdvz
<Cr(π+1)/2exp(-|x-z|2/C/)
956 S. OZAWA AND S. ROPPONGI
hold for jceM
ε
,
Therefore we get (A.7).
q.e.d.
Now we have teh following.
Lemma A.5. There exists a constant C independent of ε, t such that
Γ Γ dU(ε}( \—-(x,z,t)\dx)dσz
JdB
ε
JM
ε
 WZ
holds, for />0.
Proof. We fix an arbitrary zedBE. Then, by (A. 7) and using the
transformation of co-ordinates : x = z + (Ct)1/2y,
Γ dt/(ε) Γ
I - (x9z9t)\dx<CΓ(n+i)l2\ e\p(-\x-z\2/Ct)dxJME dvz JMε
<Cr1/2ί exp(-|j|2)^
JR»
<crίl2
hold for f>0. Here C denotes some different positive constants independent
of ε, t. Integrating this inequality on dB
ε
, we can immediately get (A. 12).
q.e.d.
References
[1] A. Friedman: Partial Differential Equations of Parabolic Type. Prentice-Hall Inc., Englewood
Cliffs, N.J., 1964.
[2] S. Ozawa: Hadamards variation of the Green kernels of heat equations and their traces
/, J. Math. Soc. Japan, 34 (1982), 455-473.
[3] S. Ozawa: Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo
Sect. IA, Math., 30 (1983), 53-62.
[4] S. Ozawa: Singular variation of domain and spectra of the Laplacian with small Robin
conditional boundary /, Osaka J. Math., 29 (1992), 837-850.
[5] S. Ozawa and S. Roppongi: Singular variation of domain and spectra of the Laplacian
with small Robin conditional boundary II, Kodai Math. J., 15 (1992), 403-429.
[6] S. Roppongi: Asymptotics of eigenvalues of the Laplacian with small spherical Robin
boundary, Osaka J. Math., 30 (1993), 783-811
[7] M.M. Schiffer and D.C. Spencer: Functionals of Finite Riemann Surfaces. Princeton Univ.
Press, Princeton 1954.
HEAT KERNEL AND VARIATION OF DOMAINS 957
Department of Mathematics
Faculty of Sciences
Tokyo Institute of Technology
O-okayama, Meguroku,
Tokyo 152
Japan



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