Heat Kernel and Analysis on Manifolds (AMS IP Studies in Advanced Mathematics)

Ok (u)

0 (u)

0,

l bk (u) - 0 (u)12 dp

fM because the integrand functions tend pointwise to 0 as k -+ oo and, by (5.7),

Vk (u) - (u) I2 < 4C2u2, whereas u2 is integrable on M. Simil//arly, we have M

f

k(u)-W(u)11I7u12dfc-40,

because the sequence of functions 10k (u) - cp (u)12 IVu12 tends pointwise to 0 as k -4 co and is uniformly bounded by the integrable function 4C2 I VuI2. EXAMPLE 5.3. Consider the functions

(t) = t+ and cp (t) _

0,

t < 0,

which can be approximated as in (5.6) as follows. Choose 01 (t) to be any smooth function on R such that 1 (t)

_

t-l, t>2, 0,

t<0

(see Fig. 5.1). Such function 01 (t) can be obtained by twice integrating a suitable function from Co (0, 2).

Wt) Wi(t) 0

2

t

0

2

t

FIGURE 5.1. Functions 0 (t) = t+ and 01 (t) and their derivatives Then define Ok by k (t) = I Yb1 (kt).

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

126

If t < 0 then bk (t) = 0. If t > 0 then, for large enough k, we have kt > 2 whence k(t) =

I

(kt - 1) = t -

1

-4 t ask -+ oo.

Hence, z//k (t) -4' (t) for all t E R. Similarly, if t < 0 then 0'' (t) = 0, and, for t > 0,

b'k(t)=

(kt)-+1 as k -4oc.

' (t) -4 co (t) for all t E R. By Lemma 5.2, we obtain that u+ E Wo and

Hence,

Vu+

_

I

Vu, u>0, 0, u < 0.

(5.9)

Applying this to function (-u), we obtain u_ E Wo and

Vu__{ -Vu, u<0.

(5.10)

Consequently, since Vu+ = Vu_ = 0 on the set {u = 0}, we obtain

Vu=0on {u=0}.

(5.11)

Of course, if the set {u = 0} has measure 0 then (5.11) is void because Vu is defined up to a set of measure 0, anyway. However, if the set {u = 0} has a positive measure then the identity (5.11) is highly non-trivial. In particular,

(5.11) implies that if u, v are two functions from Wo such that u = v on some set S then Vu = Vv on S. Similarly, u c Wo implies (u - c) (+ E Wo for any c > 0, and V (u - c)+ = {

0, 0,

u < c'

(5.12)

Since Jul = u+ + u_, it follows from (10.20), (5.10), (5.11) that V Jul = sgn (u) Vu.

(5.13)

Alternatively, this can be obtained directly from Lemma 5.2 with functions (t) = Itl and cp (t) = sgn (t). LEMMA 5.4. Let u be non-negative a function from Wo (M). Then there

exists a sequence {uk} of non-negative functions from Co (M) such that

ukW1 -+u. PROOF. By definition, there is a sequence {vk} of functions from Co (M) 1

such that vk W--> u. Let zb be a smooth non-negative function on R satisfying (5.1). By (5.3), we have

(u). Observe that 'b (vk) > 0 and '0 (vk) E Co (M). Hence, the function 0 (u) E W0 can be approximated in W1-norm by a sequence of non-negative func(vk) -+

tions from Co (M). We are left to show that u can be approximated in

5.2. CHAIN RULE IN W1

127

W1-norm by functions like 0 (u), that is, there exists a sequence {?k} functions as above such that 1

Ok (u)

Indeed, consider the functions

-*

U.

co (t) = 1(5.14)

2' (t) = t+ and

and let '%k be a sequence of non-negative smooth functions satisfying (5.5) and (5.6). Then, by (5.8), L2

'bk (u)

L2 2b (u) = u and Wk (u) -+ cp (u) 1u = Du,

which finishes the proof.

5.2. Chain rule in W1 The main result of this section is Theorem 5.7 that extends Lemma 5.2 to W1 (M).

Denote by W, (M) the class of functions from W' (M) with compact support. LEMMA 5.5. WW (M) C W0 (M)

.

PROOF. Set K = supp u and let {Ui} be a finite family of charts covering K. By Theorem 3.5, there exists a family {VQ of functions Vi E D (Ui) such that Ei 0i = 1 in a neighborhood of K. Then we have O u E WW (Ui) (cf. Exercise 4.2). Since u = EiV1iu, it suffices to prove that 2/Iiu E Wo (Ui). Hence, the problem amounts to showing that, for any chart U,

W": (U)CW0(U), that is, for any function u E W' (U), there exists a sequence {cpk} C D (U) such that cpk + u. Since U is a chart and, hence, can be considered as a part of R', also the Euclidean Sobolev space W ,',,,(U) is defined in U (see Section 2.6.1). In general, the spaces W ,',,,,(U) and W1 (U) are different. However, by Exercise 4.11(b), the fact that u E W1 (U) implies that u 1

and all distributional partial derivatives 8ju belong to Li (U). Since the support of u is compact, we obtain that u and O u belong to Leucl (U), whence u c Wd (U). By Exercise 2.30, there exists a sequence {cpk}k 1 of functions from D (U) that converges to u in We a (U), and the supports of cOk can be assumed to be in an arbitrarily small open neighborhood V of supp u. We are left to show that the convergence cpk -4 u in Wu.1 (U) implies that in W1 (U). By Exercise 4.11, for any v E Wed (U) we have 92'aZvajv'

IVgvlg = where Vv is the weak gradient of v in metric g of the manifold M. It follows that, for any fixed open set V C= U, IIVIIW1(V) <_ CIIvIIwi

(v)

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

128

where the constant C depends on the supremums of I g2j I and det g in V. Hence, choosing V C= U to contain all the supports of SPk and u, we obtain 0,

Iu - SOkIIWl(V)
that is, c'k -+ u in W1 (U). Define the space W1'0 (M) by Will"

(M)={uELiC(M):DuELic(M)}.

(5.15)

Clearly, if u E W,1, (M) then u E W1 (52) for any relatively compact open set S2 C M. Conversely, if u is a function on M such that u E W' (Qk) for an exhaustion sequence {s2k} then u E Wlo, (M). COROLLARY 5.6. If U E Wio, (M) and f E Co (M) then fu E W0 (M).

PROOF. Let S2 be any relatively compact open set containing supp f .

Then u E W1 (fl) and f E Co (92), whence we obtain by Exercise 4.3 that f u c W1 (ci). Since supp (f u) is compact and is contained in 52, we obtain f u E Wf (52) whence by Lemma 5.5 f u E Wo (0). It follows that fu EW0(M). The following result extends Lemma 5.2 to functions from W' (M). THEOREM 5.7. Let {Y'k (t)} be a sequence of C°°-smooth functions on R such that

"k (0) = 0 and sup sup 10k (t) I < 00. k

(5.16)

tE]R

Assume that, for some functions 0 (t) and cp (t) on IR,

V'k (t) - i (t) and 0k (t) - cp (t) for all t E Iii,

(5.17)

as k -+oo. (i) If u E W1 (M) then 0 (u) E W' (M) and 0' (u) = cP (u) Vu.

(5.18)

(ii) Assume in addition that function cp (t) is continuous in III \ F for some finite or countable set F. If Uk, u E W' (M) then Uk

-

U

Z(1 (Uk) -4 ?/J (u)

.

REMARK. The conditions (5.16) and (5.17) are identical to the conditions (5.5) and (5.6) of Lemma 5.2. PROOF. (i) As in the proof of Lemma 5.2, we have 0 (u) E L2 (M) and cp (u) Vu E L2 (M). The identity (5.18) means that, for any vector field WED (M), (5.19) (' (u) , div, w) = - (co (u) Vu, w) .

5.2. CHAIN RULE IN W1

129

Fix w E D (M) and let f E D (M) be a cutoff function of supp w. By Corollary 5.6, the function uo := f u is in Wo (M). Therefore, by Lemma 5.2,

o0 (u0) _'P (uo) Duo

and, hence,

(0(uo),div,.w) = Since u = uo in a neighborhood of suppw, this identity implies (5.19). (ii) It suffices to prove that a subsequence of {V (uk)} converges to 0 (u) (cf. Exercise 2.14). Since Uk --* u in L2, there is a subsequence {uk} that converges to u almost everywhere. Hence, renaming this subsequence back to {uk}, we can assume that Uk -4 u a.e.. What follows is similar to the proof of Lemma 5.1. It suffices to show

that V) (u) and 7V) (uk) Z V By (5.16), there is a constant C such that (uk)

(U).

(5.20)

10k (t)I < C and I0k (t) I < C Itl , for all k and t E R. Therefore, 110 (Uk) - 0 (u) 11 < Clink - ull

(where all norms are L2), which implies the first convergence in (5.20). Next, using (5.18), we obtain

llVV(uk) - V (u)II

<-

II CP(uk)(Duk - Vu)11 + I1((P(uk)

< C11(Duk - VU) 11 + 11(tP(uk)

- P(u))VujI

- P(u)) Vull. (5.21)

The first term in (5.21) tends to 0 by hypothesis. The second term is equal

f

to

1/2 1

(uk (x)) -

11

(u (x))121ou12 dµ (x))

(5.22)

Consider the following two sets:

S1 = {xEM:uk(x)74 u(x) ask-4oo},

S2 = {xEM:u(x)EF}, where F is set where cP is discontinuous. By construction, u (SI) = 0. Since

Vu = 0 on any set of the form {u = const} (cf. Example 5.3) and S2 is a countable union of such sets, it follows that Vu = 0 on S2. Hence, the domain of integration in (5.22) can be reduced to M \ (Si U S2). In this domain, we have

uk(x)--+u(x)VF, which implies by the continuity of ca in R \ F that cP (uk (x)) -+ cO (u (x))

Since the functions under the integral sign in (5.22) are uniformly bounded by the integrable function 4C2 l`7u12, the dominated convergence theorem

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

130

implies that the integral (5.22) tends to 0 as k -3 oo, which proves the second relation in (5.20). EXAMPLE 5.8. Fix c > 0 and consider functions

0 (t) = (t - c)+ and

(5.23)

(cf. Example 5.3). Since these functions satisfy all the hypotheses of Theorem 5.7, we obtain that if u E W 1 then (u - c)+ E W1, and V (u - c)+ is given by (5.12). Furthermore, by Theorem 5.7, Uk

u implies (Uk - c)+

(u - c)+.

Exercises. 5.1. Let fi (t) and cp (t) be functions satisfying the conditions (5.16) and (5.17) of Theorem 5.7. Prove that V)' it = co.

5.2. Let 0 E C' (1l) be such that

b (0) = 0 and sup Iii I < oc. Prove that the functions

satisfy the conditions (5.16) and (5.17) of Theorem

and cp

5.7.

5.3. Prove that if u, v E Wo (M) then also max (u, v) and min (u, v) belong to Wo (M).

5.4. Prove that if M is a compact manifold then W 1 (M) = Wo (M).

5.5. Prove that if u E W1 (M) then, for any real constant c, Vu = 0 a.e. on the set

{xEM:u(x)=c}.

5.6. Prove that, for any u E W' (M),

(u-c)+-u+ asc-+0+. 5.7. Let f E W1 (M) and assume that f (x) -+ 0 as x -+ oo (the latter means that, for any e > 0, the set {If > e} is relatively compact). Prove that f E Wo (M). 5.8. Prove that if u E Wdoo (M) and cp, 0 are functions on R satisfying the conditions of Theorem 5.7 then 0 (u) E Wtoc (M) and V (u) = cp (u) Vu. 5.9. Define the space Woo (M) by 2

Wioc

1 f E Waoc :A,,

2

E Laos} .

Prove the Green formula (4.12) for any two functions u E W0' and v E W.

5.3. Markovian properties of resolvent and the heat semigroup Set as before L = -Au,JW and recall that the heat semigroup is defined by Pt = e-t c for all t > 0, and the resolvent is defined by (5.24) Ra = (L + aid)-1 for all a > 0. Both operators Pt and Ra are bounded self-adjoint operator

on L2 (M) (cf. Theorems 4.9 and 4.5). Here we consider the properties of the operators Pt and Ra related to inequalities between functions.

5.3. MARKOVIAN PROPERTIES OF RESOLVENT AND THE HEAT SEMIGROUP 131

THEOREM 5.9. Let (M, g, p) be a weighted manifold, f c L2 (M), and

a>0.

(i) If f > O then Ra f > 0.

(ii) If f< 1 then R, f< a-1. PROOF. We will prove that, for any c > 0, f < c implies Ra f < ca-1, which will settle (ii) when c = 1, and settle (i) when c --+ 0+. Without loss

of generality, it suffices to consider the case c = a, that is, to prove that f < a implies Ra f < 1. Set u = Ra f and recall that u E dom.C = Wo and .Cu + au = f. (5.25) To prove that u < 1 is suffices to show that the function v := (u - 1)+ identically vanishes. By Example 5.3 and (5.12), we have v E W01 and Vu,

Vv = j

0

1

(5.26)

u < 1.

Multiplying (5.25) by v and integrating, we obtain (5.27)

(.Cu, v)L2 + a (u, v)L2 = (.f, v)L2 .

By Lemma 4.4 and (5.26), we have (Lu, V) L2 = - (& u, v)L2 = (Vu, Vv)L2 = J

/ {u> 1 }

whereas

(u, v)L2 = f

{v>0}

(v + 1) vdu = NvHi2 +

I Vu12 dµ >_ 0,

f

M

Hence, it follows from (5.27) and f < a that aiiv1122 + a fm vdµ < (f, v)L2 < a fm vdµ, whence we conclude IIVIIL2 = 0 and v = 0.

Many properties of the heat semigroup can be proved using the corresponding properties of the resolvent and the following identities. LEMMA 5.10. For an arbitrary weighted manifold (M, g, ,u) the following identity hold.

(i) For any a > 0,

Ra = f

e-atPtdt.

(5.28)

0

where the integral is understood in the following sense: for all f, g E L2 (M),

(R,f,9)L2 =

r e-at (ptfI9)L2

J0

dt.

(5.29)

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

132

(ii) For any t > 0,

`k

Pt= lim k-oo \t

Rkt=

(5.30)

where the limit is understood in the strong operator topology.

PROOF. Let {E,\} be the spectral resolution of the Dirichlet Laplace operator .C. Then by (4.45) we have

f

Pt f =

oQ

e-atdEaf

(5.31)

(a + A)-1 dEa f

(5.32)

0

and

R, ,,,f =

J

U

(see also Exercise A.24).

(i) Substituting (5.31) to the right hand side of (5.29) with g = f and using Fubini's theorem, we obtain

f

0°O

Cod (Ptf, f) L2

0

dt = f

00

Cat

0

f

(j00 a-atd (Eaf,

(o 0

o 0o

e

&2 ) dt

('+a)tdt) d (EA.f, .f )Lz

0 (a+A)-'d(Eaf,f)f

(Raf, f)L2

L2

,

which proves (5.29) for the case f = g. Then (5.29) extends to arbitrary

f, g using the identity

(Ptf,g)= 2(Pt(f+g),f+g)-2(Pt(f-g),f-g) and a similar identity for Rn (ii) Applying the classical identity

e-ta.kmooCl+) and the dominated convergence theorem, we obtain

I

00

e-tadEA =

km

t

I'm (l+)_kdE,

where the limit is understood in the strong sense. This implies (5.30) because the left hand side here is equal to Pt, and the integral in the right hand side

5.3. MARKOVIAN PROPERTIES OF RESOLVENT AND THE HEAT SEMIGROUP 133

is equal to 00

r (i+-)

- kdE),

kdEA

W kJ (+A)-k (t I k yd+r)

(k)k

/

k/t

THEOREM 5.11. Let (M, g, y) be a weighted manifold, f E L2 (M), and

t>0.

(i) If f > 0 then Ptf > 0. (iii) If f < 1 then Pt f < 1. PROOF. (i) If f > 0 then, by Theorem 5.9, Ra f > 0, which implies that Ra f > 0 for all positive integers k. It follows from (5.30) that Ptf > 0.

(ii) If f < 1 then, by Theorem 5.9, Ra f < a-1, which implies that Ra f < a-k for all positive k. Hence, (5.30) implies (k -k = 1,

Ptf <

lilim

on

(t) k

t)

which was to be proved.

Exercises. Let R. be the resolvent defined by (5.24). (a) Prove that, for any f E L2 and a > 0,

5.10.

Pt f=lim e -"

00

a 2ktk

L kf

Rk

f.

(5.33)

k=0

(b) Using (5.33), give an alternative proof of the fact that f < 1 implies Ptf < 1. 5.11. For all a, k > 0, define Rkk01 as cp (R,,) where cp (A) _ ak.

(a) Prove that, for all a, k > 0, R« =

f oa rtk-1 (k) e-«tPtdt

(5.34)

0

where the integral is understood in the weak sense, as in Lemma 5.10, and r is the gamma function (cf. Section A.6). (b) Write for simplicity Rl = R. Prove that Rk RI = Rk+a for all k, l > 0.

Prove that if f E L2 (M) then f > 0 implies Rkf > 0 and f < 1 implies k 'f < 1,

for allk>0.

(c) Prove that Rk = e-kL where L = log (id +G) and .G is the Dirichlet Laplace operator. REMARK. The semigroup {Rk}k>o is called the Bessel semigroup, and the operator log (id +i) is its generator.

5.12. Prove that, for any non-negative function f E L2 (M) and all t, a > 0,

PtRo.f < eatRaf 5.13. Let G be the Dirichlet Laplace operator on

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

134

(a) Prove that the resolvent R., = (G + Aid)-' is given for any A > 0 by the following formula:

-f 2 1

RAf=

e-N'_"=-vl f

(y) dy,

(5.35)

for any and f E L2 (R'). (b) Comparing (5.35) with co

RA =

e-1\tPtdt

J0

and using the explicit formula for the heat kernel in R1, establish the following identity: e-tom

exp I

=fy

-4s) e Sods

(5.36)

for all t > 0 and A > 0.

(-

REMARK. The function s exp 4s ) is the density of a probability distribution on R+, which is called the Levy distribution.

5.14. Let L be the Dirichlet Laplace operator on an arbitrary weighted manifold, and consider the family of operators Qt = exp (-tL1/2), where t > 0. (a) Prove the identity

Qt = Joy

v 4-7r83

exp (- t

4s

Pads.

(5.37)

(b) Let f E L2 (M). Prove that f > 0 implies Qt f > 0 and f < 1 implies Qt f (c) Prove that in the case M = R', Qt is given explicitly by

Qtf =

f

n

qt (x - y) f (y) dy

where 4t

(x)

=

2 t Wn+l (t2 + Ixl2)n+y

(5.38)

REMARK. The semigroup {Qt}k>0 is called the Cauchy semigroup, and the operator .112 is its generator. 5.15. Let 4 be a C°°-function on R such that' (0) = IF' (0) = 0 and 0 < T" (s) < 1 for

all s.

(a) Prove that, for any f E L2 (M), the following function F(t) := fM IF (Ptf) dot

(5.39)

i s continuous and decreasing in t E [0, -boo). (b) Using part (a), give yet another proof of the fact that f < 1 implies Ptf < 1, without using the resolvent.

5.4. WEAK MAXIMUM PRINCIPLE

135

5.4. Weak maximum principle 5.4.1. Elliptic problems. Given functions f E L2 (M), W E W1 (M), and a real constant a, consider the following weak Dirichlet problem:

f Aµu+au= f,

(5.40)

u = w mod Wo (M) , where the second line in (5.40) means that u= w + wo for some wo E Wo (M)

,

and can be regarded as the "boundary condition" for u. If a < 0 and w = 0 then the problem (5.40) has exactly one solution u = -R_a f by Theorem 4.5. Set /3 = inf spec L,

where L is the Dirichlet Laplace operator on M. Then, by Exercise 4.29, the problem (5.40) has exactly one solution for any a <,6 and w E W1 (M). Here we are interested in the sign of a solution assuming that it already exists. In fact, we consider a more general situation when the equations in (5.40) are replaced by inequalities. If u, w are two measurable functions on M then we write u < w modWo (M) if

u < w + wo for some wo E Wp (M). The opposite inequality u > w mod Wo (M) is defined similarly.

LEMMA 5.12. If u E W1 (M) then the relation

u < 0 mod Wo (M)

(5.41)

holds if and only if u+ E Wo (M). PROOF. If u+ E Wo then (5.41) is satisfied because u < u+. Conversely, we need to prove that if u < v for some v E Wo then u+ E Wo. Assume first that v E Co, and let cp be a cutoff function of supp v (see Fig. 5.2). Then we have the following identity:

u+= ((1-(p)v-E-(pu)+

(5.42)

Indeed, if cp = 1 then (5.42) is obviously satisfied. If cp < 1 then v = 0 and, hence, u < 0, so that the both sides of (5.42) vanish. By Corollary 5.6, we have cpu E Wo. Since (1 - cp) v c Co, it follows that

(1-cp)v+cpuEWo. By Lemma 5.2 and (5.42) we conclude that u+ E W01. For a general v E Wo, let {vk} be a sequence of functions from Co such

that Vk 4 v. Then we have Uk :=u+(vk-v)
5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

136

FIGURE 5.2. Functions u, v, ,p which implies by the first part of the proof that (uk)+ E Wo W. Since uk

it follows by Theorem 5.7 that (uk)+

W- U,

u+, whence we conclude that

0

uEW0.

We say that a distribution u E D' (M) is non-negative' and write u > 0 if (u, cp) > 0 for any non-negative function cp E D (M). Of course, if u E L1', C (M) then u > 0 in the sense of distributions if and only if u > 0 a.e.(cf. Exercise 4.7). Similarly, one defines the inequalities u > v and u < v between two distributions. It is possible to prove that u > v and u < v imply u = v (cf. Exercise 4.6). THEOREM 5.13. (Weak maximum principle) Set 0 = inf spec L and assume that, for some real a < 0, a function u E W' (M) satisfies in the distributional sense the inequality Amu + au > 0 (5.43) and the boundary condition u < 0 mod Wo (M) . (5.44) Then u < 0 in M.

REMARK. If a > 0 then the statement of Theorem 5.13 may fail. For example, if M is compact then 0 = 0 because the constant is an eigenfunction of L with the eigenvalue 0. Then both (5.43) and (5.44) hold with a = 0 for any constant function u so that the sign of u can be both positive and negative.

REMARK. Theorem 5.13 can be equivalently stated as a weak minimum principle: if u E W' (M) and Ottu + au < 0, u > 0 mod Wo (M) , then u > 0 in M. _

'It is known that any non-negative distribution is given by a measure but we will not use this fact.

5.4. WEAK MAXIMUM PRINCIPLE

137

REMARK. Consider the weak Dirichlet problem (5.40) with the boundary

function w = 0. By Theorem 4.5, if a < 0 then the problem (5.40) has a unique solution u = -R-,,f. By Theorem 5.13, we obtain in this case that f > 0 implies R_a f > 0, which, hence, recovers Theorem 5.9(i). PROOF OF THEOREM 5.13. By definition, (5.43) is equivalent to the inequality (Auu, co) + a (u, W) > 0

for any non-negative cp E D (M), which, in turn, is equivalent to (Vu, V W) - a (u, W) < 0.

(5.45)

Considering the round brackets here also as the inner products in L2 (M) and noticing that all terms in (5.45) are continuous in co in the norm of W1 (M), we obtain that (5.45) holds for all non-negative cp E Wo (M) (cf. Lemma 5.4).

By Lemma 5.12 we have u+ E Wo (M). Setting (5.45) cp = u+, we obtain

f (Vu, Vu+)dp - a f u+dy<0. f f Iou+12dµ _> Q f u+dp,

u u+ dot < 0.

fm

M

It follows by (5.9), that

IVu+12dµ-aJ

(5.46)

M

M

By Exercise 4.27, we have

M

M

which together with (5.46) implies

(Q - a) fm u+dµ<0. Since a < 0, we obtain IIu+IIL2 = 0 and, hence, u < 0. COROLLARY 5.14. (Comparison principle) Assume that, for some a <,8, functions u, v E Wl (M) satisfy the conditions

A/-4u +au>A,.v+av, u
PROOF. Indeed, the function u-v satisfies all the conditions of Theorem 5.13, which implies u - v < 0. COROLLARY 5.15. (The minimality of resolvent) Assume that a function u E W1 (M) satisfies the inequality

-Amu + yu > f, where f E L2 (M) and y > 0, and the boundary condition

u>0modWo(M).

(5.47)

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

138

Then u > Ry f. PROOF. Using Theorem 4.5, we obtain that

-Ap (Ryf) +7

f = f < -Au +-yu

R.y f = 0 mod Wo (M) < u mod Wo (M), whence we conclude by Corollary 5.14 with a = -ry that Rj < u.

If f > 0 then, by Theorem 5.9, Ry f > 0. The statement of Corollary 5.15 implies that u = Ry f is the minimal function satisfying (5.47) among all non-negative functions u E W1 (M).

Exercises. 5.16. Give an example of a manifold M and a non-negative function u E Wjo (M) such

that u < 0 mod WW (M)

5.4.2. Parabolic problems. Now we turn to the weak maximum principle for the heat equation. In Section 4.3, we have considered the L2-Cauchy problem related to the Dirichlet Laplace operator L. In the next statement, we consider a more general version of this problem, where the requirement to be in domL is dropped. THEOREM 5.16. (Weak parabolic maximum principle) Let u : (0, T) -3 W1 (M) be a path that satisfies the following conditions:

(i) For any t E (0, T), the strong derivative ai exists in L2 (M) and satisfies the inequality

du_&u<0 dt

,

(5.48)

where AA is understood as an operator in D' (M).

(ii) For any t E (0,T), u (t, ) < 0 mod W0 (M) .

u+ (t,)

L2

5.49)

)

0 as t -+ 0. Then u (t, ) < 0 for all t E (0, T) .

REMARK. By Theorem 4.9, the function u = Ptf satisfies all the above conditions, provided f < 0. Hence, we conclude by Theorem 5.16 that f < 0 implies Ptf < 0, which recovers Theorem 5.11(i). Let u be a solution to the L2-Cauchy problem with the initial function f as stated in Section 4.3. Then, for any t > 0, u (t, ) E dom L which ,

implies

mod W0 (M). Applying Theorem 5.16 to u and -u, we see that f = 0 implies u = 0, which recovers Theorem 4.10.

5.4. WEAK MAXIMUM PRINCIPLE

139

Similarly to Corollary 5.14, one can state in an obvious way a comparison principle associated with Theorem 5.16.

PROOF. The inequality (5.48) means that, for any fixed t E (0, T) and any non-negative function v E D (M),

(U V) <(&u, v), where u' = at , which implies (U

/' v)

(Vu, Vv)

(5.50)

.

Considering the both sides here as the inner products in L2 (M), we extend (5.50) to all non-negative functions v E W0 (M). Let a function cp E CO° (1I) be such that, for some positive constant C, s < 0, s > 0,

cp (s) = 0, cp (s) > 0,

(5.51)

0
=-

(u) Vu12 d< 0.

(5.52)

IM

Using the product rule (Exercise 4.46) and the chain rule (Exercise 4.50) for strong derivatives, we obtain dt (u, W (u)) L2

= (u', cp (u)) L2 + (u, cp (u) u) L2 (u', (p (u)) L2 + (u', 'b (u)) L2

(5.53)

where

' (s) = cp' (s) s.

(5.54)

Next, we specify function cp as follows. Define first its second derivative cp" as a non-negative smooth function on R, such that

cp"(s)=0, s<0ors>1, cp"(s)>0, O<s<1. Then co is obtained by two integrations of cp" keeping the value 0 at 0. Clearly, co satisfies (5.51). Also function 0 (s) from (5.54) satisfies (5.51), because its derivative

0'(s)=cp"(s)s+cp'(s)

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

140

is obviously non-negative and bounded. By (5.52), we conclude that the right hand side of (5.53) is non-positive, that is, (u, cp (u))L2 as a function of t is decreasing in (0, T). Since co (s) < Cs for any s > 0, we obtain that (u, c (u))L2 = (u+, W (u+))L2 G C (u+, u+)L2 .

By hypothesis, (u+, U+)L2 -3 0 as t -> 0. Hence, the function t H (u+, cp (u+))L2 is non-negative, decreasing on (0, T) and goes to 0 as t -3 0. It follows that

(u+, cp (U+))L2 = 0 for all t E (0, T), which implies that u+ (t, ) = 0 for all t E (0, T). COROLLARY 5.17. (The minimality of Pt f) Let u : (0, T) -3 W1 (M) be

a path in W' ( M) such that

dt > Op,u for all t E (0, T) u (t, ) > 0 mod Wo (M) for all t E (0, T) (t ' .) u f E LZ (M) as t -3 0

,

,

(5.55)

.

Then, for all t E (0, T),

u (t, ) > Pt f.

(5.56)

PROOF. Using Theorem 4.9, we obtain that the function v (t, ) = u (t, Pt f satisfies the conditions

dt > OAv

for all t E (0, T) ,

0 modW' (M), for all t E (0,T), V (t'.)

2

L

as t -4 0,

_> 0

whence (5.56) follows by Theorem 5.16. Corollary 5.17 implies the following minimality property of Pt f : if f > 0

then the function u (t, ) = Pt f is the minimal non-negative solution to the Cauchy problem du _ Opu, t > 0, dt 2

U (t, ) -4 f, as t -3 0.

Exercises. 5.17. Let the paths w : (0,T) -* W' (M) and v : (0,T) -* Wo (M) satisfy the same heat equation du dt_O,u

foralltE(0,T),

where de is the strong derivative in L2 (M) and A ,u is understood in the distributional sense. Prove that if )L2(M) w(t, v (t, 0 ast-*0, and w > 0 then w (t,) > v (t, ) for all t E (0, T).

5.4. WEAK MAXIMUM PRINCIPLE

141

5.4.3. The pointwise boundary condition at oo. DEFINITION 5.18. The one point compactification of a smooth manifold M is the topological space MU{oo} where cc is the ideal infinity point (that does not belong to M) and the family of open sets in M U {oo} consists of the open sets in M and the sets of the form (M \ K) U {oo} where K is an arbitrary compact subset of M.

It is easy to check that this family of open sets determines the Hausdorff topology in M U {oo} and the topological space M U {oo} is compact. Note that if M is compact then oo is disconnected from M. If M is non-compact and v (x) is a function on M then it follows from the definition of the topology of M U {oo} that, for a real c,

v(x)-+c as x-+oo if, for any E > 0, there is a compact set KE C M such that sup Iv (x) - cl < E.

(5.57)

(5.58)

xEM\KE

EXAMPLE 5.19. If M = R' then any compact set is contained in a ball IxI
{x E 1R7z :

v(x)-+c as Ix1-4oo, so that x - oo means in this case jxI -> oo. If M = 52 where E2 is a bounded open set in R then every compact set in M is contained in S2a = {x E Q: d (x, aS2) > S}

for some J > 0. Then x --> oo in M means d (x, (9s2) --* 0 that is, x --+ aQ. If M = S2 where S2 is an arbitrary open set in lR' then arguing similarly

we obtain that x -> oo in M means that x

1

oo.

+ d (x, a1) -} If va (x) is a function on a manifold M that depends on a parameter a varying in a set A, then define the uniform convergence in a as follows: v (x)

c as x --+ oo uniformly in a E A if the condition (5.58) holds

uniformly in a, that is, sup

sup

I Va (x) - cl <E.

(5.59)

aEA xEM\KE

For c = boo the conditions (5.58) and (5.59) should be appropriately modified.

The next statement is a rather straightforward consequence of Theorem 5.16 for the classical (sub)solutions to the heat equation. COROLLARY 5.20. (Parabolic maximum principle) Set I = (0, T) where

T E (0, +oo]. Let a function u (t, x) E C2 (I X M) satisfy the following conditions:

142

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

au-L u<0inIxM. u+ (x, t)

0 as x

oe in M, where the convergence is uniform

in t E I. (M) L2 0 as t - 0. u+ (t ) L Then u < 0 in I x M. PROOF. The hypothesis u+ (x, t) * 0 as x -4 oo means that, for any e> 0 there is a compact set K C M such that sup sup u (t, x) < E. tE7 xEM\K

Let S2 be any relatively compact open subset of M containing K. Then it (t, ) E Wl (S2) for any t E I and the partial derivative at coincides with the strong derivative dt in L2 (Q) (cf. Exercise 4.47). Therefore, it satisfies the hypothesis (i) of Theorem 5.16 in Q. It follows that u - e also satisfies that condition. Since (u - E)+ is supported in K C S2, we obtain by Theorem 5.7 and Lemma 5.5 that (u - e)+ E W0 (0). This constitutes the hypothesis (ii) of Theorem 5.16. L-42 (0) 0 as t - 0, which gives the hypothesis Finally, we have (u - e)+ (iii) of Theorem 5.16. Hence, we conclude by Theorem 5.16 that it - E < 0 in I x Q. Finally, letting e -+ 0 and exhausting M by sets like S2, we obtain u < 0 in I x M.

REMARK 5.21. Corollary 5.20 remains true if u+ satisfies the initial condition in the Li . sense rather than in Li sense, that is, if -)L 11-:4(M)0

u+ (t,

as t -4 0.

See Exercise 5.21.

Exercises. 5.18. Let va (x) be a real valued function on a non-compact smooth manifold M depending

on a parameter a E A, and let c E R. Prove that the following conditions are equivalent (all convergences are inform in a E A): c as x --* oo. (i) va (x) (ii) For any sequence {xk}k 1 that eventually leaves any compact set K C M, va (xk)

cask --boo. (iii) For any sequence {xk} on M that eventually leaves any compact set K C M, there is a subsequence {xk, } such that va (xki) = c as i -3 00. (iv) For any e > 0, the set

VE={xEM:suplva(x)-cl ll

aEA

>s

(5.60)

is relatively compact. Show that these conditions are also equivalent for c = ±oo provided (5.60) is appropriately adjusted.

5.5. RESOLVENT AND THE HEAT SEMIGROUP IN SUBSETS

143

5.19. Referring to Exercise 5.18, let M = St where S is an unbounded open subset of 1R'". Prove that the condition (i) is equivalent to (v) v, (xk) c for any sequence {xk} C 1 such that either Xk -> x E 00 or xkl --> Co.

5.20. Let a function v E C2 (M) satisfy the conditions: (i) -A ,v + cev < 0 on M, for some a > 0; (ii) v+ (x) -3 0 as x --> oo in M. Prove that v < 0 in M. L'2_(M)

5.21. Prove that the statement of Corollary 5.20 remains true if the condition u+ (t, ) 0 as t -+ 0 is replaced by L, '_(M)

u+

0 as t -+ 0.

5.5. Resolvent and the heat semigroup in subsets Any open subset Sl of a weighted manifold (M, g, p) can be regarded as a weighted manifold (fl, g, µ). We will write shortly L2 (SZ) for L2 (52, µ), and the same applies to W0 (S2) and other Sobolev spaces.

Given a function f on S2, its trivial extension is a function f on M defined by (x)f(x),

0,

xES2, x E M \ Q.

The same terminology and notation apply to extension of a vector field in S2

by setting it 0 in M \ Q. It is obvious that if f E Co' (Q) then f E Co (M) and

Vf = Vf

and

O,f =& f.

The space L2 (S2) can be considered as a subspace of L2 (M) by identifying any function f c L2 (St) with its trivial extension.

CLAIM. For any f E Wo (a), its trivial extension f belongs to W0 (M) and

Vf = Vf.

(5.61)

Note for comparison that if f E W1 (S2) then f does not have to be in W' (M) - see Exercise 7.7.

PROOF. If f E Co- (Q) then obviously f E Co' (M) and hence f E Wo (M). For any f E W0 (1k), there exists a sequence { fk} of functions from Co' (1) such that

fk - f in Wa (Q)

.

Clearly,

fk- f inL2(M).

144

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

On the other hand, the sequence { fk} is obviously Cauchy in W01 (M) and hence converges in Wo (M). We conclude that the limit must be f, whence it follows that f E Wo (M) and

fk -- f in WO' (M) . In particular, this implies that

Vfk-+Vf in

_E2 (M).

(5.62)

On the other hand,

Vfk-*Vf in L2(S2), whence it follows that

0 fk -3 , f in L2 (M).

5.63)

Since V fk = 0 fk, we conclude from (5.62) and (5.63) that V f =V f. Hence, the space Wo (S2) can be considered as a (closed) subspace of W0 (M) by identifying any function f E Wo (11) with its trivial extension. This identification is norm preserving, which in particular, implies that we W1 (M). In what follows, we will follow the have an embedding Wo (S2) convention to denote the trivial extension of a function by the same, letter as the function, unless otherwise mentioned. Consider the Dirichlet Laplace operator in Q -ApjW (Q) , L'11

-

as well as the associated resolvent

Ra = (Ln + a id) and the heat semigroup

P"=exp(-tEf1). A sequence {S2i}°_°1 of open subsets of M is called an exhaustion sequence

if Sli C S2i+1 for all i and the union of all sets S2j is M.

THEOREM 5.22. Let f c L2 (M) be a non-negative function, and a > 0. (i) For any open set S2 C M,

Raf
Rai f

W1

Ra f as i -+ oo.

(5.64)

Note that Raf is a short form of Rn (fin). PROOF. (i) By Theorem 5.9, the functions u = Raf and v = Raf are non-negative. By convention, v - 0 outside S2, so that we only need to prove that u > v in Q. By the definition of resolvent, u E Wo (M) and u satisfies

in M the equation

-0µu+au= f.

5.5. RESOLVENT AND THE HEAT SEMIGROUP IN SUBSETS

145

In particular, we have u c W' (S2). Applying Corollary 5.15 to the manifold SZ, we obtain u > Ra f , which was to be proved. Alternative proof of (i). This proof is longer but it uses fewer tools from the present Chapter confining them to Lemmas 5.1 and 5.4. By the definition of resolvent, we have u E Wo (M), v E Wo (0), and

L0v + av = f in Q, Gu + au = f in M.

(5.65)

By Lemma 5.4, there are sequences of non-negative functions {uk } C Co (M) and {vk } C

Co (i2) which converge, respectively, to u and v in Wl-norm. Let ') be a smooth nonnegative function on H such that

7P (t)-0 fort <0,

'(t)>0and0< '(t)<1forallt>0.

(5.66)

One can think of (t) as a smooth approximation to t+ (see Fig. 5.1). Let us show that w := (v - u) E Wo (S2). For that, set wk = V) (vk - Uk) and observe that wk E Co (M) and 0 <_ Wk < Vk. The latter implies that suppwk is contained in Q and hence wk E Co (0) (see Fig. 5.3).

FIGURE 5.3. Function wk =

(vk

- Uk)

On the other hand, by (5.3) (see Lemma 5.1) we have

b(Uk - uk) W-i)V) (v-26), which yields w E Wo (S2). Subtracting the equations in (5.65) and multiplying them by w, we obtain

(Gov,w)

L2 (S2)

-(Gu,W)L2(n)+a(v-u,w)L2(o)=0.

By Lemma 4.4, we have

(r0v' w)l L2 (Q) =

(Vv, Vw)L2(n) = J (Vv, Vw)dµ.

and

(.Cu,w)L2(n) _ (.Cu,w)L2(M) = (Vu,Vw)L2(M) = f(Vu,Vw)dP, z

where in the last equality we have used (5.61). Hence, we obtain from the above three lines that f(v (v - u), Vw)dp + ce

f,

(v - u) wdp = U.

(5.67)

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

146

By (5.2), we have

Vw=V (v-u)=0'(v-u)V(v-u)

whence, by -tbb' (t) > 0,

f(V(v -

u),Vw)d=-u)1V (v-u)12d> 0.

Since t, (t) > 0 for all E R, we obtain

Therefore, the equation (5.67) is possible only when,that

is, when (v - u) 0 (v - u) = 0 in 52, whence v - u < 0.

(ii) Set ui = Ra' f and observe that, by part (i) and Theorem 5.9, the sequence {ui} is increasing and

0
0
is in Wo (M). Let us show that the sequence {ui} is Cauchy in W0 (M). Each function ui ui

in

satisfies the equation (5.68) (©ui, V(P) + a (ui, P) = (f, W) , is the inner product in L2 (M)). Choosing for any cp E W0 (ci) (where here cp = ui, we obtain

(Vui, Vui) + a (ui, ui) _ (f, ui) Fix k > i and observe that the function cp = uk - 2ui belongs to W0 (Ilk). Therefore, by the analogous equation for Uk, we obtain (Vuk, V (Uk - 2ui)) + a (Uk, Uk - 2ui) _ (f, uk - 2ui) . Adding up the above two lines yields !IVUk112+Ilouill2-2 (Vuk, Vui)+a (IlukII2 + Iluill2 - 2 (Uk, ui))

_ (f, uk - U0,

whence

11V(Uk-ui)112+alluk-uil12=(f,Uk-ui)< IIflllluk - Uill. Since link - ui 1I -+ 0 as k, i --+ oo, we conclude that also lI V (uk - ui) 11 - 0. Therefore, the sequence {ui} is Cauchy in W01 (M) and, hence, converges in

W0 (M). Since its limit in L2 (M) is u, we conclude that the limit of {ui} in W0 (M) is also u. In particular, u E W0 (M). We are left to show that u = Raf . Fix a function co c Co' (M) and observe that the support of cp is contained in Qi when i is large enough. Therefore, (5.68) holds for this cp for all large enough i. Passing to the limit

5.5. RESOLVENT AND THE HEAT SEMIGROUP IN SUBSETS

147

as i -* oo, we obtain that the same equation holds for u instead of ui, that is,

(5.69) (Vu, VP) + a (u, (P) _ (f, P) . Since Co (M) is dense in W0 (M), this identity holds for all cp E W0 (M). By Theorem 4.5, the equation (5.69) has a unique solution u E W01 (M), and this solution is Ra f , which finishes the proof.

THEOREM 5.23. Let f E L2 (M) be a non-negative function, and t > 0. (i) For any open set S2 C M, (5.70)

P1_1 f < Pt f.

(ii) For any exhaustion sequence {Oil', in M, Poi if L2z Ptf as i -> 00. REMARK. As we will see from the proof, we have also Phi f

Pt f . It will

be shown in Chapter 7 that the functions FOif, Ptf are C°°-smooth and, Co. in fact, Pti f - -* Ptf (see Theorem 7.10 and Exercise 7.18).

PROOF. (i) For any a > 0, we have by Theorem 5.22 Ra f < Ra f . By Theorem 5.9, the operators Ra and Ra preserve inequalities. Therefore, we obtain by iteration that (Ra) k f < Ra f, whence by (5.30)

P0f

lim

kt Ik(R?) k f

(-)t Rkf=Ptf


t

(cf. Exercise 2.2 for preserving inequalities by convergence in L2). Alternative proof of (i). By Theorem 4.9, function u (t, ) := Ptf satisfies the conditions

for all t > 0, = L1,u, udt(t, ) E Wo (M) , for all t > 0,

f

as t->0,

where dt is the strong derivative in L2 (M). It follows that the restriction of u to 0 (also denoted by u) belongs to Wi (0) for any t > 0 and solves the Cauchy problem in 0 with the initial function f. Since by Theorem 5.11 u > 0, we conclude by Corollary 5.17 that u > P° f , which was to be proved.

(ii) By part (i), the sequence of functions {P0i f }°_ is increasing and is bounded by Pt f Hence, for any t > 0, the sequence {P°i f } converges almost everywhere to a function ut such that .

0
f ZL2 ut. We need to show that ut = Pt f .

5. WEAK MAXIMUM PRINCIPLE AND RELATED TOPICS

148

Fix a non-negative function co c Co (M) and observe that cp E C0 (Slj) for large enough i. It follows from (5.29) and the monotone convergence theorem that, for any a > 0,

e-at(p.f,

)=

f(R'f,

,

)dt

f e-a(ut) dt, 0

as i --3 oo. On the other hand, by Theorem 5.22 and (5.29), (Raf, P) - J

(Ra2 f, (P)

00

e-at (pt f, (P) dt.

0

We conclude that 00

00

e-at

(ut, P) dt = J

J0

e-"at (Ptf, (P) dt,

0

which, in the view of inequality (Ut, cp) < (Pt f, cp), is only possible when (ut,
(5.71)

for almost all t > 0. Let us show that, in fact, both functions t (Ptf, gyp) and t -4 (ut, co) are continuous in t > 0, which will imply that (5.71) holds for all t > 0. By Theorem 4.9, the function (Pt!, cp) is even differentiable in t > 0. The same theorem also yields dt(P"'f,(P) _

where d

dt (P

(G'-'tiP 'f, W) = (Pozf,Aup),

is the inner product in L2. Using

f, P) <

1, we obtain II f IIL2(M)11 o,,01 L2(M).

Since the right hand side here does not depend on i, we see that all the functions

f, cp) have uniformly bounded derivatives in t and, hence, are Lipschitz functions with the same Lipschitz constant. Therefore, the limit function (ut, (p) is also Lipschitz and, in particular, continuous. Finally, since (5.71) holds for an arbitrary non-negative function cp E Co (M), we conclude that ut = Pt f (cf. Exercise 4.7).

Exercises. 5.22. Let u be a function from C (M) n W0 (M). For any a > 0, set

Ua={xEM:u(x)>a}. Prove that (u - a)+ E W0 (Un). 5.23. Let S2 be an open subset of a weighted manifold M and K be a compact subset of 0. Let f be a non-negative function from L2 (M). Prove that, for all a > 0,

R.f -Rcnl f G esupR«f. M\K

(5.72)

5.24. Under the hypotheses of Exercise 5.23, prove that, for all t > 0,

Ptf -Pof < sup esupPs f. sE[O,t] M\K

(5.73)

NOTES

149

5.25. Let {ci }°_-1 be an increasing sequence of open subsets of M, S2 = u°° 1 ci, and f E L2 (SZl). Prove that the family of functions {Pt ti f }°° considered as the paths in i_ 1 L2 (SZ), is equicontinuous in t E [0, +oo) with resect to the norm in L2 (52).

5.26. Let A be the multiplication operator by a bounded,non-negative measurable function a on M.

(a) Prove that A is a bounded, non-negative definite, self-adjoint operator in L2 and, for any non-negative f E L2 and t > 0, 0 < e-tA f < f. (5.74) (b) Prove that, for any non-negative f E L2 and t > 0, 0 < e-t(c+A) f < e-tcf. (5.75) (c) Using part (b), give an alternative proof of the fact that P I f < Pt f . HINT. In part (b) use the Trotter product formula: B1\n e-t(A+a) f = urn _ t (5.76) f n-f eo Ce nAe that is true for any two non-negative definite self-adjoint operators A, B in L2.

Notes There are different approaches to the maximum principle. The classical approach as in Lemma 1.5 applies to smooth solutions of the Laplace and heat equations and uses the fact that the derivatives at the extremal points have certain signs. We will use this approach in Chapter 8 again, after having established the smoothness of the solutions. In the present Chapter we work with weak solutions, and the boundary values are also understood in a weak sense, so that other methods are employed. It was revealed by Beurling and Deny [39], [106] that the Markovian properties of the heat semigroup originate from certain properties of the Dirichlet integral fM IDu12 dµ, which in turn follow

from the chain rule for the gradient V. This is why the chain rule for the weak gradient is discussed in details in Sections 5.1 and 5.2 (see also [130]). Another useful tool is the resolvent R. The use of the resolvent for investigation of the heat semigroup goes back to the Hille-Yoshida theorem. Obtaining the Markovian properties of Pt via those of R. is a powerful method that we have borrowed from [124]. Theorems 5.16, 5.22, 5.23 in the present forms as well as their proofs were taken from [162].

The reader is referred [41], [115], [124] for the Markovian properties in the general context of Markov semigroups and Markov processes.

CHAPTER 6

Regularity theory in Ilk' We present here the regularity theory for second order elliptic and parabolic equations in R7z with smooth coefficients. In the next Chapter 7, this

theory will be transplanted to manifolds and used, in particular, to prove the existence of the heat kernel. We use here the same notation as in Chapter 2.

6.1. Embedding theorems 6.1.1. Embedding W10C * Cm. In this section, we prove the Sobolev embedding theorem (known also as the Sobolev lemma), which provides the

link between the classical and weak derivatives. Let us first mention the following trivial embedding. CLAIM. For any open set S2 C Rn and any non-negative integer m, we have an embedding C+m

A " Wio'_ A

(6.1)

PROOF. Indeed, if u E C'm' (S2) then any classical derivative &au of order a < m is also a weak derivative from L oC (S2) and, for any open set S2' C= S2, Ilaaull L2(Q') < CSUp 1aaul 0'

,

which implies

ilullwmp) < Hence, the identical mapping C' (52) Wio (S2) is not only a linear injection but is also continuous, which means that it is an embedding. CIlulIcmp).

The next theorem provides a highly non-trivial embedding of Wk (S2) to Ctm (S2).

THEOREM 6.1. (Sobolev embedding theorem) Let S2 be an open subset

of R. If k and m are non-negative integers such that

k>m+2 then u E W oC (S2) implies u c C' (S2). Moreover, for all relatively compact open sets S2', S2" such that S2' C= S2" C S2

llullcm(c,) < 151

(6.2)

6. REGULARITY THEORY IN Rn

152

where the constant C depends on Q', SZ", k, m, n.

More precisely, the first claim means that, for every u E W/ (92), there is a version of u that belongs to C' (1)), which defines a linear injection from W10C (0) to Cm (a). The estimate (6.2) means that this injection is continuous, so that we have an embedding (ci) _ Cm (Q). W1 Set

00

W°°(12)= n Wk (0) k=0

and cc

Wit (c)= nWo°(1) k=0

The topology in the space WOO (SZ) is defined by the family of seminorms IIUIIWk()),

for all positive integers k, and the topology in Wf (0) is defined by the family of seminorms IIuIIWk(s2'),

where k is a positive integer and 1' C= 52 is an open set. It follows from (6.1) and (6.3) that Wig (c) = COO (9) ,

where the equality means also the identity of the topologies. Hence, in order to prove that a function from L OC belongs to COO, it suffices to show that it has weak derivatives of all orders. Although the latter may be difficult as well, the existence of weak derivatives can be frequently proved using the tools of the theory of Hilbert spaces, which are not available for the spaces Ck. EXAMPLE 6.2. Let us show that u E L100 (SZ) implies u E W- (S2), for

any k > n/2. Indeed, fix an open set S2' C SZ and observe that, for any cp c D (S2'), we have by Theorem 6.1

(u, ) =

fudsup

I

S2'

where C depends on SZ' and n. It follows that IIUIIW-k(c') <_ CIIuIIL1(n )

and, hence, u E W10C .

PROOF OF THEOREM 6.1. We split the proof into a series of claims. Recall that BR = {x E 1Rn : IxI < R}. CLAIM 1. For any u E D (BR) and k > n/2, Iu(0)I C CIIuIIWk,

(6.4)

6.1. EMBEDDING THEOREMS

153

where the constant C depends on k, n, R. We use for the proof the polar coordinates (r, 0) centered at the origin 0 E Rr' (cf. Section 3.9), and write u = u (r, 6) away from 0. The relations between the Cartesian and polar coordinates are given by the identities

xj =rfj (0), where f are the smooth functions of 0 E Sn-1 such that

(f,)21

(6.5)

ar = P (00j,

(6.6)

3

(cf. (3.61)). This implies that whence it follows by induction that, for any positive integer k, ar = fi I .... fjka,l...a,k. Applying the Cauchy-Schwarz inequality and (6.5), we obtain 2

jaau12.

aru <

(6.7)

HaH
In particular, we see that the function ar u is bounded in Rn \ {0} (note that this function is not defined at 0), which allows to integrate aT u in r over the interval [0, R]. For any 0 E Sn-1, we have u (R, 0) = 0 whence we obtain by the fundamental theorem of calculus R

u (0)

au(r,0)dr.

fo

Integration by parts yields

U (0)

fR /R [aru (r, 0) r]0 + J raru (r, 0) dr = ra9u (r, 0) dr,

J0

0

and continuing by induction, we arrive at U (0)

fRrk_18u(r,0)dr. (k-1)!Integrating

this identity in 0 over Sn-1 and using rn-1drd9 = dp, where p is the Lebesgue measure (cf. (3.82)), we obtain Wnu (0)

(k - 1)! IER

The Cauchy-Schwarz inequality/ yields then

(0) (2 < C f

BR

J

f

BR

lar ul2 dp.

(6.8)

6. REGULARITY THEORY IN

154

The first integral in (6.8) is evaluated in the polar coordinates as follows: R

LR

r2k-2ndp = Wn f r2k-2nrn-ldr = Wn

/R //

r2k-n-lydr =

CR2k-n < 00,

0

0

where we have used the hypothesis k > n/2. Hence, this integral is just a constant depending on R. By (6.7), the second integral in (6.8) is bounded by

Ir IaI
la«u12dµIIUI12

Therefore, (6.4) follows from (6.8).

For the next Claims 2-4, 52 C R is a bounded open set. CLAIM 2. For any u c D (92) and k > n/2, we have sup Jul < CIIuIIwk

(6.9)

where the constant C depends on k, n, and' diam Q. Indeed, let x be a point of maximum of Jul and R = diam S2. Applying Claim 1 in the ball BR (x), we obtain (6.9). CLAIM 3. Assume that u c Wk (Q), where k > n/2, and let the support of u be a compact set in Q. Then u E C (S2) and the estimate (6.9) holds. Let W be a mollifier and set uj = u * cpllj where j is a positive integer. By Lemma 2.9, we have uj E D (52), provided j is large enough, and by Theorem 2.13, uj -3 u in Wk when j -* oo. Applying (6.9) to the difference

ui - uj, we obtain Suplui-u.71 CCllui - ujllwk. Since the right hand side tends to 0 as i, j -+ oo, we obtain that the sequence {uj } is Cauchy with respect to the sup-norm and, hence, converges uniformly

to a continuous function. Hence, the function u has a continuos version, which satisfies (6.9). CLAIM 4. Assume that u c Wk (S2), where k > n/2 + m and m is a positive integer, and let the support of u be a compact set in Q. Then u E Ctm (S2) and (6.10) Ilullcm < CIIullwk, where the constant C depends on k, m, n, and diam Q.

Indeed, if at < m then aau E Wk-,, which yields by Claim 3 that a«u c C (S2) and sup l aaul < Cllaaullwk-m < CNullwk,

(6.11)

whence the claim follows2. 'In fact, the constant C in (6.9) can be chosen independently of 12, as one can see from the second proof of Theorem 6.1. 2See Claim in the proof of Lemma 2.9.

6.1. EMBEDDING THEOREMS

155

Finally, let us prove the statement of Theorem 6.1. Assume u E W10C (S2)

where S2 is an open subset of Rn and k > n/2 + m. Choose open sets 52' C S2" C Sl and a function 'b E Co' (1l") such that 'b - 1 on Q. Then ?u E Wk (Q11) (cf. Exercise 2.26) and the support of u is a compact subset of a". By Claim 4, we conclude that Ou E Cm (S2"). In particular, u E Cm (S2') because u = ibu in 1'. Since .l' may be chosen arbitrarily, we conclude that u E Cm (S1). It follows from (6.10) and (2.38) that 11U11Cm(D') <

C'llullwk(si ),

CII

which finishes the proof.

SECOND PROOF. We use here the Fourier transform and the results of Exercise 2.34. Assume first that u E Wk (Rn) with k > n/2 and prove that is defined and is also u E C (Rn). Since u c L2, the Fourier transform u in L2. By (2.42) we have

f


(6.12)

n

By the Cauchy-Schwarz inequality,

(fn l u () i d

\

l

2


n

u () I2 (1 + 112) k d J n

(1 + 112) -k d

(6.13)

The condition k > n/2 implies that the last integral in (6.13) converges, which together with (6.12) yields

f

nlu(C)ld <_CIIuIIwk.

(6.14)

In particular, we see that u E L' and, hence, u can be obtain from u by the inversion formula (6.15)

(2,)n/2 fn u (C)

u (x)

for almost all x. Let us show that the right hand side of (6.15) is a continuous function. Indeed, for all x, y E Rn, we have

fn u (C) eZxld- fn

u (;) e&y6d =

fn u () (et -

eiY6)

<.

If y -3 x then the function under the integral in the right hand side tends to 0 and is bounded by the integrable function 2 (u (C) 1. We conclude by the dominated convergence theorem that the integral tends to 0 and, hence, the function u has a continuous version, given by the right hand side of (6.15). It also follows from (6.14) and (6.15) that .

Sup IuI < CIIUIIWk Rn

Since this proves Claim 3 from the first proof, the rest follows in the same way.

6. REGULARITY THEORY IN Rn

156

THIRD PROOF. We will prove here a somewhat weaker version of The-

orem 6.1, when the hypothesis k > m + n/2 is replaced by the stronger condition k > m + 21, where l is the minimal integer that is greater than n/4. This version of Theorem 6.1 is sufficient for all our applications. The advantage of this proof is that it can be carried over to manifolds under a mild assumption that the heat kernel satisfies a certain upper estimate; besides, it can be enhanced to work also for the full range of k (cf. Exercises 7.43, 7.44, 7.46).

We start with the following claim. Recall that Pt is the heat semigroup defined in Section 2.7. CLAIM. If u E D (R'), k is a positive integer, and

f = (-A+ id)k u,

(6.16)

then, for any x E llgn, U (x) = / 0

00

tk-Ie-t (k _ 1)!Ptf (x) dt.

(6.17)

By Lemma 2.17, we have in [0, +oo) x Rn the identity

Ptf = Pt ((-A + id)k u) _ (-0 + id)k Ptu. Since Ptu satisfies the heat equation and, hence,

(-0 + id)k Ptu = (-at + id)k Ptu, we obtain

Pt f = (-at + id)k Ptu. Therefore, the right hand side of (6.17) is equal to 0o tk-le-t (k - 1)! (-at + id)k Ptu dt.

Integrating by parts in (6.18) and using the identity tk-2e-t

tk-le-t

(at + id) (k - 1)! which holds for any k > 2, we obtain tk-1e-t

0o

- (k - 2)!'

(-at + id) k Ptu dt

(k - 1)!

o

00

tk-1 e-t

(-at + id)'-' Ptu o tk-le-t (_at + id)k-1 Ptu dt (at + id)

[(k - 1)! 00

+ o 0o 0

(k - 1)! tk-2e-t

(k - 2)!

(-at + id) k-1 Ptu dt,

(6.18)

6.1. EMBEDDING THEOREMS

157

where the limits at 0 and oo vanish due to k > 1 and the boundedness of the function (-at + id)k-1 Ptu (x) in [0, +oo) x Rn (cf. Lemma 2.17). Hence, the integral in (6.18) reduces by induction to a similar integral with k = 1. Integrating by parts once again and using ((9t + id) e-t = 0 and Ptu -+ u as t --- 0 (cf. Theorem 1.3), we obtain co

which proves (6.17).

Let now 1 be the minimal integer that is greater than n/4, u E D (Rn) and

f = (-A+id)lu. It is easy to see that, by (1.22),

f pt (z) dz = pt * pt (0) = P2t (0) = (87rt)-n/2. Hence, for all x E Rn and t > 0, I Ptf (x) I

<

/n pt (x - y)dp)

1/2

1/2

( JJR' f 2(y)dy

CR

= (87ft)-n/4 If IIL2,

(6.19)

which together with (6.17) yields

fCo tet <(l-1)!IPtf(x)Idt< IfIIL2J

telu(x (l-1)R(87tt)-n/4dt.

The condition l > n/4 implies that the above integral converges, whence we obtain Iu(x)I < CIIfIIL2,

where the constant C depends only on n. Since f can be represented as a combination of the derivatives of u up to the order 21, it follows that suplul < CIIuhIw21

(6.20)

The proof is finished in the same way as the first proof after Claim 2.

Exercises. 6.1. Show that the delta function b in Rn belongs to

W-k

for any k > n/2.

E

6. REGULARITY THEORY IN Rn

158

6.1.2. Compact embedding Wo -a L2. Define the space Wo (t2) as the closure of Co (0) in W1 (0). Clearly, Wo (fl) is a Hilbert space with the same inner product as in W1 (Il). THEOREM 6.3. (Rellich compact embedding theorem) If Sl is a relatively compact open subset of Rn then the identical embedding Wo (S2)

`* L2 (1)

is a compact operator.

PROOF. Any function f E Wp (1) can be extended to W" by setting f = 0 outside Q. Clearly, f E L2 fl L1 (R'). Moreover, f E W' (Rn) because f is the limit in W1 (1) of a sequence {cpk} C Co (11), and this sequence

converges also in W1 (Rn) Let { fk} be a bounded sequence in Wo (Il). Extending fk to R as above, we can assume that also fk c W' (R n). Since { fk} is bounded in L2 (Rn), there exists a subsequence, denoted again by { fk}, which converges weakly in L2 (Wi) to a function f c L2 (Rn). Let us show that, in fact, { fk} converges to f in L2 (S2)-norm, which will settle the claim.

Let us use the heat semigroup Pt as in the third proof of Theorem 6.1. For any t > 0, we have by the triangle inequality 1lfk-fIIL2(G) <_

IIfk-PtfkIIL2(n)+IjPtfk-PtfIIL2(Rn)+IIPtf-fIIL2(Rn).

(6.21)

Let C be a constant that bounds (Ifkllwl for all k. Then Lemma 2.20 yields

IIfk -PtfkllL2(an)'5 Vt

IIfklIwi(aan)

< C/.

(6.22)

Since { fk} converges to f weakly in L2 (JR'2) as k -+ oo, we obtain that, for all x E Rn,

Ptfk(x) = (fk,Pt (x On the other hand, by (6.19) sup I Ptfkl

))L2 4 (f,pt (x - .))L2 = Ptf (x)

(81rt)

IIfkjIL2(Rn) < C

(8irt)-ni4.

Hence, for any fixed t > 0, the sequence {Pt fk} is bounded in sup-norm and converges to Ptf pointwise in R'. Since p (Q) < oc, the dominated convergence theorem yields II Ptfk - Ptf IIL2(G) -* 0 as k -* oo.

(6.23)

From (6.21), (6.22), and (6.23), we obtain that, for any t > 0, lim sup I I .fk - .f I I L2 (Q) k-+oo

< C/ + II Pt f - .f I I L2 (Rn) .

The proof is finished by letting t -* 0 because II Ptf - f Lemma 2.18.

IIL2(an)

-4 0 by

6.2. TWO TECHNICAL LEMMAS

159

Exercises. 6.2. The purpose of this problem is to give an alternative proof of Theorem 6.3 by means of the Fourier transform. Let 0 be a bounded open set in R'. Recall that Wo (S2) can be considered as a subspace of Wl (R') by extending functions by 0 outside 0.

(a) Prove that, for all f E Wof(8f)9dx (fl) and g E C' (R'),

=-f faigdx.

(6.24)

= (1 + ISI2) j() ,

(6.25)

a (b) Prove that, for any f E Wo (0) and for any e E R', (f, ez£x)

W1(,)

where f is the Fourier transform of f. (c) Let { fk} be a sequence from Wo (S2) such that fk converges weakly in W' (R') to a function f E W' (Wi). Prove that fk () -i f (k), for any e E R8 . Prove that also fk

fin L2

(II8 i).

(d) Finally, prove that if {fk} is a bounded sequence in Wo (0) then {fk} contains a subsequence that converges in L2 (S2). HINT. Use Exercises 2.28 and 2.34.

6.2. Two technical lemmas LEMMA 6.4. (Friedrichs-Poincare inequality) Let S2 be a bounded open set in R°. Then, for any cp E D (il) and any index j = 1, ..., n, f cp2dp < (diam. Q)2 fn (8

)2 dµ.

(6.26)

PROOF. Set l = diam Q. Consider first the case n = 1 when we can assume that f is the interval (0,1) (note that we can always expand 0 to an interval of the same diameter since a function co E D (Sl) can be extended to a function cp E D (R') by setting cp = 0 outside f2). Since cp (0) = 0, we have, for any x E (0, 1),

(fX P2

(x) = whence, integrating in x,

W' (s)

ds) < l

fl 0

2 (s) ds,

f t W2 (x) dx < l2 f0

J0

(V,) 2 (s) ds,

which is exactly (6.26) for the case n = 1. In the case n > 1, first apply the one-dimensional Friedrichs' inequality to the function cp (x) with respect to the variable xi considering all other variables frozen, and then integrate in all other variables, which yields (6.26). Recall that, for any mollifier cp, we denote by V, the function s-"W (x/E) (see Chapter 2).

6. REGULARITY THEORY IN R'

160

LEMMA 6.5. (Friedrichs lemma) Let 52 be an open set in R' and let a E Cc (S2). Consider the operator A in S2 defined by Au = a&j u,

for some j = 1, ..., n. Then, for any function u E L2 (S2) with compact support in Sl and for any mollifier cp in Rn, we have IIA (u * 7'E) -

1 1 L 2,

(0)

- 0 as e -3 0.

(6.27)

PROOF. Let S2o be the eo-neighborhood of supp u, where eo > 0 is so small that 52o C= Q. Let us extend u to a function in Rn by setting u = 0 outside supp u. Then the convolution u * cpe is defined as a smooth function

in R' and, if e < eo then u * 7'E is supported in Q. In turn, this implies that the expression A (u * cpe) defines a function from D (a). Similarly, Au is a distribution supported by supp u, and (Au) * cpE is a function from D (0). Let us show that, for e < eo, 11A (u * 7'E) - (Au) * We 11L2 < KIInIIL2,

(6.28)

K= sup IVal Cl+ f IxI IDjcpl dx)

(6.29)

where no

.

n

The point of the inequality (6.28) is that although the constant K depends on functions a, cp and on the set S2o, it is still independent of u and e. We have, for any x E 0,

A(u * cpE)(x) = a(aj(u * co))(x) = a(u *,9jco,)(x) a(x)u(y)ajcpe(x - y)dy

(6.30)

in and

(Au) * cpe(x) _ (Au, ( p

,,

- .))

_ (aju, - (u, aj

(x - y)dy

- f u (y) aj a (y),p, (x - y)dy.

(6.31)

Setting

AEu:=A(u*cog)-(Au)*cpa, we obtain from (6.30) and (6.31)

Aeu (x) = J (a(x) - a(y)) ajcye(x - y)u(y)dy + f aja(y)W,(x - y)u(y)dy.

(6.32)

6.2. TWO TECHNICAL LEMMAS

161

Note that the domains of integration in (6.32) can be restricted to y E supp u n B, (x). (6.33) If x 0 00 then the set (6.33) is empty and, hence, AEu(x) = 0. Therefore, A,u(x) zA 0 implies x E S2o. In this case, for any y is as in (6.33), we have x E BE (y) C SZo whence it follows that

la(x) - a(y)I < sup IVal lx - yi = C Ix - yl,

(6.34)

no

where C := supoo IVal (see Fig. 6.1).

B£(y)'

B£(x)

FIGURE 6.1. If y E supp unB, (x) then x E BE (y) C SZo and, hence, the straight line segment between x and y is contained in S2o, which implies (6.34).

Since also Iaja (y) I < C, we obtain from (6.32) IAeu(x)l


f

(Ix -

(x-y)+(PE(x-y) lu(y)I dy

fe (x -

y) I u(y) I dy,

where *E(x)

Ixl

Hence, for all x E Rn, we have

(x) + 0E(x).

Ef

IA,u(x)I cCIul*7PE(x),

which implies by rescaling the inequality (2.25) of Theorem 2.11 (see also Remark 2.12) that IIAEnIIL2 < c [f.n V),(x)dxl IlullL2.

Evaluating the integral of, by changing z = x/e, we obtain

(x)dx = 1 +

EX

J

II8

=

1+j n

IEzI

dz =1 +

f

IzJ

dz,

6. REGULARITY THEORY IN R

162

which gives II AEuII L2 < Kliu1lL2,

(6.35)

that is (6.28). Let us now prove (6.27), that is, IIAeuIIL2

0 as E-a 0.

If U E D (S2) then, by Lemma 2.10, u * Te

D u and, hence,

A(u*co.)-+Au. Applying Lemma 2.10 to Au, we obtain (Au) * co, - 4 Au, which together with the previous line implies AEu 0. For an arbitrary function u E L2 (1) with compact support, choose a sequence Uk E D (S2) such that Uk -> u in L2 (for example, take Uk = u * cp1/k - cf. Theorem 2.3) and observe that

AEu=Aeuk+Ae(u-uk). To estimate the second term here, we will apply (6.35) to the difference U - Uk. If k is large enough then supp (u - uk) is contained in a small neighborhood of supp u and, hence, the constant K from (6.29) can be chosen the same for all such k. Hence, we obtain IIAeuklIL2+KIIu-ukIIL2. (u-Uk)IIL2 -< Since J U - ukIIL2 -4 0 as k --* oo and, for any fixed k, I l Aeuk II L2 -4 0 as e -+ 0, we conclude that II Aeull L2 -+ 0, which was to be proved.

IIAEUIIL2 < IIAEukllL2+II I

6.3. Local elliptic regularity Fix an open set Sl C R and L be the following differential operator in 52:

L = ai (auj (x) Off)

,

where aia (x) are smooth functions in S2 such that the matrix (air (x))ij-1 is symmetric and positive definite, for any x E Q. Any such operator with a positive definite matrix (ai3) is referred to as an elliptic operator. The fact that the matrix (aij) is positive definite means that, for any point x E 0 there is a number c (x) > 0 such that azj (x) c (x) I 12 for any E R. (6.36) The number c (x) is called the ellipticity constant of operator L at x. Clearly, c (x) can be chosen to be a continuous function of x. This implies that, for

any compact set K C 0, c (x) is bounded below by a positive constant for all x E K, which is called the ellipticity constant of operator L in K.

6.3. LOCAL ELLIPTIC REGULARITY

163

The symmetry of the matrix a'3 implies that the operator L is symmetric

with respect to the Lebesgue measure p in the following sense: for any functions u, v E D (S2),

f Luvdp = - J a2j 8iu0,vdp

uLvdµ,

(6.37)

which follows immediately from the integration-by-parts formula or from the divergence theorem. The operator L is obviously defined on D' (f2) because all parts of the expression ai (a'jaj) are defined as operators in D' (cl) (see Section 2.4). For

alluED'(Sl) andcp E V (2), we have (Lu, c)

(ai (a2'aj u) , o)

_

- (C3, u, a'jaicp)

(aZ'aj u, aiy) (u, aj

that is, (Lu, co) _ (u, Lcp).

This identity can be also used as the definition of Lu for a distribution u.

6.3.1. Solutions from Li .. LEMMA 6.6. If a function u E L2 (fl) is compactly supported in Q and Lu E W-1 (0), then

L(u*YE)W 3Lu

(6.38)

PROOF. Consider the difference

L(u*CpE)-(Lu)*7E=5i(a'j aj (u*cpe)) -ai((a2j a, u) *YE) =aiff where

a23aj (u * y) - (a"jaj u) * cpE. As follows from Lemma 6.5, 11 fE II L2 -+ 0 as e --* 0 whence IIL (u * cpE) - (Lu) *

(pEIIW-i =

IIfEIIL2 --+0.

I(atifEIIW-i

Since by Theorem 2.16 (Lu) *

7o E

Lu,

we obtain (6.38).

LEMMA 6.7. (A priori estimate) For any open set S2' C f2 and for any u c D (S2'), IIulIWI < CIILuIIW-1

,

where the constant C depends on diam f2' and on the ellipticity constant of L in f2'.

6. REGULARITY THEORY IN

164

PROOF. Lemma 6.4 implies IIuIIWI -< CllouIIL2,

where C depends on diam S2'. Setting f = -Lu, we obtain by (6.37)

(f, u) = - fn (Lu) u dp =

faui(x)au5judi.

Let c > 0 be the ellipticity constant of L in 52' so that, for any x E S2', a23 (x) azu aju > c

I Vul2

.

Combining with the previous lines, we obtain

(f,u) > c ( IDuI2 dµ ? c IIuIIWI,

(6.39)

for some c' > 0. On the other hand, by the definition of the norm W-1, (f, u) <-

IIfIIW-1IIuUIW1,

which implies IIuIIWI < IIfIIW-1IIulIW1,

whence the claim follows.

LEMMA 6.8. For any integer m > -1, if u E Woo ' (I) and Lu E

Wio (Sl) then u E Wio+2 (a). Moreover, for all open subsets S2' C= P" C= 0, IIuiIWm+2(c') <- C (IluIIwm-+i(Q,) +

(6.40)

where C is a constant depending on f2', f2", L, m.

PROOF. The main difficulty lies in the proof of the inductive basis for m = -1, whereas the inductive step is straightforward. The inductive basis for m = -1. Assuming that u E L21,,,c (Sl) and Lu E Wio (S2), let us show that u E W1 (t) and that the following estimate holds:

Ilullwl(u,) < C (IIuhIL2(n/) +

(6.41)

Let 0 E D (S2") be a cutoff function of SZ' in 1" (see Theorem 2.2). Then the function v :=,Ou obviously belongs to L2 (S2") and supp v is a compact subset of a". We claim that Lv E W-1 (Q"). Indeed, observe that

Lv =Lu + 2azjai'baju + (LO) u,

(6.42)

and, by Lemma 2.14, the right hand side belongs to W-1 (Q") because Lu, aju, u belong to WW;,, whereas 0, aijaao, Lb belong to D (Il"). Let co be a mollifier in Rn. By Lemma 6.7, we have IIv * cEllwl < CIIL (v * cE) IIW-1,

whereas by Lemma 6.38 1

L (v * cpe) -+ Lv ass -* 0,

6.3. LOCAL ELLIPTIC REGULARITY

165

which implies that limsup llv * tpellwl < CllLvllw-1. C-40

By Theorem 2.13, we conclude that v E W1 (Q") and Cl

Since u = v on ft', we obtain that u E W1 (a') and Ilullw'(ci')

By varying the set fl' we conclude u E Wloc (Il). Finally, observing that by (6.42)

C

IiLullw-1(n,,))

and combining this with the two previous lines, we obtain (6.41).

The inductive step from m - 1 to m, where m > 0. For an arbitrary distribution u E V (Q), we have

al (Lu) - L (8lu) = alai (az3aju) - ai (aaja,alu) as [al (atija,u) - aajalaju] ai [(ala2?) aju] .

(6.43)

Assuming that u E Wlo +1 (5l) and Lu E Wi ,, (SZ) and noticing that the right hand side of (6.43) contains only first and second derivatives of u, we obtain

L (alu) = al (Lu) - ai [(ala2j) 9ju] E Wlo 1.

(6.44)

Since alu E WIT, we can apply the inductive hypothesis to alu, which yields alu E Wlo +1 and, hence, u E WI +2. Finally, we( see from (6.44) that IIL (alU) IIWm-1(n,,) <

CIIUIIWm+1(Q,,) + II Lull Wm(Q,,),

whence by the inductive hypothesis II aIUII Wm+1(52')

< C (II C (ll ull

+IIL (alu) IILuJI

which obviously implies (6.40).

THEOREM 6.9. For any integer m _> -1, if u E Ll (f) and Lu E Wl

(0) then u E Wlo +2 (1). Moreover, for all open subsets S2' C= Q" C= Sl, Ilullwm+2(ci') < C (IIuliL2(Q1') + II Lull

(6.45)

where C is a constant depending on SZ', 0", L, m.

Note the hypotheses of Theorem 6.9 are weaker than those of Lemma

6.8 - instead of the requirement u E Wlo+1, we assume here only u E Li c

6. REGULARITY THEORY IN Rn

166

PROOF. Let k be the largest integer between 0 and m + 2 such that u E W We need to show that k = m + 2. Indeed, if k < m + 1 then we have also Lu E W1o° 1, whence it follows by Lemma 6.8 that u E W10C 1, thus

contradicting the definition of k.

The estimate (6.45) is proved by improving inductively the estimate (6.40) of Lemma 6.8. For that, consider a decreasing sequence of open sets {52i}m 0 such that ci0 = a", S1m+2 = S2' and S2j+1 C= Q j. By Lemma 6.8, we

obtain, for any 1 < k < m + 2, IIUIIWk(ok)

< C (IIuIwk_1k_l) +

IILuIIWk-2(Q,_,)

< C (+ II LuII which obviously implies (6.45). COROLLARY 6.10. If u E Lloc (12) and Lu E W°, (S2) where

m>l+2-2. and l is a non-negative integer then u E Cl A. Consequently, if u E Li (52) and Lu E C°° (52) then also u c CO° (S2) . PROOF. Indeed, by Theorem 6.9 u e W1,O'+2 (S2) and, since m+2 > l+ 2 Theorem 6.1 implies u E Cd (S2). The second claim is obvious.

For applications on manifold, we need the following consequence of Theorem 6.9 for a bit more general operator L. COROLLARY 6.11. Consider in 52 the following operator

L=b(x)9i(a'j(x)aj), where a'j (x) and b (x) belong to C°° (S2), b (x) > 0, and the matrix (auk (x))Z -1 is symmetric and positive definite for all x E E2. Assume that, for some positive integer k,

u, Lu, ..., Lku E Li

C

(52)

.

(6.46)

Then u E Wi j, (S2) and, for all open sets 52' C= 52" C= S2, k

IIuIIW2k(Q-) <_ C E II L1uhI LZ(n ),

(6.47)

1=0

where C depends on S2', S2", L, k.

If m is a non-negative number and

k>2+4

(6.48)

then u E Cm (52) and (6.49)

IIu1Icm(c2,) < C 1=0

6.3. LOCAL ELLIPTIC REGULARITY

167

where C depends on S-t', Q//, L, k, m, n.

PROOF. Note that Theorem 6.9 applies to operator L as well because baz (aijaju) E Wl if and only if as (aajaju) E Wl ,, and the Wm (1')-norms of these functions are comparable for any f2' C= Q.

The inductive basis for k = 0 is trivial. Let us prove the inductive step from k - 1 to k assuming k _> 1. Applying the inductive hypothesis to function v = Lu, we obtain Lu E W277-2, whence u E W2k by Theorem 6.9. To prove (6.47) observe that by (6.40) IIulIW2k(n-) < C (IluhIL2*) +

IILu!Iw2k-2(n*))

,

where f2' C f2* C Il", and, by the inductive hypothesis, k-1 IILuIIw2k-2(n*) = IIV II W2k-2(Q*)

k

<_ C E II L1vil

II L`uII

t=1

1=o

whence (6.47) follows. Finally, u e W 12k (1) and 2k > m + n/2 imply by Theorem 6.1 that U E CM (cl). The estimate (6.49) follows from (6.2) and (6.47).

6.3.2. Solutions from V. Here we extend Lemma 6.8 and Theorem 6.9 to arbitrary negative orders m. We start with the solvability of the equation Lu = f (cf. Section 4.2). For an open set U C R', consider the space Wo.(U), which is the closure of D (U) in W1 (U). Clearly, Wo (U) is a Hilbert space with the same inner product as W1 (U). LEMMA 6.12. Let U C 1 be an open set. Then there exists a bounded

linear operator R : L2 (U) -4 Wo (U) such that, for any f E L2 (U), the function u = R f solves the equation Lu = f. The operator R is called the resolvent of L (cf. Section 4.2). PROOF. Denote by [u, v] a bilinear form in Wo (U) defined by

[u, v] := f8u3jv Let us show that [u, v] is, in fact, an inner product, whose norm is equivalent

to the standard norm in Wo (U). Indeed, using the ellipticity of L, the compactness of U, and Lemma 6.4, we obtain, for any /u E D (U),

[u, u] _ / azjatiu aju dµ > c f U

I VU 12 dE.t

> c' ( u2d/.c,

U

and

[u, u] < C

f

U

(6.50)

U

I Vui2 dµ,

and these estimates extend by continuity to any u E Wo (U). It follows that [u, u] is in a finite ratio with IIUIIWl =

f u2d,ct+f IVu12dµ, U

U

6. REGULARITY THEORY IN ]fin

168

and, hence, the space Wo (U) is a Hilbert space with the inner product The equation Lu = f is equivalent to the identity

fu

a"ajua;W dp=-(f,cp) for any

V(U),

(6.51)

which can be written in the form [u, cp] = - (f, cp) for all cp E Wo (U) .

(6.52)

(f, cc) is a bounded linear functional of cp in W0 (U), because, Note that cp using again (6.50), we have I(f,cc)I < IIfIIL2IIcIIL2 -< CIIfIIL2

[cp,cp]1/2.

(6.53)

Hence, by the Riesz representation theorem, (6.52) has a unique solution U E Wo (U), which allows to define the resolvent by lZf = u. The linearity R is obviously follows from the uniqueness of the solution. Using cp = u in (6.52) and (6.53) yields [u,u]


and, hence, [u, u]1/2 < CII f IIL2, which means that the resolvent R is a bounded operator from L2 (U) to Wo (U) . REMARK 6.13. If f E C°° n L2 (U) then, by Corollary 6.10, the function u = Rf also belongs to the class C°° n L2 (U). Let us mention for a future reference that R is a symmetric operator in

the sense that (R f, g) = (f, Rg) for all f, g E L2 (U)

.

(6.54)

Indeed, setting u = R f and v = Rg, we obtain from (6.52)

[u, v] = - (f, v) = - (f, Rg) and

[v,u] = - (g, U) _ - (g,Rf), whence (6.54) follows. Since Wo (U) is a subspace of L2 (U), we can consider

the resolvent as an operator from L2 (U) to L2 (U). Since U is relatively compact and, by Theorem 6.3, the embedding of Wo (U) into L2 (U) is a compact operator, we obtain that R, as an operator from L2 (U) to L2 (U), is a compact operator. LEMMA 6.14. For any m E Z, if u E Wio+1 (S2) and Lu E Wio (0) then u E Wio+2 (12)

PROOF. If m > -1 then this was proved in Lemma 6.8. Assume m <

-2 and set k = -m so that the statement becomes: if u E Wio

+1

and It suffices to prove that u E W-k+2 (12) for any V) E D (S2). Fix such and set v = bu. Clearly, V E W-k+1 (S2) and, as it follows from (6.42), Lv E W-k (12).

Lu E Waoc then u E

WI-k12.

6.3. LOCAL ELLIPTIC REGULARITY

169

Let U be a small neighborhood of supp v such that U C Q. We need to show that II V II W-k+2(U) < oo, and this will follow if we prove that, for any f E D (U), (v, f) < CII LvII w-k IIf

(6.55)

Ilwk-2.

By Lemma 6.12, for any f E D (U) there exists a function w e C°° fl L2 (U) solving the equation Lw = f in U and satisfying the estimate I1w1lL2(U) < CII f

(6.56)

IIL2(U).

Fix a function cp E D (U) such that cp - 1 in a neighborhood of supp 0. Then cpw E D (U) and

(v, f) _

(v, Lw) _ (v, L (pw)) _ (Lv, cpw) II Lvll w-k(U) II owII Wk(U) <- CII Lvll w-k(U) II W I I Wk(U-),

where U' C= U is a neighborhood of supp cp and the constant C depends only on V. By Theorem 6.9 and (6.56), we obtain IIwIIWk(U1) < C

(IwIiL2(u) +

IILwllwk-2(U)) < C'll f llwk-2(U),

whence (6.55) follows.

Finally, we have the following extension of Theorem 6.9.

THEOREM 6.15. For any m E Z, if u E D' (S2) and Lu E Wl u E Wia+2 (c)

(0) then

PROOF. Let us first show that, for any open set U C= 52 there exists a positive integer l such that u E W-1 (U). By Lemma 2.7, there exist constants N and C such that, for all cp E D (U), (u, cp) < C max sup I aacpl laj
It follows from Theorem 6.1 (more precisely, we use the estimate (6.10) from

the proof of that theorem) that the right hand side here is bounded above by const Ilcell W, (U) provided l > N + n/2. Hence, we obtain (u, cp) < C'II cPII wt(U),

which implies, by the definition of W-1,

Ilullw-1(u) < C' < 00 and U E

W_l

(U).

In particular, we have u c W10C (U). Let k be the maximal integer between -1 and m + 2 such that u E Wio, (U). If k < m + 1 then Lu E W o- 1 (U), which implies by Lemma 6.14 u E W o±1(U), thus contradicting the definition of k. We conclude that k = m + 2, that is, u E Wo +2 (U). It follows that u E WI" +2 (12), which was to be proved.

6. REGULARITY THEORY IN Rn

170

Exercises. 6.3. Prove that, for any open set SZ' C 0, for any m > -1, and for any u E D (S2'), HHuIIwm+2 < CjjLujjwm,

(6.57)

where a constant C depends on S'2', L, m. HINT. Use Lemma 6.7 for the inductive basis and prove the inductive step as in Lemma 6.8.

6.4. Consider a more general operator L = 8a (az' (x) 8;) + b3 (x) 8; + c (x) ,

(6.58)

where a43 is as before, and b' and c are smooth functions in Q. Prove that if u c D' (1) and Lu E WI (f) for some m E 7L then u E WZ +2 (St). Conclude that Lu E C°° implies UEC°°.

6.4. Local parabolic regularity 6.4.1. Anisotropic Sobolev spaces. Denote the Cartesian coordinates in Rn+1 by t, x1, ..., x'. Respectively, the first order partial derivatives

are denoted by at = at and a; = a; for j > 1. For any (n + 1)-dimensional multiindex a = (ao, the partial derivative a« is defined by

as

= (at)a°

(ax1)at

...

(axn)an -

at jai l...an".

Alongside the order Jul of the multiindex, consider its weighted order [a], defined by

[a] := 2ao + al + ... + an. This definition reflect the fact that, in the theory of parabolic equations, the

time derivative at has the same weight as any spatial derivative a; of the second order.

Fix an open set 1 C

The spaces of test functions D (1) and distributions D' (1) are defined in the same way as before. Our purpose is to introduce anisotropic (parabolic) Sobolev spaces Vk (f2) which reflect different weighting of time and space directions. For any non-negative integer k, set II8n+1.

Vk (fl) = {u c L2 (f2) : a«u E L2 (f2) for all a with [a] < k} ,

and the norm in Vk is defined by 11a«u11L2. [a]
Obviously, Vo - L2, whereas

V1(0)={uEL2(1):a,uEL2(f) Vj=1,...,n} and

V2 (c) _ {u E L2 (f2) : atu, aju, aia3u E L2 (f2) Vi, j = 1, ..., n}

.

(6.59)

6.4. LOCAL PARABOLIC REGULARITY

171

Most facts about the spaces Vk are similar to those of Wk. Let us

Vk-[a

point out some distinctions between these spaces. For simplicity, we write Vk - Vk (Q) unless otherwise stated. CLAIM.(a) If U E Vk and [a] < k then a'u E If aau E Vk for all a with [a] _< 1 and one of the numbers k, l is even then u E Vk+t PROOF. (a) Let Q be a multiindex with [,Ci] < k - [a]. Then [a + /3] < k and hence as+Qu E L2. Therefore, & (aau) E L2, which means that aau c V k- [al

(b) It is not difficult to verify that if one of the numbers k, l is even, then any multiindex /3 with [/3] < k + l can be presented in the form 0 = a + a' where [a] < l and [a'] < k. Hence, aQu = aa' (aau) E Vk-[a'I C L2, whence the claim follows.

It follows from part (a) of the above Claim that if u E Vk then aau E Vk-1 and atu E Vk-2 (provided k > 2). We will use below only the case l = 2 of part (b). Note that if both k, l are odd then the claim of part (b) is not true. For example, if k = 1 = 1 then the condition that aau E V1 for all a with [a] < 1 means that the spatial derivatives aju are in V'. This implies that 8iu, 8iaju are in L2. However, to prove that u E V2 we need to know that also Btu E L2, which cannot be derived from the hypotheses. For any positive integer k and a distribution u E D( Q), set IIuIIv-k :=

sup

(u' `p)

(6.60)

cOED(S2)\{o} IIOIIvk

The space V -k (11) is defined by V-k (St)

{u E D' (0) :

II

Iv-k < 0-0} .

Obviously, for all u E V-k (Q) and cp c D (0), we have I(u,(P)J < IIUIIv-k(n)IIcIIvk(c).

The local Sobolev spaces Voc (i) are defined similarly to Wo' A.

The statements of Lemma 2.14 and Theorems 2.13, 2.16 remain true for the spaces Vk, and the proofs are the same, so we do not repeat them. Observing that V2 (SZ) W oc (52) and applying Theorem 6.1, we obtain

that

provided k and m are non-negative integers such that k > m + n/2. Consequently, we have

V,(0)=CO°A.

6. REGULARITY THEORY IN Rn

172

6.4.2. Solutions from Ll Fix an open set 52 C Rn+1 and consider in Q the differential operator

P = p (x) at - az (air (x) a;)

(6.61)

where p and aij are smooth functions depending only on x = (x', ..., xn) (but not on t), p (x) > 0, and the matrix (a2j )i =1 is symmetric and positive definite. The operator P with such properties belongs to the class of parabolic operators. The results of this section remain true if the coefficients

air and p depend also on t but the proofs are simpler if they do not, and this is sufficient for our applications. Setting as in Section 6.3 L = a? (a'2 (x) aj)

,

we can write

P=pat - L. The operator P is defined not only on smooth functions in S2 but also on distributions from D' (1) because all terms on the right hand side of (6.61) are defined as operators in D' (0). For all u E D' (S2) and po E D (0), we have

(Pu, p) = (pate, co) - (Lu, W) whence it follows that .

(u, patco) - (u, Lip)

(Pu, cc) _ (u, P* c0)

,

(6.62)

where

P*= -pat - L is the dual operator to P. The identity (6.62) can be also used as the definition of P on D' (S2). We start with an analog of Lemma 6.7

LEMMA 6.16. (1st a priori estimate) For any open set 12' C= S2 and for any u c D (S2'), IIuIIvi << CIIPuIIv-1,

where the constant C depends on diam S2' and on the ellipticity constant of

Lin52'. PROOF. Setting ff = Pu and multiplying this equation by u, we obtain u f dp = J/c

J fn

pu atu d1i - J uLu dp, c

where dµ = dtdx is the Lebesgue measure in Il8n+1. Since

puatu = 1Ot(pu2) ,

after integrating the function puatu in dt we obtain 0. Hence,

fpuatud= 0.

6.4. LOCAL PARABOLIC REGULARITY

173

Applying the same argument as in the proof Lemma 6.7 (cf. (6.39)), we obtain,

-J uLudµ > cllullvl(Q)

(6.63)

where the constant c > 0 depends on the ellipticity constant of L in S2' and on diam S2'. Since

- f uLudp=

f

ufdµ= (f, u) -< Ilflly-lllullvl

we obtain cllullvi < Ilflly-=Ilullvi, whence the claim follows.

LEMMA 6.17. (2nd a priori estimate) For any open set 52' C= c and for any u E D (SZ'), Ilolly2 < CIIPuIIL2,

(6.64)

where the constant C depends on S2', P.

PROOF. If follows from (6.59) that we have to estimate the L2-norm of

atu as well as that of & u and aza;u. Setting f = Pu and multiplying this equation by atu, we obtain

fatufdp = fp(atu)2d_f3tuLud.

(6.65)

Since at and L commute, we obtain, using integration by parts and (6.37),

f atu Lu dµ = - f u at (Lu) dµ = - f u L(atu) dlt = - f

Lu atu dµ,

whence it follows that

fn atuLudµ=0. Since p (x) is bounded on S2' by a positive constant, say c, we obtain

in

P ((gto)2 d t > cll atoll L2.

Finally, applying the Cauchy-Schwarz inequality to the left hand side of (6.65), we obtain Ilatul1L2(-)IIfIIL2(c) >- CIlatoliL2(Q),

whence IlatuliL2(c) <- CIIfIIL2(c).

(6.66)

To estimate the spatial derivatives, observe that the identity f = patu - Lu implies

IILuI1L2(O) <- IIfiIL2(c +CliatullL2(o) <- C'IIfIIL2(-).

(6.67)

Let Q and Q' be the projections of S2 and S2', respectively, onto the subspace Il8n C Rn+1 spanned by the coordinates x1, ..., xn. Obviously, the operator L can be considered as an elliptic operator in Q. Since Q' C= Q and u (t, ) E

6. REGULARITY THEORY IN Rn

174

D (Q') for any fixed t, the estimate (6.57) of Exercise 6.3 yields, for any fixed t, IIuIIw2(Q) < CII Lu11 L2(Q)

(a somewhat weaker estimate follows also from Theorem 6.9). Integrating this in time and using (6.67), we obtain that the L2 (Q)-norms of the derivatives 9 u and 8i8ju are bounded by Of IIL2(s) Combining with (6.66), we obtain (6.64).

LEMMA 6.18. If u E Li o (SZ) and Pu E V o (S2) then u E V oC (52)

.

Moreover, for all open subsets S2' C 52" C= S2, (6.68)

Ilullvl(n1) < C (IIuIIL2(9211) +

where C is a constant depending on Q, S2", P. PROOF. Let O E D (S2") be a cutoff function of 52' in S2", and let us prove

that the function v = bu belongs to V' (52"), which will imply u E Vol A. Clearly, v E L2 (52") and supp v is a compact subset of W. Next, we have

P (0u) _ %Pu - 2a%8iDiu + (P) u

(6.69)

(cf. (6.42)), whence it follows that Pv E V-1 (S2") and

C

IIPuIIV-1(c )) ,

(6.70)

where C depends on S2', 52", P. Fix a mollifier co in R'+1 and observe that, for small enough E > 0, v * co belongs to D (52"). By Lemma 6.16, we have liv * WEIIvi < CIIP (v * (0 llv-1,

where the constant C depends on S2" and P. Let us show that IIP (v * co) - (Pv) * cp,Ily-1 -* 0 as E -- 0. By Lemma 6.5, we have ll pat (v * 0,) - (Pate) *
(6.71)

(6.72)

(6.73)

As in the proof of Lemma 6.6, we have L (v * cpe) - (Lv) * c

= 8i ff

where

fe := a",9 (v * cp.) - (aijajv) * cpE. By Lemma 6.5,

11 Ri I

I L2 - 3 0 whence

II L (v * WE) - (Lv) * WEII v-1 = II aifftI v-1

WIL2 -* 0.

(6.74)

Combining (6.73) and (6.74), we obtain (6.72).

By extension of Theorem 2.16 to the spaces V-k, the condition Pv E V-1 (52") implies (Pv) * cp6

Pv,

6.4. LOCAL PARABOLIC REGULARITY

175

which together with (6.72) yields

(v*cM--4 Pv ase -+ 0.

(6.75)

It follows from (6.71) and (6.75) that limsup Ilv * 0'Ilvl < CII Pvll v-I. e--r0

By extension of Theorem 2.13 to Vk, we conclude that v c V1 (S2") and llvllv1(s2") < CII'Pvlly-1(a ).

Combining this estimate with (6.70) and Ilulivl(cl,) = llvilv1(Q,), we obtain (6.68).

El

LEMMA 6.19. If u E V1', (52) and Pu E Li (S2) then u E V2, (S2). Moreover, for all open subsets S2' C 12" C= 52, (6.76)

IIUIlV2(Q,) < C

where C is a constant depending on S2', S2", P. PROOF. Let z/b E D (S2") be a cutoff function of a in S2", and let us prove that the function v = -%u belongs to V2 (S2"), which will imply u E VoC (S2).

It follows from (6.69) that Pv E L2 (0") and C

II PvII

NPuiI L2(c,-))

,

(6.77)

where C depends on 52', 0", P. Function v belongs to V1 (12") and has a compact support in W. For any mollifier cp in Rn+1 and a small enough e > 0, we have v * D (S2"). By Lemma 6.17, we obtain

cpe E

(6.78)

liv*coSIIV2

where C depends on S2", P. Let us show that ---+

(6.79)

For that, represent the operator P in the form P = -a2jaza1 - bias + p5t, where bj = alai?. The part of the estimate (6.79) corresponding to the first order terms bjaj and pat, follows from Lemma 6.5 because v E L2 (S2). Next, applying Lemma 6.5 to function acv, which is also in L2 (S2), we obtain Ila"a2 (acv * cps) - (aijaza,v) * cPEII L2 _4 0,

whence (6.79) follows. Since Pv E L2, we have by Theorem 2.11 (Pv) * cpE

Pv

6. REGULARITY THEORY IN R"

176

which together with (6.79) yields L2 P(v*co)-*Pv ase-+0.

Combining with (6.78), we obtain Jim SUP IIv * E-O

EIIv2 <

CIIPvIIL2.

Therefore, by extension of Theorem 2.13 to Vk, we conclude that v E V2 (S2") and
Combining this with (6.77) and Ilullv2(s') = Ilvllv2(12'), we obtain (6.76).

LEMMA 6.20. For any integer m > -1, if u E Va +1 (52) and Pu c Vo (12) then u E Vo +2 (12). Moreover, for all open subsets S2' C 12" C= 12, (6.80)

IluliVm+2(Q,) < C (Ilullv-+1(Q") +

where the constant C depends on S2', 0", P, m.

PROOF. The case m = -1 coincides with Lemma 6.18, and the case m = 0 coincides with Lemma 6.19. Let us prove the inductive step from m - 2 and m - 1 to m, assuming m > 1. To show that u E Vo+2, it suffices to verify that

atu, a,u, aia,u E Vo

.

Since atu E Vj-1 and 10C P (atu) = atPr E V,-2.

(6.81)

the inductive hypothesis yields atu E V m . It follows from (6.43) that

al (Pu) - P ((9au) _ (alp) atu - ai [(ala2j) aju]

,

(6.82)

which implies V--1.

P (alu) E Since alu C V the inductive hypothesis yields alv E V0C+1 Consequently, all the second0corder derivatives aiajv are in Vo , which was to be proved. Let us now prove (6.80). It follows from (6.81) and the inductive hypothesis for m - 2 that I1atU11vm(Q1)


117' (atu)

C

IlPullVm(si")) .

(6.83)

it follows from (6.82) that

C IT (alu) whence, by the inductive hypothesis for m - 1, Ilalull vm+1(c')

IlPullVm(Q )) ,

< C (II alulivm(Q") + IIP (alu)

C

(6.84)

6.4. LOCAL PARABOLIC REGULARITY

177

Combining (6.83) and (6.84) yields (6.80).

THEOREM 6.21. For any integer m > -1, if u E Li (S2) and Pu E C

Vo (52) then u E Vo+2 (a) Moreover, for all open subsets Q' C= Q" C= S2, IIuIIVm+2(o1)

(6.85)

<- C

where C depends on SZ', Sl", P, M.

PROOF. Let k be the largest number between 0 and m + 2 such that u E V0C (Q). If k < m + 1 then we have Pu E VoC 1, which implies by Lemma 6.20 that u E 10C 1, thus contradicting the definition of k. Hence, k = m + 2 and u E Vo +2 which was to be proved. To prove the estimate (6.85), consider a decreasing sequence of open sets {Sli} o2 such that Slo = Sl", 1m+2 = Q' and S1j+1 C SZi. By Lemma 6.20, we obtain, for any 1 < k < m + 2, IuIIVk(ok)


(IIuIIvk-1(Ok_l) +

IIPuIIVk-2(Qk_1))

(Iluvk_1(kl) +

which obviously implies (6.85).

(i) If U E L 10C (S2) and Pu = f where f E

COROLLARY 6.22.

C°° (Q) then also u E C°° (Q). (ii) Let {uk} be a sequence of smooth functions in S2, each satisfying

the equation Puk = f where f c C' (S2). If uk u c L210° (il) then Pu = 0, u E C°° (SZ), and uk

L2

u where

u.

PROOF. (i) Since Pu E V o (c) for any positive integer m, Theorem 6.21 yields that also u E Vo (Sl) for any m. Therefore, u E Wlo (Q) for any m and, by Theorem 6.1, we conclude u E C°° (52).

(ii) Let us first show that u satisfies the equation Pu = f in the distributional sense, that is, for all cp E D (sl) ,

(u, P*(p)

(6.86)

where P* _ -pat - L is the dual operator (cf. (6.62)). Indeed, Puk = f implies that (uk,

(f, c )

whence (6.86) follows by letting k -+ co. By part (i), we conclude that UGCOOA. Setting Vk = u - uk, noticing that Pvk = 0, and applying to vk the estimate (6.85), we obtain, for all open subsets Sl' C= S2" C i and for any positive integer m, IIVkIIVm(c1) < CllvkI L2(Q1/),

Since vk -- 0 in L2 (S2"), we obtain that vk --+ 0 in Vm (SZ').

6. REGULARITY THEORY IN Rn

178

Hence, Vk - 0 in Vr (0) for any m, which implies that also vk - 0 in Wzo (1) for any m and, by the estimate (6.2) of Theorem 6.1, Vk -+ 0 in C°° (fl), which was to be proved. 11

Exercises. 6.5. Let S2' C 52 be open sets and m > -1 be an integer. (a) Prove that, for any u E V (SZ'), Ilullvm+2(sa) S Cl)PuIIvm(sa),

(6.87)

where a constant C depends on £T, P, m. (b) Using part (a), prove that, for any u E C°° (1), Iku11vm+2(ca') < C (JIUJIL2(sa) + IIPUIIVm(o)) .

(6.88)

REMARK. The estimate (6.88) was proved in Theorem 6.21. In the case u E C°°, it is easier to deduce it from (6.87).

6.4.3. Solutions from D. We start with a parabolic analogue of Lemma 6.12.

LEMMA 6.23. Let U C- SZ be an open set of the form U = (0, T) x Q where T > 0 and Q is an open set in Rn. Then, for any f E L2 (U), there exists a function u E L2 (U) solving the equation Pu = f and satisfying the estimate IIUIIL2(U) < T II f/PII L2(u).

(6.89)

REMARK 6.24. As it follows from Corollary 6.22, if f c CO° n L2 (U) then the solution u also belongs to C°° f1 L2 (U). PROOF. By Lemma 6.12 and Remark 6.13, the resolvent R of the equation Lu = f is a compact self-adjoint operator in L2 (Q). The multiplication operator by the coefficient p (x) is a bounded self-adjoint operator in L2 (Q). Therefore, Rop is a compact self-adjoint operator in L2 (Q). By the HilbertSchmidt theorem, there exists an orthonormal basis {Vk} in L2 (Q), which consists of the eigenfunctions of the operator R op. Since 1Z (pv) = 0 implies pv = LO = 0 and, hence, v = 0, zero is not an eigenvalue of R op. Therefore, each Vk is also an eigenfunction of the inverse operator p-1L, and let the corresponding eigenvalue be .Xk, that is, Lvk = akPvk. (6.90) Since ran R is contained in W0 (Q), we have vk E Wo (Q). Using the identity (6.51) with u = cp = Vk and f = .Akpvk, we obtain

I aijaavk ajvk dx = -\k J

JU

pvk dx,

U

whence it follows that .Ak < 0.

Given f E D (U), expand function f /p in the basis {vk}: f (t, x) f,k (t) vk (x) P (x)

L-'

6.4. LOCAL PARABOLIC REGULARITY

179

where fk (t) = (f (t, ) /p, vk), and set Uk

(t) =

Jt

e-Aksfk (s) ds.

(6.91)

We claim that the function u (t, x)

Aktuk (t) Vk (x)

(6.92)

k

belongs to L2 (U) and solves the equation Pu = f. Indeed, since Ak G 0 and

= I t eak(t-s)fk (s) ds,

eaktuk (t)

0

we see that, for any t E (0,T), e,\ktuk

f

(t)

0

T

Ifk (s) I ds,

whence by the Parseval identity z

1: eAktuk (t)

T

<

k

Tf

k

I

(s)12

ds

fT

=TJ

IIf (s,) Ili2(Q)ds = TII f/pll L2(U).

Therefore, the series (6.92) converges in L2 (Q) and, for any t E (0,T), Ilu (t,')

2

112

< TIIf/p11L2(U)'

Integrating in t, we obtain u E L2 (U) and the estimate (6.89). The same argument shows that the series (6.92) converges in L2 (U). Using (6.90) and (6.91) we obtain

P (etuv) = peAkt (atUk) Vk + pAkeAktukvk - eAktukLvk = pfkvk Using the convergence of the series (6.92) in D' (U), we obtain

Pu = E pfkvk = f, k

which finishes the proof.

In the proof of the next statement, we will use the operator

P* :_ -pat - L, which is dual to P in the following sense: for any distribution u e D' (0) and a test function cp E D (a), (Pu, co) = (u, P*(P) (cf. Section 6.4.2). Let T be the operator of changing the time direction, that is, for a test function cp E D (52),

Tcp (t, x) = cp (-t, x)

6. REGULARITY THEORY IN 1Rn

180

and, for a distribution u E D' (a), (Tu, co) _ (u, Tco) for all co E D ('re)

.

Clearly, we have P* = P o T. Using this relation, many properties of the operator P* can be derived from those for P. In particular, one easily verifies that Theorem 6.21 and Lemma 6.23 are valid3 also for the operator P*.

THEOREM 6.25. For any m E Z, if u c D' (1) and Pu E V a (S2) then u E Vo+2 (c).

PROOF. As in the proof of Theorem 6.21, let us first show that u E Vo +' (Q) and Pu E Vo (1k) imply u E Vo +2 (S2). For m > -1, this was proved in Theorem 6.21, so assume k := -m > 2. It suffices to prove that ou E V-k+2 (S2) for any z/ E D (11) with sufficiently small support. Fix a cylindrical open set U C= S2, a function 0 E D (U), and set v = bu. It follows from the hypotheses that V E V-k+l (0) and Pv (z- V-k (S2) (cf. (6.69)).

We need to show that IIvIIv-k+2(u) < oo, and this will be done if we prove that, for any f E D (U), (v, f) < CIIPvIIv-k

IfIIVk-2.

(6.93)

By Lemma 6.23, for any f E D (U), there exists a function w E C°° n L2 (U) solving the equation P*w = f in U and satisfying the estimate IIWIIL2(U) < CII f IIL2(U).

(6.94)

Fix a function co E D (U) such that co - 1 in a neighborhood of supp 0. Then cow E D (U) and

(v, f) _

(v, P*w) = (v, P* (cow)) _ ('PV' WW) IIPvIIv-k(U)lcowllvk(U) <-CIIPvIIv-k(U)IIWIIvk(U/),

where U' C U is a neighborhood of supp co and the constant C depends only on co. Using the estimate (6.85) of Theorem 6.21 (or the estimate (6.88) of Exercise 6.5) and (6.94), we obtain Ilwllvk(U) < c (IIwIL2u) + IIP*WI!vk-2(U)) <- C'IIfIlvk-2(U), whence (6.93) follows.

Assume now that u E D' (12) and Pu E V c (1), and prove that u E V e+2 (fl). As was shown in the proof of Theorem 6.15, for any open set U C 0 there exists l > 0 such that u E W-1 (U). Since II 11W, < II - IIv21 and, hence, II Ilv-21 < II II w-c, this implies u E V-21 (U). Let k < m + 2 be the maximal integer such that u E Vk' (U). If k < m + 1 then Pu E Joe 1(U) .

-

whence by the first part of the proof u E V o 1 (U) . Hence, k = m + 2, which was to be proved. 3Let us emphasize that the solvability result of Lemma 6.23 is not sensitive to the time direction because we do not impose the initial data.

NOTES

181

Combining Theorems 6.25 and 6.1, we obtain the following statement that extends the result of Corollary 6.22(i) from Li to D` (1k) . COROLLARY 6.26. If u E V' (ft) and Pu E C' (S2) then u E C' (0).

Exercises. 6.6. Consider a more general parabolic operator

P=p8t-82(ai,(s)88)-b3(x)8g-c(x), where a'j and p are as before, and bi and c are smooth functions in f2. Prove that if u e D' (f2) and Pu E VC (fl) for some m E Z then u E Vo +2 (0). Conclude that Pu E C°° (f2) implies u E C°° (cl).

Notes All the material of this chapter is classical although the presentation has some novelties. The theory of distributions was created by L. Schwartz [327]. The Sobolev spaces were introduced by S. L. Sobolev in [328], where he also proved the Sobolev embedding theorem. We have presented in Section 6.1.1 only a part of this theorem. The full state2n ment includes the claim that if k < n/2 then W o°y L° 2k , and similar results hold for the spaces Wk'' based on LP. The modern proofs of the Sobolev embedding theorem can be found in [118], [130]; see also [1] and [269] for further results. One of the first historical result in the regularity theory (in the present sense) is due to H. Weyl [357], who proved that any distribution u E D' solving the equation Au = f

is a smooth function provided f E C°° (Weyl's lemma). This and similar results for the elliptic operators with constant coefficient can be verified by means of the Fourier transform (see [309]). The regularity theorem for elliptic operators with smooth variable coefficients was proved by K. O. Friedrichs [123], who introduced for that the techniques of mollifiers in [122]. Alternative approaches were developed concurrently by P. D. Lax [245] and L. Nirenberg [293], [294]. Nowadays various approaches are available for the regularity theory. The one we present here makes a strong use of the symmetry of the operator (via the Green formula) and of the mollifiers. The proofs of the Friedrichs lemma (Lemma 6.4) and the key Lemmas 6.7, 6.8 were taken from [208]. The reader may notice that Lemma 6.4 is the only technical part of the proof. Other frequently used devices include elementary estimates of the commutators of the differential operators with the operators of convolution and multiplication by a function. The parabolic regularity theory as presented here follows closely its elliptic counterpart, with the Sobolev spaces Wk being replaced by their anisotropic version.

Different accounts of the regularity theory can be found in [118], [121], [130], [241],

[241], [273], [308], although in the most sources the theory is restricted to solutions from L1 ° or even from W11.° as opposed to those from V. A far reaching extension and unification of the elliptic and parabolic regularity theories was achieved in L. Hormander's theory of hypoelliptic operators [208] (see also [297], [348]). Another branch of the regularity theory goes in the direction of reducing the smoothness of the coefficients - this theory is covered in [130], [118], [242], [241]. If the coefficients are just measurable functions then all that one can hope for is the Holder continuity of the solutions. The fundamental results in this direction were obtained by E. De Giorgi [103] for the elliptic case and by J. Nash [292] for the parabolic case. For the operators in non-divergence form, the Holder regularity of solutions was proved by N. Krylov and

182

6. REGULARITY THEORY IN Rn

M. Safonov [236], partly based on the work of E. M. Landis [243]. See [130] and [230] for a detailed account of these results.

CHAPTER 7

The heat kernel on a manifold This is a central Chapter of the book, where we prove the existence of the heat kernel and its general properties. From Chapter 6, we use Corollaries 6.11, 6.22, and 6.26.

7.1. Local regularity issues Let (M, g, µ) be a weighted manifold. The only Sobolev space on M we have considered so far was Wr (M). In general, the higher order Sobolev spaces Wk cannot be defined in the same way as in 1R' because the partial derivatives of higher order are not well-defined on M. Using the Laplace operator, we still can define the spaces of even orders as follows. For any non-negative integer k, set W2k (M) = Wax

(M, g, t1) _ {'u : u, A ,u, ..., 0µu E L2 (M) }

and k

11AI U112

Ilullw2k =

(7.1)

L2.

1=0

It is easy to check that W2k (M) with the norm (7.1) is a Hilbert spacer. Define the local Sobolev space Wi k (M) by Wi C (M) _ {u : u, OP u, ..., 0µu E LiC (M)}

.

Equivalently, u E W o, (M) if u E W2k (f2) for any open set Q C= M. The topology in W oc (M) is determined by the family of seminorms IIuIIw2k(a). The following theorem is a consequence of the elliptic regularity theory of Section 6.3.1. THEOREM 7.1. Let (M, g, µ) be a weighted manifold of dimension n, and let u' be a function from )/Vi c (M) for some positive integer k. 'By considering in addition the gradient of 0µu, one could define W2k+1 (M) similarly to W1 (M), but we have no need in such space (cf. Exercise 7.1). The reader should be warned that if M is an open subset of R then W2k (0) need not match the Euclidean Sobolev space W2k (Q), although these two spaces do coincide if M = R' (cf. Exercise 2.33(d)). 183

7. THE HEAT KERNEL ON A MANIFOLD

184

(i) If k > n/4 then u E C (M). Moreover, for any relatively compact

open set 52 C M and any set K C 12, there is a constant C = C(K, 52, g, y, k, n) such that (7.2)

sup Jul < CIIujIw2k(Q). K

(ii) If k > m/2 + n/4 where m is a positive integer then u E Cm (M). Moreover, for any relatively compact chart U C M and any set K C= U, there is a constant C = C(K, U, g, p, k, n, m) such that II uIl C"(x) <_

CIIUIIw2k(U).

(7.3)

PROOF. Let U be a chart with coordinates x1, ..., xn, and let A be the Lebesgue measure in U. Recall that by (3.21) dp = p (x) dA, where p = T det g and T is the density function of measure I.L. Considering U as a part of Rn, define in U the following operator L = P-10i (Pg2'8j) .

(7.4)

Lcp = Dµco for all co E D (U) .

(7.5)

By (3.45), we have

Now let us consider the operators L and A. in D' (U). Since we will apply the results of Chapter 6 to the operator L, we need to treat it as an operator in a domain of Rn. Hence, we define L on D' (U) using the definitions of 8ti and the multiplication by a function in D' (U) given in Section 2.4 (cf. Section 6.3).

However, we treat Dµ as an operator on M, and Dµ extends to D' (U) by means of the identity (4.3). Then L and A. are not necessarily equal as operators on D' (U) because their definitions as operators in D' (U) depend on the reference measures, which in the case of Dµ is g and in the case of L is A. Indeed, for any u E D' (U) and co E D (U), we have (p-'az(Pg''8j),W) = (8z(Pg2'8;),p-1W)

- (pgiiaj, ai(P-1S°)) _ (u, aj(Pgziai (P-1cP))) (u, PL (P-1W))

(7.6)

whereas

Aµu, 0) = (u, A/M

(7.7)

Obviously, we have in general Lu A.U. Nevertheless, when the distributions ON,u and Lu are identified with L210C (or L oc) functions, the reference measures are used again and cancel, which leads back to the equality Lu = Aµu. More precisely, the following it true2.

CLAIM. If u c Llo (U) and 0µu E Liar (U) then Lu = 0µu in U. 2Compare this to Exercise 4.11, where a similar identity is proved for the weak gradient.

7.1. LOCAL REGULARITY ISSUES

185

As follows from (7.6), if u E L2 , (U) then, for any cp E D (U)

(Lu, gyp) = fuPL

(p)

dA.

To prove the claim it suffices to show that

fu Aµu dA

= fu u p L

(p-1W) dA.

Since both u and O,Lu are in Ll (M), the identity (7.7) becomes

fudkt = fud.

Using this identity and (7.5), we obtain

f uL(p 1c )pdA,

JLucoda= f.pu (p lip) dµ = f u Ap(p U

U

U

U

which was to be proved. The hypothesis u E 11V1 10k

and the above claim imply that, in any

chart U,

u, Lu, ..., Lku E L

C

(U)

.

If k > m/2 + n/4 then we conclude by Corollary 6.11 that u E C"2 (U) and, hence, u c C72 (M). The estimate (7.3) follows immediately from the estimate (6.49) of Corollary 6.11 and the definition (7.1) of the norm Ilullw2k. To prove (7.2) observe that there exist two finite families {Uz} and {Uz} of relatively compact charts such that K is covered by the charts U and V C Uj C= Q (cf. Lemma 3.4). Applying the estimate (6.49) of Corollary 6.11 in each chart UZ for the operator L = AA and replacing L2 (Ui, A)-norm by L2 (U2 , µ)-norm (which are comparable), we obtain sup Jul < C Vi

IlA' UllL2(Ui,),) < C' i IloullL2(c2,,) 1=0

1=0

Finally, taking maximum over all i, we obtain (7.2). If we define the topology in Ctm (M) by means of the family of seminorms llUllCm(K) where K is a compact subset of a chart then Theorem 7.1 can be shorty stated that we have an embedding Wloc (M)

Cm (M)

provided k > m/2 + n/4. Let us introduce the topology in C°° (M) by means of the family of seminorms

sup l Gaul , K

(7.8)

7. THE HEAT KERNEL ON A MANIFOLD

186

where K is any compact set that is contained in a chart, and a« is an arbitrary partial derivative in this chart. The convergence in this topology, denoted by C°°

Vk ---3 V,

means that vk converges to v locally uniformly as k --3 oo and, in any chart and for any multiindex a, aavk converges to a°v locally uniformly, too. Denote by WIT (M) the intersection of all spaces Wi 1 (M), and define the topology in Wl (M) by means of the family of seminorms IIufjw21(O),

(7.9)

where l is any positive integer and St is a relatively compact open subset of M. The convergence in WWo° (M), denoted by Vk -4 v,

means that Vk converges to v in L10C (M) and, for any positive integer 1, Duvk converges to A' v in L oC (M). COROLLARY 7.2. The natural identity mapping

I : C°° (M) - Wf (M) (7.10) is a homeomorphism of the topological spaces C°° (M) and Wlo° (M).

PROOF. If f E C°° then I (f) is the same function f considered as an element of L,,,. Clearly, I (f) E Wlo° so that the mapping (7.10) is welldefined. The injectivity of I is obvious, the surjectivity follows from Theorem 7.1(ii) . The inequality (7.3) means that any seminorm in C°° is bounded by a seminorm in )4) . Hence, the inverse mapping I-1 is continuous. Any seminorm (7.9) in W,',, can be bounded by a finite sum of seminorms (7.8) in C°°, which can be seen by covering Il by a finite family of relatively compact charts. Hence, I is continuous, and hence, is a homeomorphism. El

It is tempting to say that the spaces C°° and WIT are identical. However,

this is not quite so because the elements of CO° are pointwise functions whereas the element of Wic.* are equivalence classes of measurable functions.

COROLLARY 7.3. If a function u E Li ,(M) satisfies in M the equation -D,u + an = f where a E lR and f E C°°(M), then u E COO (M). More precisely, the statement of Corollary 7.3 means that there is a C°° smooth version of a measurable function u.

PROOF. By Corollary 7.2, it suffices to prove that 0µu E L for all k = 1, 2,.... It is obvious that an - f c L oC and, hence,

AAu = an - f E L°. Then we have

OAu = a0Au - Du f E L °.

7.1. LOCAL REGULARITY ISSUES

187

Continuing by induction, we obtain

0µu = aDµ-lu -

Ak-1 f

E Ll c,

which finishes the proof.

Now we consider the consequences of the parabolic regularity theory of Section 6.4.3. Fix an open interval I C R and consider the product manifold N = I x M with the measure dv = dt d1t. The time derivative at is defined on D' (N) as follows: for all u E D' (N) and cp E D (N),

- (u, ate) , and the Laplace operator Aµ of M extends to D' (N) by (atu, W) =

(Dµu, cP) = (u, AIM

Hence, the heat operator at - A. is naturally defined on D' (N) as follows: (7.11)

((at-A,)u,(P) =-(u,8tcp+Dµcp).

THEOREM 7.4. Let N = I x M. (i) If u E D' (N) and Btu - Dµu E C°° (N) then u E CD° (N). (ii) Let {uk} be a sequence of smooth functions on N, each satisfying the same equation

atuk - Aµuk = f, where f E C°° (N). If Uk

L`°)uEL2loc (N)

then (a version of) function u is C°°-smooth in N, satisfies the equation

atu-Aµu=f, and Uk

C-

U.

PROOF. (i) As in the proof of Theorem 7.1, let U be a chart on M with coordinates x1, ..., xn, and A be the Lebesgue measure in U. Then we have dp = p (x) dA, where p (x) is a smooth positive function in U, and the Laplace operator A. on D (U) has the form Dµ = P-laz (P9ziaj) = P-1L, where

L=ai(pg2iaj) Note that U := I _x U is a chart on N. Using the definition of the operators Dµ and bj in D'(U), we obtain, for all u E D(U) and cp E D(U), (0µu, P)

_ (u, DµcP) = (u, P-lai (P9ti'a,(p)) =

- (az

(P-lu)

,

(p-1u,

ai (P9iia;w))

P92iajc,) _ (a, (P92iaj (p -'u)) cP) = (Lv, V), ,

7. THE HEAT KERNEL ON A MANIFOLD

188

where v = p-lu E D'(U). Hence, (atu - 0µu, (P) = (pats - Lv,
so that we have the identity

Btu-0µu=paty-Lv. The hypothesis atu - 0µu E C°° (N) implies pats - Lv E C°O(U), and Corollary 6.26 yields v E C°O(U). Hence, we conclude u E C°°(U), which finishes the proof. (ii) It follows from (7.11) that if {uk} is a sequence of distributions on N each satisfying the same equation atuk - Dµuk = f and uk -+ u then u also

satisfies atu - 0µu = f. In particular, this is the case when Uk u as we have now. By part (i), we conclude that u E C' (N), and the convergence Uk -- u follows from Corollary 6.22. LL *°

COROLLARY 7.5. The statement of Corollary 7.3 remains true if the hypothesis u E L a'(M) is relaxed to u E Li c (M).

PROOF. Indeed, consider the function v (t, x) = eatu (x), which obviously belongs to L10 (N) where N = R x M. In particular, v E D' (N). We have then atv - zµv = aeatu - eat&µu = eat f. Since eat f E C°° (N), we conclude by Theorem 7.4 that v E C°° (N), whence

uECO°(M). Exercises. For any real s > 0, define the space Wo (M) as a subspace of L2 (M) by )/V' (M) = dom (L + id)112

where L is the Dirichlet Laplace operator. The norm in this space is defined by IIfl1w8 := II (C+id)312 fIIL2.

7.1. Prove that Wo is a Hilbert space. 7.2. Prove that Wo = Wo and Wo = Wo including the equivalence (but not necessarily the identity) of the norms. 7.3. Prove that if k is a positive integer then f E Wok if and only if f, L f, ..., Lk-a f E W0 (M) and Ckf E L2 (M) . 7.4. Prove that Wok C W2k and that the norms in

W02k and W2k

(7.12)

are equivalent.

7.5. Prove that if f E Wok then, for all integer 0 < l < k, II.`fIIL2 _< IIfIIL2-`'1kilLkfllL2

(7.13)

7.6. Let M be a connected weighted manifold. Prove that if f c L2 10, (M) and V f = 0 on M then f = const on M.

7.1. LOCAL REGULARITY ISSUES

189

7.7. Let M be a connected manifold and 0 be an open subset of M such that and M \ S2 is non-empty. Prove that In 0 W1 (M) and In 0 Wo (Q). REMARK. If in addition u(S2) < oc then clearly lo E L2 (12) and V10 = 0 in 0 whence In E W1 (2). In this case we obtain an example of a function that is in W' (S2) but not in Wo (52).

7.8. (The exterior maximum principle)Let M be a connected weighted manifold and 0 be a non-empty open subset of M such that M \ ) is non-empty. Let u be a function from

C (M) n Wo (M) such that 0µu = 0 in Q. Prove that sup u = sup U. asa

Q

Prove that if in addition S2 is the exterior of a compact set, then the hypothesis u E C (M) n Wo (M) can be relaxed to u E C (S2) n Wo (M).

7.9. Assume that u E L210° (M) and 0µu E Li ° (M). Prove that u E WW'. (M) and, moreover, for any couple of open sets S2' C= S2" C M, (7.14)

1IUIIWl(sz') < C (IIuIIL2(ni,) + 1AAUIIL2(n/'))

where the constant C depends on S2', S2", g, p, n. The space Wio° (M) is defined in Exercise 5.8 by (5.15).

7.10. Prove that if u E D' (M) and 0µu E C°° (M) then u E C°° (M) . 7.11. A function u on a weighted manifold M is called harmonic if u c C°° (M) and 0µu = 0. Prove that if {uk}k 1 is a sequence of harmonic functions such that uk

L2

u e LCoc (M)

then (a version of) u is also harmonic. Moreover, prove that, in fact, uk

U.

7.12. Let {uk} be a sequence of functions from La ° (M) such that

-Dµuk + akuk = fk,

(7.15)

for some ak E lI and fk E W,2,, (M), with a fixed non-negative integer m. Assume further

that, as k -4 oo, 1/V2_

ak -+ a, fk _ f Prove that function u satisfies the equation

L2 and uk -4 u.

-0µu+au= f, and that

(7.16)

w2-+2 Uk

U.

c

(7.17)

Prove that if in addition fk E C°° (M) and fk -+ f then (versions of) uk and u belong to C°° (M) and uk U. 7.13. Let {uk} be a sequence of non-negative functions from C°° (M), which satisfy (7.15)

with ak E R and fk E C°° (M). Assume further that, as k -4 oo,

ak -4 a, fk 274 f and Uk (x) fi u (x) for any x c M, where u (x) is a function from L210° that is defined pointwise. Prove that u E C°° (M) and c°°

uk -+ U.

7.14. Prove that, for any relatively compact open set S2 (Z M, for any set K C= S2, and for sulp any a E 1R, there exists a constant C = C (K, S2, a) such that, for any smooth solution to

the equation -0µu + au = 0 on M,

IuI <- C1IuJIL2(SZ).

7. THE HEAT KERNEL ON A MANIFOLD

190

7.15. Let Ra be the resolvent operator defined in Section 4.2, that is, Ra = (L + aid) where a > 0. Prove that if f E L2 n COO (M) then also Ra f E L2 n C°° (M). 7.16. Let {f ,} be an exhaustion sequence in M. Prove that, for any non-negative function

f EL2nC°°(M) and any a> 0, Rni f

HINT. Use that R°i f -

2

Ra f as i --> oo.

Raf (cf. Theorem 5.22).

7.2. Smoothness of the semigroup solutions The next theorem is a key technical result, which will have may important consequences.

THEOREM 7.6. For any f e L2 (M) and t > 0, the function Pt f belongs to C°° (M). Moreover, for any set K C= M, the following inequality holds sup I Pt f I <_ FK (t) 11f 42(M),

(7.18)

K

where

FK (t) = C (1 + t-o) , (7.19) Q is the smallest integer larger than n/4, and C is a constant depending on

K,g,p,n. Furthermore, for any chart U C= M, a set K C= U, and a positive integer m, we have IIPtfilcm(K) <_ FK (t) IIfIIL2(M),

(7.20)

where FK (t) is still given by (7.19) but now a is the smallest integer larger than m/2 + n/4, and C = C(K, U, g, A, n, m). The estimate (7.18) is true also with a = n/4, which is the best possible exponent in (7.19) (cf. Corollary 15.7). However, for our immediate applications, the value of o- is unimportant. Moreover, we will only use the fact that the function FK (t) in (7.18) and (7.20) is finite and locally bounded in t E (0, +co). PROOF. Let {EA} be the spectral resolution of the operator L = -OJ,Iyl o Xke-tA, where t > 0 and k is a in L2 (M). Consider the function (A) = positive integer. Observe that by (4.50) Lke-tc =

45 (G) = f Ake-tadEa.

(7.21)

0

Since the function (A) is bounded on [0, +oo), the operator 4) (G) is bounded and so is L e-t,c. Hence, for any f E L2 (M), we have

Lk (e-tL f) EL 2 (M),

that is,

0 (Ptf) E L2 (M)

7.2. SMOOTHNESS OF THE SEMIGROUP SOLUTIONS

191

Since this is true for any k, we obtain Pt f E W°° (M). By Theorem 7.1 (or Corollary 7.2) we conclude that Ptf E C°° (M). Let us prove the estimates (7.18) and (7.20). Observe that the function A

,\ke-ta takes its maximal value at A = k/t, which implies, for any

f E L2, A II

Ilike-tCf

APtf 11L2

11L2

1/2 ( e-ta12 1 dllEa.f II il I l Uo':*

< sup (Ake_tx) ( \f

0

dEaf

2)

(tk lk

l

1/2

(7.22)

e-k11f1lL2.

Using the definition (7.1) of the norm in W2' and (7.22), we obtain, for any positive integer (7, v

IIPtfIIw2Q =

IIakPtfIIL2 k=0

k < C 1+E (t)

k

e-k

< C'(1+t-a)IIfIIL2.

IIf 11L2

(7.23)

By the estimate (7.2) of Theorem 7.1, we have sup IPtf I <_ CII Ptf IIW2a(M), K

provided o- > n/4, which together with (7.23) yields (7.18). In the same way, (7.20) follows from (7.3).

Initially Pt f was defined for as e-tc f, which is an element of L2 (M). By Theorem 7.6, this function has a C°°-version. From now on, let us redefine Ptf to be the smooth version of e-t"f . Now we are in position to prove

that, on any weighted manifold M, the operator Pt possesses an integral kernel.

THEOREM 7.7. For any x E M and for any t > 0, there exists a unique function pt ,x E L2 (M) such that, for all f c L2 (M), Ptf (x) = fMPtX (y) f (y) dµ (y) .

(7.24)

Moreover, for any relatively compact set K C M and for any t > 0, we have sup II pt,x II L2 (M) <_ FK (t) , xEK

(7.25)

where FK (t) is the same function as in the estimate (7.18) of Theorem 7.6.

7. THE HEAT KERNEL ON A MANIFOLD

192

REMARK 7.8. The function pt,x (y) is defined for all t > 0, x E M but for almost all y e M. Later on, it will be regularized to obtain a smooth function of all three variables t, x, y.

PROOF. Fix a relatively compact set K C M. By Theorem 7.6, for all t > 0 and f E L2 (M), the function Ptf (x) is smooth in x E M and admits the estimate

IPtf(x)I 0 and x c K, the mapping f H Ptf (x) is a bounded linear functional on L2 (M). By the Riesz representation theorem, there exists a function pt,x E L2 (M) such that Ptf (x) = (Pt,x, f )L2 for all f c L2 (M),

whence (7.24) follows. The uniqueness of pt,x is clear from (7.24). Since for any point x e M there is a compact set K containing x (for example, K = {x}), the function pt,x is defined for all t > 0 and x E M. Taking in (7.26) f = pt,s and using

Ptf (x) = (f, f)L2 = If IIL2 we obtain IIfIIL2 < FK (t) 11f 11L2,

whence (7.25) follows.

EXAMPLE 7.9. Recall that the heat semigroup in R' is determined by (4.62), which implies that in this case

pt,x (y) = pt (x - y) =

1

Ix_yI2

exp

4t

Using the identity pt * pt = pet (see Example 1.9), we obtain

IIpt22

2

pt

(x_y)dy=(pt*pt)(0)=p2t(0)=

1

(8,t)n/2

whence IIPt,xIIL2 = (8nt)-n/4

In particular, we see that the estimate (7.25) with FK (t) = C (1 + t--) and a > n/4 is almost sharp for small t.

Now we prove that the function Ptf (x) is, in fact, smooth jointly in t, x. Consider the product manifold N = R+ x M with the metric tensor gnr = dt2 + gm and with measure du = dtdp. The Laplace operator A. of (M, t) , which is obviously defined on C°° (N), extends to D' (N) as follows: (& v, 0) = (v, Dµ(P) for all v E D' (N) and co E D (N). The time derivative at is defined in on D' (N) by

(ata`

)

7.2. SMOOTHNESS OF THE SEMIGROUP SOLUTIONS

193

Hence, for a function u E L10C (N), it makes sense to consider the heat equation at = & u as a distributional equation on N.

THEOREM 7.10. For any f c L2 (M), the function u (t, x) = Ptf (x) belongs to C°° (N) and satisfies in N the heat equation at = Au.

More precisely, the statement means that, for any t > 0, there is a pointwise version of the L2 (M)-function u (t, ) such that the function u (t, x) belongs to C°° (N).

PROOF. We already know by Theorem 7.6 that the function u (t, x) is

C°° smooth in x for any fixed t > 0. To prove that u (t, x) is continuous jointly in t, x, it suffices to show that u (t, x) is continuous in t locally uniformly in x. In fact, we will prove that, for any t > 0, (7.27) u (t + e, ) -4 u (t, ) as e - 0, which will settle the joint continuity of u. By Corollary 7.2), to prove (7.27) it suffices to show that, for any non-negative integer k,

u(t+e,.)

(7.28)

We know already from the proof of Theorem 7.6 that, for any non-negative integer m, u (t, ) E dom L'n and, hence, A' u = (-L)' u. Therefore, it suffices to prove that, Lm (Pt+,F7 f) ---Z L' (Ptf)

(7.29)

Since by (7.21) 00

(Pt+Ef) =

Ame-(t+E)adEaf

J0 \,,e-(t+,)A and the function remains uniformly bounded in A as e -+ 0, Lemma 4.8 allows to pass to the limit under the integral sign, which yields (7.29).

Since u (t, x) is continuous jointly in (t, x), it makes sense to consider u as a distribution on N. Let us show that the function u (t, x) satisfies on N the heat equation in the distributional sense, which amounts to the equation

(u, a +

0,

(7.30)

for any cp E D' (N). Using Fubini's theorem, we obtain (u,

a+z uo) = f u(a+Auco)dv N

f

+ (U,

a )L2(M) dt + J + (u, OOP)L2(M) dt.(7.31)

Considering co (t, ) as a path in L2 (M), observe that the classical partial derivative at coincides with the strong derivative - (cf. Exercise 4.47).

7. THE HEAT KERNEL ON A MANIFOLD

194

The product rule for the strong derivative (cf. Exercise 4.46) yields (u, ddt,) d (u, u, (7.32) ) - (dt P)' dt (u'

where all the brackets mean the inner product in L2 (M), Since cp (t, ) vanishes outside some time interval [a, b] where 0 < a < b, we obtain

J

+

dt (u, W) dt = 0.

(7.33)

To handle the last term in (6.66), recall that, by Theorem 4.9, du = OAu'

dt which yields

(du, )dt R+

dt

= f (& u, co)dt = f (u, Atzw)dt. +

(7.34)

+

Combining (7.32), (7.33), and (7.34), we obtain that the right hand side of (7.31) vanishes, which proves (7.30). Applying Theorem 7.4 to function u (t, x), which satisfies the heat equation in the distributional sense, we conclude that u E CO° (N) and u satisfies the heat equation in the classical sense. SECOND PROOF. In this proof, we do not use the parabolic regularity theory (Theorem 7.4). However, we still use the first part of the first proof, namely, the convergence (7.27). Let us fix a chart U C M so that we can consider the partial derivatives as with respect

to x in this chart. By Theorem 7.6, a'u is C°°-smooth in x. By (7.27), we have -C-74 a"U (t, -)

as E-}0,

which implies that a"u is jointly continuous in t, x. To handle the time derivative atu, let us first prove that, for any t > 0,

u(t+E, -v,

9-!

ase

(7.35)

By Corollary 7.2, it suffices to prove that, for any non-negative integer k,

-Lu and this, in turn, will follows from Lm a (t + E, ) - u (t, ')

L2

E

+ -Lm+lu

(7.36)

provided (7.36) holds for all non-negative integers rn,. It follows from (7.21) that Lm

u (t + E, ) - u (t, ) = /'°° .,m. E

Jo

E

- 1 e-"dEAf

Since the function under the integration remains uniformly bounded in A as e -* 0 (cf the estimate (4.60) from the proof of Theorem 4.9), by Lemma 4.8 we can pass to the limit under the integral sign, which yields (7.36). It follows from (7.35) that atu exists in the classical sense for all t > 0 and x E M, and atu = &,,u. (7.37)

7.2. SMOOTHNESS OF THE SEMIGROUP SOLUTIONS

195

Moreover, (7.35) also yields that, for any partial derivative as in x,

aau (t + r, ) - aau (t, )

aaoA,u ( t ,

E

),

which implies that

at (aau) = aa0',u.

7.38)

In particular, we obtain that at (aau) is continuous in t, x. Observe that the function v = ON,u can be represented in the form v = Pt-sg where g E L2 (M), which follows from the identity -Le-(t-s)Ge-sG Ge-sG f=_e -(t-s)G v = -Lu = f) Therefore, all the above argument works also for function v instead of u and, hence, for Dku instead of u, for any positive integer k. Replacing in (7.38) u by A.u we obtain (aa0A_lu)

at

= as

(AA,u)

(7.39)

Iterating (7.39) for a decreasing sequence of values of k, we obtain

a-Aku = at

(aaAµ-1Zl) l1

= at 1 aa0k-2u) _ ... = ay -1 (a'o,,,u) = at aau.

Hence, we have the identity

at u = Dku, whence applying aa,

aaatu

=aa Ak

Au.

Finally, using the above two identities, any partial derivative at 1 aa1 ai 2 aa2 ....u

can be brought to the form aaAµu and, hence, it exists and is continuous in t, x, which finishes the proof.

THIRD PROOF. Let ' (A) be a continuous function on [0, +oo) of a subexponential growth; that is for any e > 0 I'D (A)l = o (eel as .l -4 +oo. (7.40)

Fix f E L2 (M) and, for any t > 0, consider the function v (t .)

f

7 (A) e-tadEAf,

(7.41)

0

where {E,\} is the spectral resolution of the Dirichlet Laplace operator L on M. We will prove that v (t, x) belongs to C°° (N) and satisfies the heat equation on N (obviously, this contains Theorem 7.10 as a particular case for 4) = 1). Fix the numbers 0 < a < b and consider the open set Na,b C N defined by Na.,b = (a, b) x M.

LEMMA 7.11. For any t E (a, b), function v (t, .) can be modified' on a subset of pmeasure 0 of M so that v (t, x) E L2 (Na,b). Furthermore, the weak derivatives at and A,,v exist in L2 (Na,b) and satisfy the identity (A) e-t)dEaf. av = Q,v = - f°° A

(7.42)

J0

3Such a modification is necessary because there are non-measurable subsets of N that have ,u-measure 0 for any fixed t.

7. THE HEAT KERNEL ON A MANIFOLD

196

PROOF. By (7.40), the function (b (A) e-t ' is bounded for any t > 0, which implies that the right hand side of (7.41) is defined for all f E L2 (M) and determines a function from L2 (M). As in the proof of Theorem 4.9, one shows that the mapping t H v (t, ) as a path in L2 (M) is strongly differentiable and satisfies the equation dv

(t, ) = -Gv (t, ) _ - J

Ab (A)

(7.43)

0

(cf. Exercise 4.51).

Consequently, the path t H v2 (t, .) is continuous in L' (M). By Exercise 4.49, the function v (t, ) can be modified for any t E (a, b) on a set of u-measure 0 in M so that v2 (t, x) E L' (Na,b). Hence, v (t, x) E L2 (Na.b). Since the function .qD (A) also satisfies the condition (7.40), we conclude by the above

argument that dt (t, x) E L2 (Na,b). Let us show that the distributional derivative at coincides with the strong derivative, that is, 8v

dv

(7.44)

8t= at

Indeed, applying the product rule of the strong derivative (see Exercise 4.46), we

obtain,

for any cP E Co (Na,b) d dt (v) 0)L2(M) Since

_

v,

dW + dt L2(M)

dv)L2(M) 1

( dt

i

fb (v, )L2(M) = 0,

it follows that

fb( d

fb (dv

dt ,

Since d =

e(cf.

t

P)L2(M)dt =v, d t )L2(M)dt.

Ja Exercise 4.47), we conclude that

= -(v,

dt , W )L2(N)

)L2(N),

which proves (7.44).

Let us prove that

Dµv = -CV. (7.45) By definition of L, for any fixed t > 0, 0µv (t, ) as a distribution on M coincides with -,Cv (t, ), which implies -(Gv,W)L2(N)

=-

jb (

v(t,')

(t,))L2(M)dt= f (v(t,),(t,'))dt a

f

(v, Al-P)L2(N),

a

whence (7.45) follows. Combining (7.43), (7.44), and (7.45), we obtain (7.42).

Let o be the (distributional) Laplace operator on the manifold N, that is, 2

8t2

+D

By Theorem 7.1, in order to prove that v E C°° (Nab), it suffices to show that 1kv E L2 (Na,b) for all k > 1. Since the function A) (A) also satisfies condition (7.40), Lemma 7.11 applies to function (A) e_tadEAf 8v = - f°°

at

J

A

7.2. SMOOTHNESS OF THE SEMIGROUP SOLUTIONS a2

and yields that the weak derivative

197

' exists in L2 (Na,b) and

a2v = r°° A2,D C7t2

(A)

0

Since ate and A ,v belong to L2 (Na,b), we obtain that also Av E L2 (Na.,b) and

av = f (A2 - A)

(A)

e-tadEAf

0

Applying the same argument to the function (A2 - A) (D (A) instead of (D (A) and then continuing by induction, we obtain, that for all integers k > 1, Okv = f °° (A2

- A)k (D (A) e

tadEAf,

0

and, hence, ']jkv E L2 (Na,b). We conclude that v E C°° (Na b) and the equation at = 0µv (which follows from (7.42)) is satisfied in the classical sense.

Exercises. 7.17. Prove that, for any compact set K C M, for any f E L2 (M, µ), and for any positive integer m, sup IA (Pt.f)I

Ct-"" (1 I

K

t Q) Jjf 112,

(7.46)

where a is the smallest integer larger than n/4. 7.18. Let f be a non-negative function from L2 (M) and {Il } be an exhaustion sequence

in M. Prove that

-'

Pt f S2;

Ptf asi -+oo.

HINT. Use the fact that, for any t > 0,

Ptntf -+Ptf asz -foo (cf. Theorem 5.23).

7.19. Prove that if f E Co (M) then Pt f

C-b fast -* 0.

7.20. Consider the cos-wave operator

Ct = cos (tL'12

(cf. Exercise 4.52). Prove that, for any f e Co (M) , the function 'u (t, x) = Ct.f (x)

belongs to COO (H x M) and solves in H x M the wave equation a2U C7t2

Dµu

with the initial conditions

u (0, x) =f (x) and at (0, x) = 0.

7. THE HEAT KERNEL ON A MANIFOLD

198

7.3. The heat kernel By Theorem 7.7, for any x E M and t > 0, there exists a function pt,x E L2 (M, p) such that, for all f E L2 (M, µ), Ptf (x) = (Pt,x, f )L2

.

(7.47)

Note that the function pt,x (y) is defined for all x but for almost all y. Here we construct a regularized version of pt,x (y), which will be defined for all y. Namely, for any t > 0 and all x, y E M, set Pt (x, y)

(Pt/2,x, Pt/2,y) L2

(7.48)

DEFINITION 7.12. The function pt (x, y) is called the heat kernel of the weighted manifold (M, g, /2).

The main properties of pt (x, y) are stated in the following theorem. THEOREM 7.13. On any weighted manifold (M, g, p) the heat kernel satisfies the following properties.

Symmetry: pt (x, y) = pt (y, x) for all x, y E M and t > 0. For any f E L2, and for all x E M and t > 0, Ptf (x) = im Pt (x, y) f (y) dlt (y) .

(7.49)

pt (x, y) > 0 for allfM x, y E M and t > 0, and Pt (x, y) dl-t (y)

1,

(7.50)

for all xEM andt>0. The semigroup identity: for all x, y E M and t, s > 0, Pt+s (x, y) = Im Pt (x, z) PS (z, y) dµ (z) .

(7.51)

For any y E M, the function u (t, x) := pt (x, y) is C°° smooth in (0, +oo) x M and satisfies the heat equation 8u _&u (7.52) 8t For any function f E Co (M), fM Pt (x, y) f (y) dµ (y) -+ f (x) as t -4 0,

(7.53)

where the convergence is in Coo (M). REMARK 7.14. Obviously, a function pt (x, y), which satisfies (7.49) and

is continuous in y for any fixed t, x, is unique. As we will see below in Theorem 7.20, the function pt (x, y) is, in fact, CO6 smooth jointly in t, x, y. Note also that pt (x, y) > 0 provided manifold M is connected (see Corollary 8.12).

7.3. THE HEAT KERNEL

199

PROOF. Everywhere in the proof, stands for the inner product in L2 (M). The symmetry of pt (x, y) is obvious from definition (7.48). The latter also implies pt (x, y) > 0 provided we show that pt,x > 0 a.e.. Indeed, by Theorems 5.11 and 7.6, Pt f (x) > 0 for all non-negative f E L2 and for all t > 0, x > 0. Setting f = (pt,x)_, we obtain

0 < Ptf (x) = (pt,x,f) = ((pt,x)+, f) - ((Pt,x)_,f) _

(f, f),

whence f = 0 a.e. and pt,x > 0 a.e.

The proof of the rest of Theorem 7.13 will be preceded by two claims. CLAIM 1. For all x E M, t, s > 0, and f E L2 (M),

Pt+sf (x) =

IM

(7.54)

(pt,z,ps,x) f (z) dy (z) .

Indeed, using Pt+s = P3Pt, (7.47), and the symmetry of Pt, we obtain

Pt+sf (x) = Ps (Ptf) (x) (ps,x, Ptf) = (Ptps,x, f )

f MPtps,x(z)f(z)dµ(z) fm (pt,z,ps,x) f (z) dp (z)

,

whence (7.54) follows.

CLAIM 2. For all x, y E M and t > 0, the inner product (ps,x, pt_s,y) does not depend on s E (0, t). Indeed, for all 0 < r < s < t, we have, using (7.47) and applying (7.54) with f = pr,x, (ps,x,pt-8'Y) = Pspt-s,y (x) = Pr (Ps-rpt-s,y) (x)

fM' x (z) (Ps-r,z, pt-s,y) dp (z) Pt-rpr,x (y) (pt-r,y,Pr,x)

which was to be proved. Proof of (7.49). Combining (7.54) and (7.48), we obtain

Ptf (x) = fm (pt/2,x,Pt/2,y) f (y) dlt (y) = fMpt

(x,

y) f (y) dp (y) .

(7.55)

Proof of (7.50). By Theorem 5.11, f < 1 implies Pt f (x) < 1 for all x E M and t > 0. Taking f = 1K where K C M is a compact set, we obtain

whence (7.50) follows.

f

pt,(x,y)dp(y)

< 1,

7. THE HEAT KERNEL ON A MANIFOLD

200

Proof of (7.51). It follows from Claim 2 that, for all x, y E M and

0<s
Pt (x, y) = (ps,x, pt-"Y)

(7.56)

Indeed, (7.56) holds for s = t/2 by definition (7.48), which implies that it holds for all s E (0, t) because the right hand side of (7.56) does not depend on s. Comparison of (7.47) and (7.49) shows that (7.57) pt (x, ) = pt,x a.e. Using (7.56) and (7.57), we obtain, for all x, y E M and t, s > 0,

r.. pt (x, z) ps (z, y) dp (z) = (pt (x, ) ps (y, )) _ (pt,x,ps,y) = pt+s (x) y)

(7.58)

Proof of (7.52). Fix s > 0 and y E M and consider the function v (t, x) Pt+s (x, y). We have by (7.58) v (t, x) _ (Pt,x, ps,y) = Ptps,y (x) .

(7.59)

Since ps,y E L2 (M), Theorem 7.10 yields that the function v (t, x) is smooth

in (t, x) and solves the heat equation. Changing t to t - s, we obtain that the same is true for the function pt (x, y). Proof of (7.53). If f E Co (M) then also Aµ f E Co (M) whence it follows by induction that f E dom £m for any positive integer m, where L is the Dirichlet Laplace operator. By (A.48), this implies that /°°A2mdllEEfll2 <

f

oo.

0

00 ,C- f = o AmdEaf 0

Lmpt f =

f

Ame-tadEAf

0

IIrm (Ptf - f) IIL2 = f A2m (1

- e-ta)2 dilEAfll2.

Since the function .2m (1 - e-t)) 2 is bounded for all t > 0 by the integrable function A2,, and ,\2m (1

- e-tA)

2

--+ 0ast-30,

the dominated convergence theorem implies that IiLm(Ptf-f)IIL2-30ast-40.

We see that Ptf - f -+ 0 in VVW (M), which implies by Corollary 7.2 that Pt f

f,

(7.60)

7.4. EXTENSION OF THE HEAT SEMIGROUP

201

which was to be proved.

Exercises. 7.21. Prove that, for all x, y E M and t > 0, (7.61) Pt (x, y) _ pt (x, x) pt (y, y). 7.22. Prove that, for all x E M, the functions pt (x, x) and EIpt,x 112 are non-increasing in t.

7.23. Let K C M be a compact set. (a) Prove that the function S (t) := sup pt (x, y) x,yEK

is non-increasing in t > 0. (b) Prove that, for all t > 0,

S(t) 0, where C depends on K.

7.24. Let J be an isometry of a weighted manifold M (see Section 3.12). Prove that

pt (Jx, Jy) ° Pt (x, y)

7.4. Extension of the heat semigroup So far the operator Pt has been defined on functions f E L2 so that Ptf E L2 n C°°. Using the identity (7.49), we now extend the definition of Pt as follows: set

Ptf (x)

LA

pt (x, y) f (y) dµ (y) ,

(7.62)

f or any function f such that the right hand side of (7.62) makes sense. In

particular, Ptf (x) will be considered as a function defined pointwise (as opposed to functions defined up to null sets).

7.4.1. Heat semigroup in Li c THEOREM 7.15. If f E Ll° (M) is a non-negative function on M then the function Ptf (x) is measurable in x E M (for any t > 0) and in (t, x) E

R+xM.

If, in addition, Pt f (x) E Ll (I x M) where I is an open interval in

R+, then the function Pt f (x) is C°° smooth on I x M and satisfies the heat equation at(Ptf)=AA(Ptf).

PROOF. Let {SZk} be a compact exhaustion sequence in M, that is, an increasing sequence of relatively compact open set fZk C M such that SZk C Qk+l and the union of all sets S2k is M. Set

fk = min (f, k) In,

202

7. THE HEAT KERNEL ON A MANIFOLD

and observe that functions fk are bounded, compactly supported, and the sequence { fk} is monotone increasing and converges to f a.e.. Since fk E L2 (M), by Theorems 7.10 and 7.13, the function uk (t, x) = Ptfk (x) is smooth in N := I x M and satisfies the heat equation in N. Set also u (t, x) = Pt f (x) and observe that, by (7.62) and the monotone convergence theorem, (7.63) uk (t, x) -+ u (t, x) for all (t, x) E N. Hence, u (t, x) is a measurable function both on M and N (note that so far u may take value oo). If, in addition, u E Li C (N) then u can be considered as a distribution on N. The heat equation for uk implies the identity

a

IN ( at + °j"P

uk dpdt = 0,

uk is unifor all co E D (N). Since the sequence of functions ( + formly bounded on N by the integrable function CIsUPP Pu, where C = sup l atcp + °NCol, we can pass to the limit under the integral sign as k -4 00 and obtain that u satisfies the same identity. Hence, u solves the heat equation in the distributional sense and, by Theorem 7.4, u admits a C°° (N)modification, which we denote by u (t, x). The sequence {uk} is increasing and, by (7.63), converges to ii a.e.. Since u is smooth and, hence, is E Li C (N), we obtain by the dominated (N)

u. By the second part of Theorem 7.4, convergence theorem that uk La°` u. Finally, since Uk -4 u pointwise, we see that we conclude that Uk u (t, x) = ii (t, x) for all (t, x) E N, which finishes the proof.

For applications, Theorem 7.15 should be complemented by the conditions ensuring the finiteness of Pt!. It is also important to understand whether Pt f converges to f as t -- 0 and in what sense. We present in the next subsections some basic results in this direction. 7.4.2. Heat semigroup in Cb. Denote by Cb (M) the class of bounded continuous functions on M. The following result extends Theorem 1.3 to arbitrary weighted manifold. THEOREM 7.16. For any f E Cb (M), the function Ptf (x) is finite for all t > 0 and x E M, and satisfies the estimate

inf f < Pt f (x) < sup f.

(7.64)

Moreover, Pt f (x) is Coo smooth in + x M, satisfies in 1+ x M the heat equation, and lim Pt f (x) = f (x) , (7.65) 19

where the limit is locally uniform in x E M.

7.4. EXTENSION OF THE HEAT SEMIGROUP

203

In particular, we see that Pt f E Cb (M) for any t > 0 so that Pt can considered as an operator in Cb (M). The statement of Theorem 7.16 can be rephrased also as follows: for any f E Cb (M), the function u (t, x) = Pt f (x) is a bounded solution to the Cauchy problem

f

= 0u in II8+ x M,

l ul t=0 understood in the classical sense. The question of uniqueness is quite subtle and will be first addressed in Section 8.4.1.

PROOF. By treating separately f+ and f_, we can assume that f > 0. By (7.50), we obtain, for all t > 0 and x E M, Pt f (x) < sup f

pt (x, y) dµ (y) < sup f ,

(7.66)

fM

which proves the finiteness of Pt f and (7.64). By the first part of Theorem 7.15, Ptf (x) E LO° (IR+ x M), and by the second part of Theorem 7.15, Pt f (x) E C°° (R+ x M) and Pt f satisfies the heat equation. The initial condition (7.65) was proved in Theorem 7.13 for f E Co (M).

Assume next that f E Co (M), where Co (M) is the class of continuous functions with compact supports. Since Co (M) is dense in Co (M) (cf. Exercise 4.5), there exists a sequence { fk} of functions from Co (M) that converges to f uniformly on M. Obviously, we have

Ptf -f =(Ptf -Ptfk)+(Ptfk-fk)+(fk-f) For a given e > 0, choose k large enough so that

sup fk - f I < e.

(7.67)

M

By (7.64) we have, for all t > 0,

suplPt(fk -A < E. M

By the previous step, Ptfk -+ fk as t -a 0 locally uniformly; hence, for any compact set K c M and for small enough t > 0, SuplPtfk - fkI < 6. K

Combining all the previous lines yields sup K

IPtf-fI<3s,

whence the claim follows.

Let now f E Cb (M). Renormalizing f, we can assume 0 < f _< 1. Fix a compact set K c M and a let V) E Co (M) be a cutoff function of K in M, that is, 0 < V < 1 and - 1 on K (cf. Theorem 3.5). Since f b = f on

K, we have the identity

Ptf - f = (Ptf - Pt (f V))) + (Pt (f ,)

- f0) on K.

7. THE HEAT KERNEL ON A MANIFOLD

204

Since

E Co (M), we have by the previous step

supIPt(f b)- fbl -+ 0 ast-40.

(7.68)

K

To estimate the difference Ptf - Pt (f 0), observe that, by 0 < f < 1 and (7.64),

0
By the previous part, we have Pb t-+ b as t -4 0 locally uniformly. Since 1 on K, we obtain that Pb t-* 1 uniformly on K, which implies that

supIPt(.f -fo)I -#0 ast-+0, K

which together with (7.68) implies

supIPtf -# fI -+ ast -+ 0, k

which was to be proved. REMARK 7.17. Consider the function u(t,x)

Ptf(x), t>0,

t=0.

.f (x) ,

It follows from Theorem 7.16 that if f E Cb (M) then u is continuous in [0, +oo) x M. The main difficulty in the proof of Theorem 7.16 was to- ensure that the convergence (7.65) is locally uniform. Just pointwise convergence is much simpler - see Lemma 9.2.

7.4.3. Heat semigroup in L'. Our next goal is to consider Pt f for f E L' (M). We will need for that the following lemma. LEMMA 7.18. Let {v2k} be a double sequence of non-negative functions

from Ll (M) such that, for any k, l

v2k

L* uk E L' (M) as i -4 oo

and 1

uk-4uEL'(M) ask -+ oo. Let {w2} be a sequence of functions from LI (M) such that, for all i, k, vik <wi and I1willL1 < IIuIIL1. 1

Then w24uasi-400. PROOF. All the hypotheses can be displayed in schematic form in the following diagram:

<

Uk + Ll

where all notation are self-explanatory.

w2


u

7.4. EXTENSION OF THE HEAT SEMIGROUP

205

Given e > 0, we have, for large enough k,

IIu-ukIIL1 <E. Fix one of such indices k. Then, for large enough i, we have Iluk

-VikllLl <- 6

so that IIu - vik II Ll < 2e.

Let us show that, for such i, IIu

- wiljL1 < 4e,

(7.69)

which will settle the claim. By condition vik < wi, we have

U-wi
(u - wi)+ < (u - vik)+ and, hence, II (u - wi)+ IILi < 2e.

Next, write

JM (u-wi)d1i = fu>wi} (u-wi)dA+J{u<wi} (u-wi)dµ = II (u - wi)+ IILi - II (u - wi)- IILi. By hypothesis,

fM

(u - wi) du = IIUIIL1 - IIWjIIL1 > 0,

whence it follows that II (u - wi)- IILi < II (u - wi)+ IILi < 2e, and which proves (7.69). and

THEOREM 7.19. For any f E L' (M) and t > 0, we have Pt f E L' (M)

(7.70) IIPtfIIL'
Ptf

Lip)

f ast-*0.

(7.71)

7. THE HEAT KERNEL ON A MANIFOLD

206

PROOF. Without loss of generality, we can assume f _> 0 (otherwise, use

f = f+ - f_). Note that by Theorem 7.15, Pt f (x) is a measurable function of x and of (t, x). Using (7.50), we obtain

f Ptf dp = f 1(1 f

(fM Pt (x, y) f (y) dp (y)/ du (x)

M

M

Pt (x, y)

M

<

M

df.L

(x))

f (Y) dlu (y)

f(y)dp(y)=IIfIILI,

which implies Pt f E Li (M) and the estimate (7.70). Integrating the latter in dt, we obtain that Ptf (x) E L1 (R+ x M). By Theorem 7.15, we conclude that Ptf E C°° (R+ x M) and Pt f satisfies the heat equation. Let us now prove the initial condition (7.71). Let {Slk} be a compact exhaustion sequence in M. Set C

fk=min(f,k)1Ok and observe that fk E L2 (SZk), which implies by Theorem 4.9 that

tkk f L4 ) A

Pt

as t -4 0.

Since A (1 k) < oo and, hence, L2 (SZk) y L' (SZk), we obtain also

Ptkfk

L1SZ

(4k) fk

ast-4 0.

Extending function Pnk fk (x) to M by setting it to 0 outside 12k, we obtain Pflk fk

L1) fk

ast-f0.

Obviously, fk L1) f as k -4 oo, so that we have the diagram Ptok fk

ILl

A

Ptf

<


f

1

-L4 1

and conclude by Lemma 7.18 that Ptf -+ f.

11

Exercises. 7.25. Prove that, for any two non-negative measurable functions f and g on M,

(Pt (fg))2 < Pt (f2) Pt (g2). Prove that

(Ptf)2 < Pt (f2). 7.26. Prove that the following dichotomy takes place: either sup Pt 1 = 1 for all t > 0 or there is c > 0 such that sup Pt 1 < exp (-ct) for all large enough t.

7.4. EXTENSION OF THE HEAT SEMIGROUP

207

7.27. Prove that, for any fixed t > 0 and x c M, the heat kernel pt (x, y) is a bounded function of y E M.

7.28. Let F be a set of functions on M such that f E F implies If I E F and Pt f E F. (a) Prove that the semigroup identity PtPs = Pt+s

holds in F. (b) Assume in addition that F is a normed linear space such that, for any f E .F, IIPtfII.F < IIfII.F and

IIPtf-flIF-0as t-0. Prove that, for any s > 0,

as ters. 7.29. Let f E WWo, (M) be a non-negative function such that At,, f < 0 in the distributional

sense. Prove that Pt f < f for all t > 0. 7.30. Let f E L10C (M) be a non-negative function such that Ptf f for all t > 0. (a) Prove that Ptf (x) is decreasing in t for any x c M. (b) Prove that Pt f is a smooth solution to the heat equation in IR+ x M. (c) Prove that Pt f L-`° f as t -+ 0. (d) Prove that A, f < 0 in the distributional sense.

7.31. Under the conditions of Exercise 7.30, assume in addition that A, f = 0 in an open set U C M. Prove that the function

u (t x) =

Pt f (x) ,

f( ),

t>0'

t<0,

is C°° smooth in R x U and solves the heat equation in R x U. REMARK. The assumption Pt f < f simplifies the proof but is not essential - cf. Exercise 9.8(c).

7.32. Let f E L10c (M) be a non-negative function such that Pt f E Li°, (M) for all t E (0,T) (where T > 0) and Pt f > f for all t E (0,T). (a) Prove that Pt f (x) is increasing in t for any x E M. (b) Prove that Pt f is a smooth solution to the heat equation in (0, T) x M. (c) Prove that Pt f L=am f as t

0.

(d) Prove that A ,f > 0 in the distributional sense. (e) Show that the function f (x) = exp (L4) in R- satisfies the above conditions.

7.33. Let f c L°° (M). Prove that Ptf E L°° (M) for any t > 0, IIPtfIIL- < IfIIL00, and the function u (t, x) = Pt f (x) is C°° smooth in I[8+ x M and satisfies the heat equation.

7.34. Let 0 C M be an open set, and consider the function 01, XEQ f (x) = in (x) { x C M \ 52. ,

Prove that

lim Pt f (x) = f (x) for all x E M \ 852, t--ro

and the convergence is locally uniform in x.

(7.72)

7. THE HEAT KERNEL ON A MANIFOLD

208

7.35. Prove that if a function f E L°° (M) is continuous at a point x E M then Pt f (x) -3 f (x) as t -* 0.

(7.73)

7.36. Let I < r < oo and f E L' (M). (a) Prove that Ptf E L' (M) for any t > 0, and (7.74)

IIfIILr-

(b) Prove that Ptf (x) is a smooth function of (t, x) E R+ x M and satisfies the heat equation.

7.37. Prove that if 1 < r < oo and f E L' (M) then Pt f

f as t -4 0.

7.38. Assume that

F (t) := sup pt (x, x) < 00. xEM

Prove that, for all 1 < r < s < +oo, f E L'` (M) implies Pt f E Ls (M) and

IIPtfIIL' < F

(t)lir-"S IIfIILr.

(7.75)

7.5. Smoothness of the heat kernel in t, x, y In this section, we prove the smoothness of the heat kernel pt (x, y) jointly in t, x, y.

THEOREM 7.20. The heat kernel pt (x, y) is C'-smooth jointly in t > 0

and x, y E M. Furthermore, for any chart U C M and for any partial differential operator Da in t c- R+ and x E U, Dapt (x, ) E L2 (M)

(7.76)

and, for any f E L2 (M),

DaPtf (x) =

IM

-Dapt (x, y) f (y) dµ (y)

(7.77)

PROOF. Fix a relatively compact chart U C M, and let X be a closed

ball in U. We. will assume that x varies in X and denote by as partial derivatives in x in chart U. Recall that, by Theorem 7.13, aapt (x, y) is a smooth function in t, x for any fixed y. Let us first prove the following claim, which constitutes the main part of the proof. CLAIM. Function aapt (x, y) is continuous in x locally uniformly in t, y. By Theorem 7.6, we have, for any f c L2 (M) and any multiindex a, sup 1a0Ptf 1 X

FX,lal (t) if 11L2

(7.78)

where Fx,k (t) is a locally bounded function of t E R+. Since (7.78) can be also applied to the derivatives 9jaa, it follows that, for all x, x' E X,

-

I aaPtf (x) a0Ptf (x) I c FX,101+1 (t) 1!f !IL2 IX - x'I , where Ix - x'J is the Euclidean distance computed in the chart U.

(7.79)

7.5. SMOOTHNESS OF THE HEAT KERNEL IN t, x, y

209

For all t, s > 0, y E M, and x, x' E X, we have by (7.56) pt+s (x) y) - Pt+s W, Y) = Ptps,y (x) - Ptps,y W) which implies by (7.79). Rapt+s (x, y) - 8apt+s (x', y) I C FX,I aI +l (t) II ps,y 11L2 IX - X' J . Restricting y to a compact set Y C M and applying the inequality (7.25) of Theorem 7.7 to estimate I IPs,y I I L2 , we obtain I aapt+s (x, y)

- 8apt+s (x', y) I

FX,lal+1 (t) Fy (s) I x - x'I

,

(7.80)

for all x, x' E X and y E Y. Hence, aapt+s (x, y) is continuous in x locally uniformly in t, y. Since s > 0 is arbitrary, the same holds for aapt (x, y), which was claimed. By Theorem 7.13, pt (x, y) is a continuous function in t, y for a fixed x.

By the above Claim, pt (x, y) is continuous in x locally uniformly in t, y, which implies that pt (x, y) is continuous jointly in t, x, y. Denote by Ax the operator 0,, with respect to the variable x. It follows from the above Claim that Oxpt (x, y) is continuous in x locally uniformly in t, y. Since by Theorem 7.13 Oxpt (x, y) =apt (x, y) = Dypt (x, y)

(7.81)

and AYPt (x) y) is continuous in t, y, we conclude that all three functions in (7.81) are continuous jointly in t, x, V. Now consider the manifold N = M x M with the product metric tensor and the product measure dv = djt dlt. Since pt (x, y) and its derivatives

(7.81) are continuous functions on R+ x N, all these derivatives are also the distributional derivatives of pt (x, y) on 118+ x N. Hence, we have the following equation 8

1

= 2 (Ox + Dy) pt,

atpt which is satisfied in the distributional sense in R+ x N. Since

Ox + AY = 0v, where Ov is the Laplace operator on (N, v), the function pt (x) y) satisfies the heat equation on R+ x N (up to the time change t H 2t). By Theorem 7.4, we conclude that pt (x, y) is CO° smooth on R+ x N, which was to be proved.

Let Da be any partial derivative in t and x. By the previous part of the proof, Dapt (x, y) is a smooth function in t, x, y, which implies that, for any

f E Co (M)

Da f pt (x, y) f (y) dp (y) = J m

Dapt (x, y) f (y) dp (y) ,

(7.82)

because the the function pt (x, y) f (y) is Coo-smooth in t, x, y and the range of t, x, y can be restricted to a compact set.

210

7. THE HEAT KERNEL ON A MANIFOLD

Observe that the estimate (7.78) holds also for the derivative D« in place of a«, because by (7.81) the time derivative operator at on pt (x, y) can be replaced by Ox and, hence, by a combination of operators a«. Then (7.78) implies that, for fixed t, x, the left hand side of (7.82) is a bounded linear functional on f E L2 (M). By the Riesz representation theorem, there exists a function ht,x E L2 (M) such that this functional has the form (ht,x, f). It follows from (7.82) that, for all f E C0 (M), fM

Dpt (x, y) f (y) dµ (y) = (ht,x, f

By Lemma 3.13, we conclude that

D pt x, = ht,x a.e., whence (7.83) D«pt (x, ) E L2 (M). Finally, to prove the identity (7.82) for all f E L2 (M), observe that, by (7.83), the right hand side of (7.82) is also a bounded linear functional on f E L2 (M). Hence, the identity (7.82) extends by continuity from C0 (M) to L2 (M) (cf. Exercise 4.4), which finishes the proof. In what follows we will give an alternative proof of Theorem 7.20, without the parabolic regularity theory. As we will see, the joint smoothness of the heat kernel in t, x, y follows directly from the smoothness of Ptf (x) for any f E L2 (M), by means of some abstract result concerning the differentiability of functions taking values in a Hilbert space.

Consider an open set ) C R', a Hilbert space W, and a function h : f2 --* W. Denote the inner product in 71. We say that the function h is weakly Ck if, for any cp E 7-l, the numerical function by

x'-+ (h(x),gyp)

belongs to Ck (Sl). The function h is strongly continuous if it is continuous with respect to the norm of 7-l, that is, for any x E 0,

Ilh(y)-h(x)II -4 0asy-4 x. The Gateaux partial derivative 8zh is defined by h (x)

h(x+ses) - h(x) =lim S

where eti is the unit vector in the direction of the coordinate xi and the limit is understood in the norm of 7-L. One inductively defines the Gateaux partial derivative 8«h for any multiindex a. We say that the function h is strongly Ck if all Gateaux partial derivatives 8°h up to the order k exist and are strongly continuous. Since the norm limit commutes with the inner product, one easily obtains that if h is strongly Ck then h is weakly Ck and

8' (h (x) ,,p) = (8'h (x) ,:p) for any cp E 7-L,

(7.84)

provided Ia! < k. It turns out that a partial converse to this statement is true as well. LEMMA 7.21. For any non-negative integer k, if h is weakly Ck+1 then h is strongly Ck. Consequently, h is weakly C°° if and only if h is strongly C'.

7.5. SMOOTHNESS OF THE HEAT KERNEL IN t, x, y

211

PROOF. We use induction in k.

Inductive basis for k = 0. Fix a point x E 1 and prove that h is strongly continuous at x. Fix also cp G 'H and consider a numerical function f (x) _ (h (x), cp),

which, by hypothesis, belongs to C' (Il). Choose e > 0 so small that the closed Euclidean ball BE (x) lies in Q. Then, for any vector v E WL such that Jul < e, the straight line segment connecting the points x and x + v lies in Q. Restricting function f to this segment and applying the mean-value theorem, we obtain

If(x+v)-f(x)I -< sup IVfIlvl BEW

Rewrite this inequality in the form

(h(s+v)_h(x),) Jul

where C (x, cp) := supBE(.) I V f I, and consider h(x+Ivl h(x) as a family of vectors in 71 parametrized by v (while x is fixed). Then (7.85) means that this family is weakly bounded. By the principle of uniform boundedness, any weakly bounded family in a Hilbert space

is norm bounded, that is, there is a constant C = C (x) such that

h(x+v)-h(x)
11

11-

(

786 )

1

for all values of the parameter v (that is, Ivl < e and v that h is strongly continuous at x.

0). Obviously, (7.86) implies

Inductive step from k - 1 to k. We assume here k > 1. Then, for any cp E 'H, the function (h (x), cp) belongs to C' (il), and consider its partial derivative 8i (h (x) , cp) at a fixed point x E 11 as a linear functional of cp E 7-l. This functional is bounded because by (7.86)

(h(s+sei)_h(s)ca)

< C (x) IIWIi

and, hence, jai (h(x),cc)l _
By the Riesz representation theorem, there exists a unique vector hi = hi (x) E 7-l such

that

8i (h (x) , cp) = (hi (x) , cp) for all cp E W.

(7.87)

The function hi (x) is, hence, a weak derivative of h (x). The condition that (h (x) , cp) belongs to Ck+i (1) implies that (hi (x) , cp) belongs to C" (Q), that is, hi is weakly Ck. By the inductive hypothesis, we conclude that hi is strongly Ck-1 To finish the proof, it suffices to show that the Gateaux derivative 8ih exists and is equal to hi, which will imply that h is strongly Ck. We will verify this for the index i = n; for i < n, it is done similarly. Consider a piecewise-smooth path y : [0, T] -3 52 such that 'Y (0) = xo and y (T) = x, and show that

f 0

2

hi (y ( t ) )

(t) dt = h (x) - h (xo)

.

(7.88)

7. THE HEAT KERNEL ON, A MA NifOLD

212

Denoting the integral in (7.88) by I and using (7.87) and the fundamental theorem of calculus, we obtain, for any cp E 71, (I,A

f

T

(hi (7 (t)) , w) 7' (t) dt

0

T

J

6i (h

(t) dt

d

(h'(7(t)),ca)dt JoT dt (h(x),:p) - (h(xo), p), whence (7.88) follows.

Now fix a point x E 0 and choose e > 0 so that the cube (x - s, x - e)' lies in 1. For simplicity of notation, assume that the origin 0 of 1R' is contained in this cube, and consider the polygonal path 7 connecting 0 and x inside the cube, whose consecutive vertices are as follows:

0 0)

(0 0

(xl 0,

0 0)

...

xn-1 0), (x' x'

(xl x2

x°

n)

By (7.88), we have T

h (x) = h (0) + f hi (7 (t)) Y' (t) dt.

(7.89)

0

The integral in (7.89) splits into the sum of it integral over the legs of ^r, and only the last one depends on x'2. Hence, to differentiate (7.89) in xn, it suffices to differentiate the integral over the last leg of 7. Parametrizing this leg by 7 (t) _ (xl, x2, ..., Xn-1, t) , 0 < t < xn, we obtain x"

hi (7 (t))' i' (t) dt = n f

C7nh (x) = aan f which was to be proved.

yn hn (XI.

t) dt = h, (x)

/

SECOND PROOF OF THEOREM 7.20. Let S2 be a chart on the manifold R- x M, and

consider pt,,. as a mapping Q -> L2 (M). By Theorem 7.10. for any f E L2 (M), the function Ptf (x) = (pt,x, f )L2 is C°°-smooth in t, x. Hence, the mapping pt.x is weakly C°°. By Lemma 7.21, the mapping pt, is strongly C°°. Let Sl' be another chart on + x M which will be the range of the variables s, y. Since ps, is also strongly C°° as a mapping from S2' - L2 (M), we obtain by (7.56) pt+s (x, y) = (Pt,m, ps,Y)L2 = C°°

(n x 9'):

which implies that pt (x, y) is C°°-smooth in t, x, y. Let D'' be a partial differential operator in variables (t, x) E Q. By (7.84), we have, for any f E L2 (M),

D' (pt,., f) = (Daps,.if ) (7.90) where Dapt, is understood as the Gateaux derivative. Since the left hand sides of (7.82) and (7.90) coincide, so do the right hand sides, whence we obtain by Lemma 3.13

Dapt (x, ) = Daps,. a.e. Consequently, Dapt (x, ) E L2 (M) and, for any f E L2 (M),

Da f pt (x, y) f (y) dp = Da (pt,., f) _ (Dapt,x, f) = f Dapt (x, y) f (y) dA, M

which finishes the proof.

M

7.5. SMOOTHNESS OF THE HEAT KERNEL IN t, x, y

213

Exercises. 7.39. Let f : M -* [-co, +oo] be a measurable function on M. (a) Prove that, if f > 0 then the function Ptf (x) = fMPt (x, y) f (y) dl-t (y)

7.91)

is measurable on M for any t > 0. (b)

(c)

Prove that if f is signed and the integral (7.91) converges for almost all x then Ptf (x) is measurable on M. Prove the identity

Pt+Sf = Pt(Psf) for any non-negative measurable function f. 7.40. For any open set l C M, denote by pt' (x, y) the heat kernel of the manifold (fl, g, lc).

(a) Prove that pt (x, y) < pt (x, y) for all x, y E 0 and t > 0. (b) Let {l } be an exhaustion sequence in M. Prove that C°°(R+xMxM)

" A (x, y)

-#

pt (x, y) as i --> oo.

(c) Prove that, for any non-negative measurable function f (x),

,

Pif (x) -+ Pt.f (x) asi--3oo, for any fixed t> 0 and XE M. (d) Prove that if f c Cb (M) then

f (x)

C°O(-- M)

Ptf (x) as i --+ oo.

7.41. Let (X, gx, Ax) and (Y, gy, iy) be two weighted manifold and (M, g, lt) be their direct product (see Section 3.8). Denote by pt and pt the heat kernels on X and Y, respectively. Prove that the heat kernel pt on M satisfies the identity

pt ((x, y) , (x , y')) = px (x, x) pe (y, y) , for all t > 0, x, x' E X, y, y' E Y (note that (x, y) and (x', y') are points on M).

(7.92)

7.42. For any t > 0, consider the quadratic form in L2 (M), defined by

Et(f)=

(f_ftf,f)2

(cf. Exercise 4.38). Prove that if the heat kernel is stochastically complete, that is, for all

XE M and t> 0, IM

pt (x, y) dl-t (y) = 1,

(7.93)

then the following identity holds:

Et (f) = 2t M f M(f(x) - f(y))2pt(x,y)dl-,(y)dFt(x), for allt > O and f EL 2 (M).

(7.94)

7.43. Prove that, for any real k > 0 and for any f E L2 (M), k-1

(G + id)-k f (x) = f 0

rt (k) e -'P, f (x) dt,

for almost all x E M, where r is the gamma function. HINT. Use Exercise 5.11.

(7.95)

7. THE HEAT KERNEL ON A MANIFOLD

214

7.44. Assume that the heat kernel satisfies the following condition pt(x,x)
for allxEMandO
(7.96)

where -y, c > 0. Fix a real number k > ry/2.

(a) Prove that, for any f E L2 (M), the function (L + id)-l' f is continuous and suupI(L+id)-k fl
(7.97)

where C = C (c, ry, k).

(b) Prove that, for any u E dom Gk, we have u E C (M) and sup Jul <_ C

(lItIIL2 + II LkuII

M

L2)

.

(7.98)

7.45. Prove that if (7.98) holds for all u E dom Lk with some k > 0 then the heat kernel satisfies the estimate (7.96) with 'y = 2k. 7.46. The purpose of this question is to give an alternative proof of Theorem 6.1 (Sobolev embedding theorem). (a) Prove that if u E Wk (R') where k is a positive integer then u E dom Lk12, where L

is the Dirichlet Laplace operator in R'. Prove also that, for any u E Wk (R'), II (L+id)k/2uIIL2 < CIIujjWk, where C is a constant depending only on n and k. (b) Prove that if u E Wk (R") where k is an integer such that k > n/2 then u E C (R") and

-

sup IuI < CII"ullwk. Rn (c)

Prove that if k > m + n/2 where m is a positive integer then u E Wk (III") implies

u E C' (Wi) and IIUIIC-(Ra)< CIIUIIWk(R.).

(d) Prove that if 12 is an open subset of R and k and m are non-negative integers such that k > m + n/2 then u E WWo (12) implies u E Ctm (0). Moreover, for any open sets S2' Cc 12" C E2,

IlullCm > G C with a constant C depending on S2', f2", k, m, n. HINT. Use Exercise 4.25 for part (a) and Exercise 7.44 for part (b) 7.47. (Compact embedding theorems)

(a) Assume that p (M) < oo and sup pt (x, x) < oo for all t > 0. xEM

(7.99)

Prove that the identical embedding Wo (M) -*L2 (M) is a compact operator. (b) Prove that, on any weighted manifold M and for any non-empty relatively open compact set 12 C M, the identical embedding Wo (S2) " L2 (Sl) is a compact operator (cf. Theorem 6.3 and Corollary 10.21). HINT. Use for part (a) the weak compactness of bounded sets in L2 and Exercises 7.36, 4.40.

7.48. Let I be an open interval in JR and 1l be a Hilbert space. Prove that if a mapping h : I -3 7-l is weakly differentiable then h is strongly continuous.

7.6. NOTES

215

7.6. Notes One of the main results of Chapter 7 is the existence of the heat kernel satisfying various nice properties (Theorems 7.7, 7.13, and 7.20). The classical approach to construction of the heat kernel on a Riemannian manifold, which originated from [275], [276], uses a parametrix of the heat equation, that is, a smooth function which satisfies the necessary conditions in some asymptotic sense. The parametrix itself is constructed using as a model the heat kernel in R'. A detailed account of this approach can be found in many sources, see for example [36], [37], [51], [58], [317], [326]. This approach has certain advantages as it gives immediately the short time asymptotics of the heat kernel and requires less of abstract functional analysis. On the other hand, the theory of elliptic and parabolic equations with singular (measurable) coefficients, developed by de Giorgi [103], Nash [292], Moser [279], [280], and Aronson [9], has demonstrated that the fundamental solutions for such equations can be constructed using certain a priori estimates, whereas the parametrix method is not available. The method of construction of the heat kernel via a priori estimates of solutions has been successfully applied in analysis on more general spaces - metric measure spaces with energy forms.

In our approach, the key a priori estimate (7.18) of the heat semigroup is given by Theorem 7.6. The proof of (7.18) uses the elliptic regularity theory and the Sobolev embedding theorem. As soon as one has (7.18), the existence of the heat kernel follows as in Theorem 7.7. This approach gives at the same token the smoothness of Ptf (x) in variable x. There are other proofs of (7.18) based only on the local isoperimetric properties of manifolds, which can be used in more general settings (cf. Corollary 15.7 in Chapter 15).

The heat kernel obtained as above is not yet symmetric. Its symmetrization (and regularization) is done in Theorem 7.13 using a general method of J.-A. Yan [360]. The smoothness of the heat kernel pt (x, y) is proved in three installments: first, smoothness of Pt f (x) in x (Theorem 7.6), then smoothness of pt (x, y) in (t, x) (Theorems 7.10 and 7.13) and, finally, smoothness of pt (x, y) in (t, x, y) (Theorem 7.20; the second proof of this theorem uses the approach from [96, Theorem 5.2.1] and [92, Corollary 1.42]). Other methods for construction of the heat kernel are outlined in Section 16.4. An somewhat similar approach for construction of the heat kernel via the smoothness of Ptf was used by Strichartz [330], although without quantitative estimates of Pt!. That method was also briefly outlined in [96]. After the heat kernel has been constructed, the heat semigroup Pt f can be extended from L2 to other function classes as an integral operator. We consider here only extensions to L' and Cb. A good account of the properties of the heat semigroup in spaces L4 can be found in [330].

CHAPTER 8

Positive solutions This Chapter can be regarded as a continuation of Chapter 5. However, the treatment of the Markovian properties is now different because of the use of the smoothness of solutions.

8.1. The minimality of the heat semigroup We say that a smooth function u (t, x) is a supersolution of the heat equation if it satisfies the inequality

at

-

in a specified domain. The following statement can be considered as an extension of Corollary 5.17.

THEOREM 8.1. Set I = (0, T) where T E (0, +oo]. Let u (t, x) be a non-negative smooth supersolution to the heat equation in I x M such that L2

u (t, ) f as t --3 0, (8.1) for some f E Ll (M). Then Pt f (x) is also a smooth solution to the heat equation in I x M, satisfying the initial condition (8.1), and u (t, x) > Pt f (x) , (8.2)

for all tEI andxEM. PROOF. Note that f _> 0. Let (8.2) be already proved. Then Ptf (x) is locally bounded and, by Theorem 7.15, it is a smooth solution to the heat equation. Let us verify the initial condition 2

Pt f

L° f

as t -+ 0.

(8.3)

Indeed, for any relatively compact open set S2 C M, we have

Pt (fin)
we conclude that Ptf -4 f, which implies (8.3). In order to prove (8.2), we reduce the present setting to the L2-Cauchy problem.(5.55) of Corollary 5.17. Choose an open set 92 C M. The smoothness of u implies that u (t, ) E W1 (S2) and the strong derivative at in L2 (12) obviously coincides with the classical derivative -aFut. Hence, u (t, ) as a path 217

8. POSITIVE SOLUTIONS

218

in W1 (St) satisfies the conditions (5.55), and we conclude by Corollary 5.17,

that' (8.4)

Let {1 k} be a compact exhaustion sequence in M, and set fk = f lQk. It follows from (8.4) that

u (t, ) > Pt"kfi, for all i and k. Since fz c L2 (M), by Theorem 5.23 we obtain z

P"kfi- Ptfz as k-4oo, whence it follows that

Letting i -3 oo, we obtain

u (t, ) > Pt f a.e. (8.5) This implies Pt f E Li c (I x M), and we conclude by Theorem 7.15 that the function Pt f (x) is smooth in t, x. Hence, (8.5) implies the pointwise estimate (8.2). COROLLARY 8.2. Let u (t, x) be a non-negative smooth solution to the

heat equation in I x M such that Lz

ast-+0

for some f E Li c (M), and 0 as x -- oo in M, where the convergence is uniform in t E I. Then u (t, x) = Ptf (x). u (t, x)

(8.6)

The hypotheses of Corollary 8.2 are exactly those of Theorem 8.1 except for the additional condition (8.6), which leads to the identity of u (t, x) and

Ptf W. PROOF. It follows from Theorem 8.1 that the function v (t, x) := u (t, x)L2

Pt f (x) satisfies the heat equation in I x M and the initial condition v (t, ) ° 0 as t -+ 0. Besides, (8.6) implies v (t, x) 0 as x -* oo. Hence, by Corollary 5.20, v - 0, which was to be proved. COROLLARY 8.3. For any non-negative f E Cb (M), the function u (t, x) = Ptf (x) is the minimal non-negative solution to the following Cauchy problem

f at = 0µu, ult=o = f,

in ]R

x M,

(8 7)

where ujt=o = f means that u (t, ) -* f as t -* 0 locally uniformly in x. 'Corollary 5.17 says that (8.4) holds almost everywhere on M (for any t E I). However, since by Theorem 7.10 Pn f is a smooth function, (8.4) holds, in fact, everywhere on M.

8.2. EXTENSION OF RESOLVENT

219

PROOF. Indeed, by Theorem 7.16, the function Pt f (x) does solve (8.7), and by Theorem 8.1, u (t, x) > Pt f (x) for any other non-negative solution U.

Exercises. 8.1. Prove that if h is a non-negative function satisfying on M the equation

-A ,h + ah = 0, where a is a real constant, then Pth < e"h for all t > 0.

8.2. Extension of resolvent Our goal here is to extend the resolvent Ra f to a larger class of functions

f and to prove the properties of R« similar to the properties of the heat semigroup Pt given by Theorems 7.15 and 8.1.

Recall that, for any a > 0, the resolvent Ra is a bounded operator in L2 (M) defined by

Ra = (L+aid)-1 where L is the Dirichlet Laplace operator (cf. Section 4.2). For any f E L2 (M), the function u = Ra f satisfies the equation

-AAu + au = f

(8.8)

in the distributional sense. As an operator in L2, the resolvent is related to the heat semigroup by the identity -at (R«f) 9)L2 = rn

e

(Pt.f, 9)L2 dt,

(8.9)

for all f, g E L2 (M) (cf. Theorem 4.5 and Lemma 5.10). Now we extend Ra f to a more general class of functions f by setting 00

Raf (x)

e-atPtf (x) dt = fo,

1M

e-atPt (x, y) f (y) dp (y) dt, (8 .10)

whenever the right hand side of (8.10) makes sense. Note that the function Ra f (x) is defined by (8.10) pointwise rather than almost everywhere. If f is a non-negative measurable function then the right hand side of (8.10) is always a measurable function by Fubini's theorem, although it may take value oo. If, in addition, f E L2 (M) then substituting Ra f from (8.10) into (8.9), we obtain, again by Fubini's theorem, that (8.9) holds for all nonnegative g E L2 (M), which implies that the new definition of Ref matches the old one. For a signed f E L2 (M), the same conclusion follows using

f = f+ - f-. THEOREM 8.4. Fix a non-negative function f c Li (M) and a constant

a>0.

S. POSITIVE SOLUTIONS

220

(a) If u E Ll (M) is a non-negative solution to the equation

-AAn + au = f

(8.11)

then u>Raf. (b) If Raf E Ll (M) then the function u = Raf satisfies the equation (8.11).

PROOF. Let {Ilk} be a compact exhaustion sequence in M and set fk = min (k, f )1c2k.

(a) It follows from (8.11) that ANu E L oC (M), and we conclude by Exercise 7.9 that u E W110 (M). Then u E Wl (Slk) and, applying Corollary 5.15 in S2k, we obtain u > R«k f . Consequently, we have, for all indices i, k, u > Rak f2.

Since fZ E L2 (M), Theorem 5.22 yields R«k f2 L--* Rf, as k -* oo, whence it follows u > Ra f2. Passing to the limit as i --* oo, we finish the proof. (b) Since fk E L2 (M), the function and Uk = R« fk belongs to L2 (M) and satisfies the equation

-AAuk + auk = fk (cf. Theorem 4.5). Hence, for any co E D (M), we have

J u(-+ a) d=f fd.

(8.12)

M

llM

Since the sequence fk is monotone increasing and converges to f a.e., we obtain by (8.10) and the monotone convergence theorem that Uk (x) t u (x) pointwise. Since u and f belong to Ll (supp cp), we can pass to the limit in (8.12) by the dominated convergence theorem and obtain that u also satisfies this identity, which is equivalent to the equation (8.11).

REMARK 8.5. Part (a) of Theorem 8.4 can be modified as follows: if u E Wioc (M), u > 0, and u satisfies the inequality

-Aµu + an > f, then u > Raf.. This is proved in the same way because the only place where the equality in (8.11) was used, is to conclude that A,,u E L o, and, hence,

u E W. COROLLARY 8.6. If f E L°° (M) and a > 0 then Raf is bounded, sup I Raf I < a-I Iif IIL°°,

(8.13)

and u = Raf is a distributional solution to the equation (8.8). PROOF. The estimate (8.13) follows from (8.10) and sup I Ptf I c Iif IIL°° (cf. Exercise 7.33).

If f > 0 then the fact that Raf solves (8.8) follows

from Theorem 8.4(b). For a signed f, the same follows from Raf = Raf+ Ra f- .

-

8.2. EXTENSION OF RESOLVENT

221

Some cases when one can claim the smoothness of Ra f are stated in the following theorem.

THEOREM 8.7. Let f E C°° (M) and a > 0. (i) I f f > 0 and and RC f E L aC (M) then RC f E C°° (M) . (ii) If f is bounded then R, ,f E C°° (M).

PROOF. Consider first a special case when f > 0 and f c Co (M), and prove that Ra f E C°O (M). Since the function u = RC f satisfies the equation (8.8), it is tempting to conclude that u c COO applying Corollary 7.3. Indeed, the latter says that every distributional solution to (8.8) is a smooth function, which means that u as a distribution is represented by a C°° function, that is, there is a function u E C°° such that u = u a.e.. However, our aim now is to show that u (x) itself is C. Recall that, by Theorem 7.13, function Ptf is C°° smooth in [0, +oo) x M (cf. (7.53)). Therefore, the function ul (x) =

f

l

e-CtPtf (x) dt

0

is C°° smooth on M for any finite 1 > 0 and, moreover, any partial derivative of ul can be computed by differentiating under the integral sign. Using the properties of the heat semigroup, we obtain

rt

t

& ul

= J e-Ct/m (Ptf) dt = J 0

e-at_ (Ptf) dt

0

e-atpt.f] o + a I

t

e-CtPtf dt, 0

which implies that

-Dul + aul = .fl := f - e-a*Plf By the estimate (7.20) of Theorem 7.6, we have, for any compact set K that is contained in a chart, and for any positive integer m, II PIf IICm(K) < FK,m (1) II f II L2(M),

where FK,,m (1) is a function of 1 that remains bounded as l -* oo. This implies e -«1 P1f

0 asl-3oc

and, hence, A -- f . The sequence {ui (x)} increases and converges to u (x) pointwise as l -+ oo. Since u E L2 (M), this implies by Exercise 7.13 that u (x) belongs to Coo, which finishes the proof in the special case. (i) Let {SZk} be a compact exhaustion sequence in M and let Vik be a cutoff function of Stk in SZk+l. Set fk = Ok f so that fk E Co (M). By the special case above, the function Uk = Ra fk belongs to C°°. Since the sequence { fk } increases and converges pointwise to f, by (8.10) the sequence

222

8. POSITIVE SOLUTIONS

Uk (x) also increases and converges pointwise to u = Ra f . Since u E Li ° coo and fk ---4 f, we conclude by Exercise 7.13 that u E C. (ii) By Corollary 8.6, Ra f is bounded and, hence, belongs to L1 '. If

f > 0 then Ra f is smooth by part (i). For a signed f, the smoothness of Ra f follows from the representation

f=of - (ef - f) because both function of and of - f are bounded, smooth, and non-negative. REMARK 8.8. If f is a non-negative function from C°° fl L2 then Ra f E

L2 and we obtain by Theorem 8.7 that Ra f E C°°. This was stated in Exercise 7.15, but using the definition of R,, as an operator in L2. In other words, the statement of Exercise 7.15 means that the L2-function Ra f has a smooth modification, whereas the statement of Theorem 8.7 means that the function Ra f which is defined pointwise by (8.10), is C°° itself. ,

Exercises. 8.2. If u E L2 ,2 (M) is a non-negative solution to the equation

-0µu + au = f where a > 0 and f E L 102': (M), f > 0. Prove that if

u(x)-*0as x- oo, then u=Raf. 8.3. Let U E L2 (M) satisfy in M the equation

Dµu+au=0, where A E R, and

u(x)->0as x-3oo. Prove that u E Wo (M). REMARK. Since by the equation 0µu E L2 (M), it follows that u E dom (L) and, hence, u satisfies the equation Lu = -Au. Assuming that u 0 0 we obtain that u is an eigenfunction of the Dirichlet Laplace operator.

8.3. Strong maximum/minimum principle 8.3.1. The heat equation. As before, let (M, g, µ) be a weighted manifold. For an open set SZ C R x M, define its top boundary OtopI as the set of points (t, x) E 852 for which exists an open neighborhood U C M of x and e > 0 such that the cylinder (t - e, t) x U is contained in 1 (see Fig. 8.1). For example, if t2 = (a, b) x Q where a < b and Q is an open subset of M, then Btp1 = {b} x Q. If M = 1I81 and SZ is a Euclidean ball in Rn+1 then BtopSI = 0.

8.3. STRONG MAXIMUM/MINIMUM PRINCIPLE

223

ato

t

(t-E. t) x U ; .

M

FIGURE 8.1. The top boundary of a set Q. DEFINITION 8.9. The parabolic boundary apse of an open set S2 C R x M is defined by apS2 := aS2 \ at',s2.

If 52 is non-empty and the value of t in 0 has a lower bound then apt is non-empty - indeed, a point (t, x) E S2 with the minimal value of t cannot

belong to atps2. The importance of this notion for the heat equation is determined by the following theorem, which generalizes Lemma 1.5.

THEOREM 8.10. (Parabolic minimum principle) Let S2 be a non-empty relatively compact open set in R x M, and let a function u E C2 (S2) satisfy in 52 the inequality (8.14) Then

inf u = inf u. 'a

(8.15)

apse

REMARK. Any function u E C2 satisfying (8.14) is called a supersolution to the heat equation, while a function satisfying the opposite inequality

is called a subsolution. Obviously, Theorem 8.10 can be equivalently stated as the maximum principle for subsolutions: sup u = sup U. apf

In particular, if u < 0 on the parabolic boundary of 1 then u < 0 in Q. Let 52 = (0, T) x Q where Q is a relatively compact open subset of M. Then the condition u < 0 on apQ can be split into two parts: u (t, x) < 0 for all x E aQ and t E (0, T)

u(0,x)<0forallxEQ,

8. POSITIVE SOLUTIONS

224

which imply the following (assuming u e C (M): u+ (t, x) = 0 as x --> oo in Q (considering Q is a manifold itself)

u+(t,x)z3 0 ast-30. Then the conclusion that u < 0 in SZ follows from Corollary 5.20. Hence,

for cylindrical domains 11, Theorem 8.10 is contained in Corollary 5.20. However, we will need Theorem 8.10 also for non-cylindrical domains, and for this reason we provide below an independent proof of this theorem. PROOF OF THEOREM 8.10. The proof is similar to that of Lemma 1.5. Assume first that u satisfies a strict inequality in 1:

at

> & u.

(8.16)

Let (to, xo) be a point of minimum of function u in fZ. If (to, xo) E O 2 then

(8.15) is trivially satisfied. Let us show that, in fact, this point cannot be located elsewhere. Assume from the contrary that (to, xo) is contained in SZ or in atpQ. In the both cases, there exists an open neighborhood U C M of

xo and E > 0 such that the cylinder r :_ (to - E, to) x U is contained in Q. Since function t H u (t, xo) in [to - E, to] takes the minimal value at (to, xo), we necessarily have

at (to, xo) <

(8.17)

0.

By the choice of U, we can assume that U is a chart, with the coordinates xl, ..., xn. Let g be the matrix of the metric tensor g in the coordinates xl, ..., xn and g be the matrix of g in another coordinate system y', ..., yn in U (yet to be defined). By (3.25), we have

g=JTgJ, where J is the Jacobi matrix defined by k

Jz =

aX

It is well known from linear algebra that any quadratic form can be brought to a diagonal form by a linear change of the variables. The quadratic form

is positive definite and, hence, can be transform to the form (1)2 + ... + (Sn)2 by a linear changei = A is a numerical non-singular matrix. This implies that gig (xo)

ATg(xo)A=id. Defining the new coordinates yi by the linear equations x2 = Ajy-7, we obtain

that J (xo) = A and, hence, g (xo) = id. So, renaming y2 back to xi, we obtain from (3.46) that the Laplace operator A. at point xo has the form A1,1.0

-

a2

(axi)2 +

ia

b

axi

8.3. STRONG MAXIMUM/MINIMUM PRINCIPLE

225

for some constants b'. Since xo is a point of minimum of function x -u (to, x) in U, we obtain that z

(to, xo) = 0 and (Sx2)z (to, xo) > 0. axz

This implies A, u (to, xo) >_ 0,

which together with (8.17) contradicts (8.16). In the general case of a non-strict inequality in (8.14), consider for any e > 0 the function uE = u + et, which obviously satisfies (8.16). Hence, by the previous case, we obtain

inf (u + et) = inf (u + Et)

,

whence (8.15) follows by letting e -* 0.

El

THEOREM 8.11. (Strong parabolic minimum principle) Let (M, g, ,u) be

a connected weighted manifold, and I C R be an open interval. Let a nonnegative function u (t, x) E C2 (I x M) satisfy in I x M the inequality

at

OMu.

If u vanishes at a point (t', x') E I x M then u vanishes at all points (t, x) E

IxMwith t
Under the conditions of Theorem 8.11, one cannot claim that u (t, x) = 0 for t > t' - see Remark 9.22 in Section 9.3. Note that the function u in Theorem 8.11 is a supersolution to the heat equation. Since u+const is also supersolution, one can state Theorem 8.11 as follows: if u is a bounded below supersolution then u (t', x') = inf u at some point implies u (t, x) = inf u for all t < t' and x. Equivalently, Theorem 8.11 can be stated as the strong parabolic maximum principle for subsolutions: if u is a bounded subsolution in I x M then u (t', x') = sup u at some point implies u (t, x) = sup u for all t < t' and x. PROOF. The main part of the proof is contained in the following claim. CLAIM. Let V be a chart in M and xo, xj be two points in V such that the straight line segment between xo, xj is also in V. If u is a function as in the hypotheses of Theorem 8.11 then u (to, xo) > 0

u (ti, xi) > 0 for all tj > to,

(8.18)

assuming that to, tj E I. For simplicity of notation, set to = 0. By shrinking V, we can assume that V is relatively compact and its closure V is contained in a chart. Let r > 0 be so small that the Euclidean 2r-neighborhood of the straight line

226

8. POSITIVE SOLUTIONS

segment [xo, xl] is also in V. Let U be the Euclidean ball in V of radius r centered at X. By further reducing r, we can assume that also inf u (0,x) > 0.

(8.19)

xEU

Setting = tl (x1 - xo) we obtain that the translates U+t are all in V for any t c [0, ti]. Consider the following tilted cylinder (see Fig. 8.2)

F={(t,x):0
FIGURE 8.2. Tilted cylinder r. This cylinder is chosen so that the center of the bottom is (0, xo) while the center of the top is (ti, xi). We will prove that, under condition (8.19),

u is strictly positive in r, except for possibly the lateral surface of r; in particular, this will imply that u (ti, XI) > 0. To that end, construct a non-negative function v E C2 (P) such that av < A AV in 1',

(8.20)

and

(8.21) v = 0 on the lateral surface of I', and v > 0 otherwise. Assuming that such a function v exists, let us compare v and Eu, where E > 0 is chosen so small that

inf u (0, x) > E sup v (0, x) .

xEU

xEU

Due to this choice of e, we have u > ev at the bottom of r. Since v = 0 on the lateral surface of r and u is non-negative, we conclude that inequality

8.3. STRONG MAXIMUM/MINIMUM PRINCIPLE

227

u > Ev holds on the whole parabolic boundary of P. The function u - Ev satisfies the hypotheses of Theorem 8.10, and we conclude by this theorem that u > Ev in P. By (8.21), this implies that u is positive in f except for the lateral surface, which was to be proved. Assume for simplicity that xo = 0 is the origin in the chart V. Let us look for v in the form v (t, x) = e-atf (I x - t12)

where a > 0 and function f are to be specified, and 1.1 stands for the Euclidean length of a vector. Note that (t, x) E P implies x - t E U, which means IX - t12 < r2. Hence, to satisfy conditions (8.21), function f should be positive in [0, r2) and vanish at r2. Let us impose also the conditions

f' < 0 and f">0 in [0, r2],

(8.22)

which will be satisfied in the final choice of f. Denoting by xi, ..., x' the coordinates in V and setting n

w(t,x) := i=1

we obtain, for (t, x) E P, av = -ae-at f (w) + e-at f' (w)

at

(fat - xi) < -e-at (a f (w) + C f' (w))

where C is a large enough constant (we have used that the ranges of t and x are bounded, 6 is fixed, and f _< 0). Observe that, by (3.46),

a 2w

09w + bi axi ,

°Aw ^ 9u axia x where bi are smooth functions, which yields

°A,w = 2gZZ + 2bz (x2 - eit) < C,

where C is a large enough constant. Computing the gradient of w and using the positive definiteness of the matrix (gij), we obtain Vw

2

= = g'3

Ig

z

aw (9W >

axi axe -

i=1

(ax )

= cw,

where c' and c = c'/4 are (small) positive constants. Using the chain rule for °N, (see Exercise 3.9) and (8.22), we obtain from the above estimates

°µv = e-«t (fn

(w) IVwlg + f' (w) °uw) > e-at (cw f" (w) + Cf' (w)) ,

which yields `

Y - °v <

-e-at (a f

(w) + C f' (w) + cw f" (w))

,

8. POSITIVE SOLUTIONS

228

where we have merged two similar terms Cf' (w). Now specify f as follows:

f (W) = (r2 -'W)2. Obviously, this function is smooth, positive in [0, r2), vanishes at r2, and satisfies (8.22). We are left to verify that, for a choice of a, c if (w) + C f' (w) + cw f" (w) > 0 for all w c [0, r2], which will imply (8.20). Indeed, we have

of (w)+Cf'(w)+cwf"(w) = a(r2-w)2-2C(r2-w)+2cw = az2-2(C+c)z+2cr2, where z = r2 - w. Clearly, for large enough a, this quadratic polynomial is positive for all real z, which finishes the proof of the Claim. The proof of Theorem 8.11 can now be completed as follows. Assuming u (t', x') = 0, let us show that u (t, x) = 0 for all (t, x) E I x M with t < t'. By the continuity of u, it suffices to prove that for t < t'. Since M is connected, it is possible to find a finite sequence {xz}k o so that xo = x, Xk = x', and any two consecutive points xi and xti+l are contained in the same chart together with the straight line segment between them. t

t= to

x=xp

x1

x2

x3

xk=x

M

FIGURE 8.3. If non-negative supersolution u vanishes at (t', x') then it vanishes also at any point (t, x) with t < t'. Choosing arbitrarily a sequence of times (see Fig. 8.3)

t=to 0 then also u (tj, xi) > 0 and, continuing by induction, u (tk, xk) > 0, which contradicts the assumption u (t', x') = 0.

The strong maximum/minimum principle has numerous applications. Let us state some immediate consequences.

8.3. STRONG MAXIMUM/MINIMUM PRINCIPLE

229

COROLLARY 8.12. On a connected manifold (M, g, µ), the heat kernel pt (x) y) is strictly positive for all t > 0 and x, y E M.

PROOF. Assume that pt (x', y') = 0 for some t' > 0, x', y' E M. Since the function u (t, x) = pt (x, y') satisfies the heat equation (see Theorem 7.20), we obtain by Theorem 8.11 that pt (x, y') = 0 for all 0 < t < t' and all x E M. Consider any function f E Co (M) such that f (y') zA 0. By Theorem 7.13, we have fMPt (x, y') f (x) dy (x) --+ f (y')

as t - 0,

which, however, is not possible if pt (x, y') = 0 for small t > 0. Note that, by Example 9.10 below, if M is disconnected then pt (x, y) = 0 whenever x and y belong to different connected components of M.

8.3.2. Super- and subharmonic functions. DEFINITION 8.13. Let a E R. A function u E C2 (M) is called asuperharmonic on M if it satisfies the inequality -Dµu+au > 0. It is called a-subharmonic if -D,,,u + au < 0, and a-harmonic if -0µu + au = 0.

Of course, in the latter case u E C°° (M) by Corollary 7.3. If a = 0 then the prefix "a-" is suppressed, that is, u is superharmonic if AMU < 0, subharmonic if A Au < 0, and harmonic if 0µu = 0. COROLLARY 8.14. (Strong elliptic minimum principle) Let M be a connected weighted manifold and u be a non-negative a-superharmonic function

on M, where a E R. If u (xo) = 0 at some point x0 E M then u (x) = 0. PROOF. Consider function v (t, x) = eatu (x). The condition

-0µu + au > 0 implies that v is a supersolution to the heat equation in R x M, because av - 0µv = aeatu eat0µu > 0.

at

-

If u (x0) = 0 then also v (t, xo) = 0 for any t, which implies by Theorem 8.11

that v (t, x) = 0 and, hence, u = 0. There is a direct "elliptic" proof of the strong elliptic minimum principle,

which does not use the heat equation and which is simpler than the proof of Theorem 8.11 (see Exercise 8.4).

COROLLARY 8.15. Let M be a connected weighted manifold. If u is a

superharmonic function in M and u (x0) = inf u at some point xo then u = inf u. If u is a subharmonic function in M and u (xo) = sup u as some point xo then u -=sup u. PROOF. The first. claim follows from Corollary 8.14 because the function

u - inf u is non-negative and superharmonic. The second claim trivially follows from the first one.

8. POSITIVE SOLUTIONS

230

COROLLARY 8.16. (Elliptic minimum principle) Let M be a connected weighted manifold and SZ be a relatively compact open subset of M with non-

empty boundary. If u E C (S2) fl C2 (52) is a superharmonic function in Sl then inf u = inf u. an n PROOF. Set m = infa u and consider the set

S={xES2:u(x)=m}. We just need to show that S intersects the boundary 852. Assuming the contrary, consider any point x E S. Then x E it and, in any connected open neighborhood U C SZ of x, function u takes its minimal value at the point x. By Corollary 8.15, we conclude that u - m in U, which means that U c S and, hence, S is an open set. Since set S is also closed and non-empty, the connectedness of M implies S = M, which contradicts to

Sc52CM\852. A companion statement to Corollary 8.16 is the maximum principle for subharmonic functions: under the same conditions, if u is subharmonic then sup u = sup U. an ?i

Exercises. 8.4. Let M be a connected weighted manifold and E, F be two compact subsets of M. Prove that, for any real a there is a constant C = C (a, E, F) such that, for any nonnegative a-superharmonic function u on M, inf u < C inf u. E

-

F

8.5. (A version of the elliptic minimum principle) Let M be a non-compact connected weighted manifold and let u (t, x) E C2 (M) be a superharmonic function. Prove that if lim sup u (xk) > 0

(8.23)

k-4oo

for any sequence {xk} such that xk -4 oo in M, then u (x) > 0 for all x E M.

8.6. (A version of the parabolic minimum principle) Fix T E (0, +oo] and consider the manifold N = (0, T) x M. We say that a sequence {(tk,xk)}k 1 of points in N escapes from N if one of the following two alternatives takes place as k -* oc:

1. xk--+ ooinMandtk -+tE [0,T];

2. xk-4xEMand tk- 0.

Let u (t, x) E C2 (N) be a supersolution to the heat equation in N. Prove that if lim sup u (tk, xk) > 0 k-+oo

for any sequence {(tk, xk)} that escapes from N, then u (t, x) > 0 for all (t, x) E N.

(8.24)

8.4. STOCHASTIC COMPLETENESS

231

8.4. Stochastic completeness DEFINITION 8.17. A weighted manifold (M, g, p) is called stochastically complete if the heat kernel pt (x, y) satisfies the identity

IM

pt (x, y) d,u (y) = 1,

(8.25)

for all t>0and xEM. The condition (8.25) can also be stated as Ptl - 1. Note that in general we have 0 < Ptl < 1 (cf. Theorems 5.11 and 7.16). If the condition (8.25) fails, that is, Ptl 0- 1 then the manifold M is called stochastically incomplete. Our purpose here is to provide conditions for the stochastic completeness

(or incompleteness) in various terms.

8.4.1. Uniqueness for the bounded Cauchy problem. Fix 0 < T < oo, set I = (0, T) and consider the Cauchy problem in I x M

j at =A u, inIxM, u1t=o = f

(8.26)

where f is a given function from Cb (M). The problem (8.26) is understood in the classical sense, that is, u c CO° (I x M) and u (t, x) --+ f (x) locally uniformly in x E M as t --+ 0. Here we consider the question of the uniqueness of a bounded solution of (8.26). THEOREM 8.18. Fix a > 0 and T E (0, oo]. For any weighted manifold (M, g, p), the following conditions are equivalent. (a) M is stochastically complete. (b) The equation O,-,v = av in M has the only bounded non-negative solution v = 0. (c) The Cauchy problem in (0, T) x M has at most one bounded solution.

REMARK 8.19. As we will see from the proof, in condition (b) the assumption that v is non-negative can be dropped without violating the statement.

PROOF. We first assume T < oo and prove the following sequence of implications

- (b)

- (c) -, (a) , where -, means the negation of the statement. Proof of - (a) -, (b). So, we assume that M is stochastically incomplete and prove that there exists a non-zero bounded solution to the equation --A,v + av = 0. Consider the function (a)

u (t, x) = Ptl (x) =

fm

pt (x, y) dp (y) ,

8. POSITIVE SOLUTIONS

232

which is C°° smooth, 0 < u < 1 and, by the hypothesis of stochastic incompleteness, u # 1. Consider also the function 00

W (x) = Ra1(x) = J

e-atu (t, x) dt,

o

(8.27 )

which, by Theorem 8.7 and Corollary 8.6, is C°°-smooth, satisfies the estimate 0 < w < a-1 (8.28) and the equation -&, w + caw = 1. (8.29) It follows from u 0 1 that there exist x E M and t > 0 such that u (t, x) < 1.

Then (8.27) implies that, for this value of x, we have a strict inequality w (x) < 01-1. Hence, w # a-1. Finally, consider the function v = 1 - aw, which by (8.29) satisfies the equation Acv = av. It follows from (8.28) that 0 < v < 1, and w # a-1 implies v 0 0. Hence, we have constructed a non-zero non-negative bounded solution to A v = av, which finishes the proof. Proof of (b) = (c). Let v be a bounded non-zero solution to equation

A,v = av. By Corollary 7.3, v E C°° (M). Then the function u (t, x) = eats (x)

(8.30)

satisfies the heat equation because DtLu = eatLµv = aeaty = at

Hence, u solves the Cauchy problem in R+ x M with the initial condition u (0, x) = v (x), and this solution u is bounded on (0, T) x M (note that T is finite). Let us compare u (t, x) with another bounded solution to the same Cauchy problem, namely with Ptv (x). By Theorem 7.16, we have sup IPtvj < sup IvI, whereas by (8.30)

sup Ju (t, )I = eat sup lvl > sup jvj.

Therefore, u # Ptv, and the bounded Cauchy problem in (0, T) x M has two different solutions with the same initial function v. Proof of -, (c) (a). Assume that the problem (4.43) has two different bounded solutions with the same initial function. Subtracting these solutions, we obtain a non-zero bounded solution u (t, x) to (4.43) with the

initial function f = 0. Without loss of generality, we can assume that 0 < sup u < 1. Consider the function w = 1 - u, for which we have 0 < inf w < 1. The function w is a non-negative solution to the Cauchy problem (4.43) with the initial function f = 1. By Theorem 8.1 (or Corollary 8.3), we conclude that w (t, ) > Pt 1. Hence, inf PtI < 1 and M is stochastically incomplete.

8.4. STOCHASTIC COMPLETENESS

233

Finally, let us prove the equivalence of (a), (b), (c) in the case T = cc.

Since the condition (c) with T = oo is weaker than that for T < oo, it suffices to show that (c) with T = oc implies (a). Assume from the contrary

that M is stochastically incomplete, that is, Ptl 0- 1. Then the functions ul - 1 and u2 = Ptl are two different bounded solutions to the Cauchy problem (4.43) in R x M with the same initial function f - 1, so that (a) fails, which was to be proved.

Exercises. 8.7. Prove that any compact weighted manifold is stochastically complete.

8.8. Prove that R' is stochastically complete (cf. Exercise 8.11).

8.9. Prove that ifPtl(x)=lforsome t>0,xEMthen Ptl(x)=1forallt>0,xEM. 8.10. Fix a > 0. Prove that M is stochastically complete if and only if

a-1.

8.4.2. a-Super- and subharmonic functions. We prove here convenient sufficient conditions for stochastic completeness and incompleteness

in terms of the existence of certain a-super- and a-subharmonic functions (see Sections 8.3.2 for the definitions).

THEOREM 8.20. Let M be a connected weighted manifold and K C M be a compact set. Assume that, for some a > 0, there exists an asuperharmonic function v in M \ K such that v (x) -3 +oo as x -+ 002. Then M is stochastically complete.

PROOF. By enlarging K, we can assume v is defined also on aK and

that v > 0 in M \ K. Then v is also 0-superharmonic in M \ K for any a so we can assume a > 0. By Theorem 8.18, in order to prove that M is stochastically complete, it suffices to verify that any non-negative bounded solution M to the equation AAu = au is identical zero. Assume that 0 < u < 1 and set

m = maxu. K

Then, for any e > 0, we have

Ev>0>u-m on8K.

(8.31)

By hypothesis v (x) -3 +oo as x - oo and Exercise 5.18, the set {v < E-1} is relatively compact; therefore, there exists a relatively compact open set 1 C M that contains {v < e-1} and K. Compare the functions Ev and u-m in S2 \ K. By the choice of S2, we have v > e-1 on 8SZ and, consequently,

ev > 1 > u - m on 0SZ. In SZ \ K, the function ev satisfies the equation -Ou, (EV) + a (ev) = 0, 2See Definition 5.18.

(8.32)

8. POSITIVE SOLUTIONS

234

whereas

-Aµ(u-m)-I-a(u-m)=-am<0, whence it following that the function ev-(u - m) is superharmonic in 11\K. By (8.31) and (8.32), we have ev > u - m on the boundary a (SZ \ K), which

implies by Corollary 8.16 that ev > u - m in 12 \ K. Exhausting M by a sequence of sets 2, we obtain that ev > u - m holds in M \ K; finally, letting

e- 0, we obtain u < m in M \ K. Hence, m is the supremum of u on the entire manifold M, and this supremum is attained at a point in K. Since &, u = au > 0, that is, the function u is subharmonic, Corollary 8.14 implies that u - m on M. The equation Au = au then yields m = 0 and u = 0, which was to be proved.

O

THEOREM 8.21. Let M be a connected weighted manifold. Assume that there exists a non-negative superharmonic function u on M such that u coast and u E Ll (M). Then M is stochastically incomplete. PROOF. Let us first construct another non-negative superharmonic func-

tion v on M such that v E LI (M) and Dµv # 0. Fix a point x0 E M such that Vu (xo) 0 and set c = u (x0). Then the function u := min (u, c) is not differentiable at xo. Consider the function Ptu. Since u is superharmonic, we have by Exercise 7.29

Ptu
Ptu
Ptu
8.33)

Therefore,

Pt+su = P3(Ptu) < Ptu,

that is,

at

Ptu < 0.

8.34)

Since Ptii is a smooth function for any t > 0, and ii is not, we see that there

ist>0andxeMsuch that

at

Ptu (x) < 0.

(8.35)

Set v = Ptii and observe that, by (8.33), (8.34), and (8.35), v E LI (M) , -0µv > 0 and Dµv # 0.

Therefore, there exists a non-negative function f E Co (M) such that f 0 0

and -0µv > f on M. Since v E W' (M) and v satisfies for any a > 0 the inequality

-&µv -}- av > f,

8.4. STOCHASTIC COMPLETENESS

235

Theorem 8.4 yields v > Ra f a.e.(cf. Remark 8.5). Since R, ,f is smooth by Theorem 8.7, we obtain v > Ra f pointwise. Letting a -9 0 and using (8.10), we obtain, for all x E M,

f

00

v (x) >

0

Ptf (x) dt =

ff 0

M

pt (x, y) f (y) dµ (y) dt.

Integrating in x and using the condition v E L1 (M), we obtain

f ff M

M

pt (x, y) f (y) dµ (y) dtdp (x) < oo,

whence it follows by interchanging the order of integration that 00

f J However, if M is stochastically complete and, hence, Ptl = 1, this integral Pt1(y) f (y) dµ (y) dt < co.

M

should be equal to

f

( f°° t\ M f (y) d(y) dt = oo. This contradiction finishes the proof.

REMARK 8.22. The hypothesis u # const cannot be dropped because it can happen that 1 E L' (M) and M is stochastically complete, for example, if M is a compact manifold (see Exercise 8.7).

THEOREM 8.23. Assume that, for some a > 0, there exists a non-zero non-negative bounded a-subharmonic function v on M. Then M is stochastically incomplete. PROOF. By hypothesis, we have Dµv > av and, without loss of generality, we can assume that 0 < v < 1. Let {1 k} be a compact exhaustion sequence in M. Consider in each S2k the following weak Dirichlet problem

-Dµuk + auk = 0 uk = 1 mod Wo (Slk) .

(8.36)

Since 1 E W1 (92k), this problem has a unique solution uk, by Exercise 4.29 (or by Theorem 4.5). We will show that V C Uk+1 < uk < 1

(8.37)

(see Fig. 8.4), which will imply that the sequence {uk} has a limit u that,

hence, satisfies the equation 0u = au on M and v < u < 1. The latter means that u is bounded but non-zero, which implies by Theorem 8.18 that M is stochastically incomplete. To prove (8.37), observe that function v belongs to W1 (&2k) and obviously satisfies the conditions

-0µv + av < 0,

v<1.

(8-38)

8. POSITIVE SOLUTIONS

236

FIGURE 8.4. Functions v, uk, uk+l

Comparing (8.36) and (8.38) and using Corollary 5.14, we conclude v < uk. Since the constant function 1 satisfies the condition

-L h1 + al > 0, the comparison with (8.36) shows that Uk < 1. Of course, the same applies also to Uk+l. Noticing that uk+1 satisfies in S2k the conditions -& Uk+1 + auk+1 = 0, 1 uk+1 <_ 1,

and comparing them to (8.36), we obtain Uk+1 < uk, which finishes the proof.

Exercises. 8.11. Prove the following claims.

(a) 118" is stochastically complete for all n > 1. (cf. Exercise 8.8).

(b) 1R' \ {0} is stochastically complete if n > 2, whereas 1[81 \ {O} is stochastically incomplete. (c) Any open set f2 C ]R such that n 54 R', is stochastically incomplete.

8.12. Let Sl be an open subset of 1IV and h be a positive smooth function in ) such that

Ah=0in0, h (x) = e°(I-I) as jxj -+ oo

Prove that P°h = h for all t > 0. 8.13. Let f be a non-negative superharmonic function on M. (a) Prove that the function v (x) . tlm Pt f (x) +00

(8.39)

satisfies the identity Ptv = v for all t > 0 and, hence, is harmonic on M (the limit in (8.39) exists and is finite because by Exercise 7.29 the function Pt f (x) is finite and decreases in t). (b) Assume in addition that manifold M is stochastically complete and f is bounded. Prove that, for any non-negative harmonic function h on M, the condition h < f implies h < v.

8.4. STOCHASTIC COMPLETENESS

237

REMARK. The maximal non-negative harmonic function that is bounded by f is called the

largest harmonic minorant of f. Hence, the function v is the largest harmonic minorant of f. 8.14. Set v (x) = limt..r Ptl (x). Prove that either v - 0 or sup v = 1. Prove also that either v - 1 or inf v = 0.

8.15. Let SI be the exterior of the unit ball in R', n > 2. Evaluate limt, Pt 1 (x).

8.4.3. Model. manifolds. Let (M, g, E.i) be a weighted model based on M = ][8' as it was defined in Section 3.10. This means that, in the polar coordinates (r, 0) in R', the metric g and measure p are expressed as follows: g = dr2 + b (r) 2 gsn-1,

(8.40)

where V) (r) is a smooth positive function on (0, +oo), and dp. = T (r)

(r)'-1 drd0,

where dO is the Riemannian measure on function on (0, +oo). Recall that

(8.41)

S'z-1 and T (r) is a smooth positive

S (r) := w,zT (r) 2bn'-1 (r)

is the area function of M, and

V(Br)=fs(t)dt is the volume function of M. By (3.93), the weighted Laplace operator of (M, g, y) has in the polar coordinates the following form:

0A _ 3r2 + S

)

ar +

y2

1r)

(8.42)

It is important to observe that, away from a neighborhood of 0, the functions

(r) and T (r) can be chosen arbitrarily as long as they are smooth and positive. Near 0 some care should be taken to ensure that the metric g, defined by (8.40), extends smoothly to the origin. If 0 (r) and T (r) are prescribed for large r then it is always possible to extend them to all r > 0 so that 0 (r) = r and T (r) = 1 for small enough r. This ensures that the metric and measure in a neighborhood of the origin are exactly Euclidean and, hence, can be extended to the origin. It follows from this observation that any function S (r) can serve as the area function for large r, as long as S (r) is smooth and positive. Furthermore, setting T 1, we can realize S (r) as the area function of a Riemannian model.

Our main result here is the following criterion of the stochastic completeness of the model manifold.

8. POSITIVE SOLUTIONS

238

THEOREM 8.24. The weighted model (M, g, u) as above is stochastically complete if and only if O° V (r)

J

S (r)

dr = oo.

(8.43)

For example, if V (r) = exp r' for large r then M is stochastically com-

plete if a < 2 and incomplete if a > 2. The manifolds R' and En with their canonical metrics satisfy (8.43) because S (r) = wnrn-1 for Rn and S (r) = w, sinhn-1 r for IH[n (see Section 3.10). Hence, both R and Hn are stochastically complete. Note that 5n is stochastically complete by Exercise 8.7.

PROOF. Let us show that (8.43) implies the stochastic completeness of M. By Theorem 8.20, it suffices to construct a 1-superharmonic function v = v (r) in the domain {r > 1} such that v (r) -> +oo as r -* oo. In fact, we construct v as a solution to the equation AAv = v, which in the polar coordinates has the form

v+v'-v=0. " -

(8.44)

So, let v be the solution of the ordinary differential equation (8.44) on [1, +oc) with the initial values v(1) = 1 and v'(1) = 0. The function v(r) is monotone increasing because the equation (8.44) after multiplying by Sv and integrating from 1 to R, amounts to /R

Svv'(R) = J S (v12 + v2) dr > 0. 1

Hence, we have v _> 1.

Multiplying (8.44) by S, we obtain

(Sv')' = Sv, which implies by two integrationsf

v(R) = 1+ I R S(r)

f I r S(t)v(t)dt.

Using v > 1 in the right hand side, we obtain, for R > 2, JR (V(r) - V(1))dr > c J2R V(r)dr v(R) > R dr JRS(t)dt S(r) S(r) S(r)

-

--

-

where c = 1 - V(2) > 0. Finally, (8.43) implies v (R) --+ oo as R - oo. Now we assume that

S (r) dr < oo,

(8.45)

and prove that M is stochastically incomplete. By Theorem 8.21, it suffices to construct on M a non-negative function u E L1 (M) such that -AY U = f, (8.46)

8.4. STOCHASTIC COMPLETENESS

239

where where f E Co (M), f > 0 and f 0 0. Both functions u and f will depend only on r so that (8.46) in the domain of the polar coordinates becomes U11+

S

u'= f.

(8.47)

Choose f (r) to be any non-negative non-zero function from Co (1, 2), and set, for any R > 0, 00

U (R) = IR

dr jor S (t) f (t) dt. S (r)

(8.48)

Since f is bounded, the condition (8.43) implies that u is finite. It is easy to see that u satisfies the equation

(Su')' = -Sf, which is equivalent to (8.47). The function u (R) is constant on the interval

0 < R < 1 because f (t) - 0 for 0 < t < 1. Hence, u extends by continuity to the origin and satisfies (8.46) on the whole manifold. We are left to verify that u E Ll (M). Since f (t) - 0 for t > 2, we have

forR>2 u (R) = C

/r°°

dr

JR S (r)

where C = fo S (t) f (t) dt. Therefore,

f

00

udtC = R>2}

u (R) S (R) dR

J2

dr C J `0 (fR S (r) 00

/

S (R) dR

= C /2'f

(f' S (R) dR)// Sdr(r)


s

J

dr < oo,

which gives u E L1 (M). EXAMPLE 8.25. Let us show that, for any continuous positive increasing

function F (r) on (0, +oo) such that F (r) -+ oo as r -+ oo, there exists a stochastically complete model M for which

F (r) - 1 < V (r) < F (r)

(8.49)

,

for large enough r. Indeed, for large r, the volume function V (r) of a weighted model M may be any smooth positive increasing function. Choose first any such function V (r) satisfying

F (r) - 1/2 < V (r) < F (r)

,

(8.50)

240

8. POSITIVE SOLUTIONS

and then modify V (r) as follows. Select a sequence of disjoint intervals (ak, bk) such that ak ---+ oC and

F (bk) = F (ak) + 1/2. (8.51) Now reduce V (r) on each interval (ak, bk) to create inside (ak, bk) the points with very small derivative V' (r) while keeping the values of V (r) at the ends and the monotonicity, which together with (8.50) and (8.51) will ensure (8.49) for the modified V (see Fig. 8.5).

ak

bk

r

FIGURE 8.5. Modification of function V (r) on the interval (ak, bk).

By doing so, we can make the value of integral fak v ) dr arbitrarily large, say, larger than 1, which implies ao

J

V,()dr = oo. r

Therefore, M is stochastically complete by Theorem 8.24.

Exercises. 8.16. (A -model with two ends) Set M = R x S'-1 (where n > 1) so that every point x E M can be represented as a couple (r, 0) where r E R and 0 E S"-1. Fix smooth positive functions 0 (r) and T (r) on II8, and consider the Riemannian metric on M g = dr2 + y)2 (r) gsn-1,

and measure µ on (M, g) with the density function T. Define the area function S (r) by S (r) = wnT (r) on-1 (r) and volume function V (R) by

V (R) = f

S (r) dr,

[o,R]

so that V (R) > 0. (a) Show that the expression (8.42) for Du remains true in this setting.

NOTES

241

(b) Prove that if function V (r) is even then the following are equivalent: (i) (M, g, µ) is stochastically complete. (ii) There is a non-constant non-negative harmonic function u E L' (M, µ). (r) dr = oo. (iii) f o VV(r) (c) Let S (r) satisfy the following relations for some a > 2: S (r)

-{

eexp (ral)

xp

la), r < 1 1.

Prove that (M, g,,u) is stochastically incomplete. Prove that any non-negative harmonic function u E L' (M,,u) is identical zero.

Notes The material of Sections 8.1 and 8.2 extends the Markovian properties of Chapter 5. The proofs of the parabolic maximum/minimum principles (Theorems 8.10 and 8.11) are taken from [243]. Theorems 8.18, 8.20, 8.23 are due to Khas'minskii [223] (see also [93], [155]), Theorems 8.21, 8.24 were proved in [142] (see also [155]).

CHAPTER 9

Heat kernel as a fundamental solution Recall that the heat kernel was introduced in Chapter 7 as the integral kernel of the heat semigroup. Here we prove that the heat kernel can be characterized as the minimal positive fundamental solution of the heat equation. This equivalent definition of the heat kernel is frequently useful in applications.

9.1. Fundamental solutions DEFINITION 9.1. Any smooth function u on R+ x M satisfying the following conditions 1

at = Ofu

in 1+ x M, by

9.1

as t--0,

is called a fundamental solution to the heat equation at the point y. If in addition u > 0 and, for all t > 0, 1,

(9.2)

then u is called a regular fundamental solution. As it follows from Theorem 7.13, the heat kernel pt (x, y) as a function of t, x is a regular fundamental solution at y. The following elementary lemma is frequently useful for checking that a given solution to the heat equation is a regular fundamental solution. LEMMA 9.2. Let u (t, x) be a smooth non-negative function on R+ x M satisfying (9.2). Then the following conditions are equivalent: (a)

-*by ast-+0.

(b) For any open set U containing y,

as t-}0.

(9.3)

(c) For any f E Cb (M), fd/-t -4 f (y) as t-++0. 243

(9.4)

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

244

PROOF. The implication (c) = (a) is trivial because u (t, ) J. is equivalent to (9.4) for all f E D (M). (a) = (b). Let f c D (U) be a cutoff function of the set {y} in U. Then (9.4) holds for this f. Since f (y) = 1 and

f

Mu(t,fdu<

fuu(t, )dl-t<1,

(9.3) follows from (9.4).

(b) = (c) . For any open set U containing y, we have

fM

u (t, x) f (x) dµ (x)

=

f

M\U u

(t, x) f (x) dp (x)

+ J u (t, x) (f (x) - f (y)) dp (x) U f +f (y) u (t, x) dp (x) JU The last term here tends to f (y) by (9.3). The other terms are estimated as follows:

fM\U

u (t, x) f (x) dp <supIfI

fM\uu(t,x)dµ(x)

(9.5)

and

fu(t,x)(f(x)_f(Y))dsup I.f (x) - f (y)I f u (tx) d(x) XEU U

sup If (x) - f (y)I

xEU

(9.6)

Obviously, the right hand side of (9.5) tends to 0 as t --} 0 due to (9.2) and (9.3). By the continuity of f at y, the right hand side of (9.6) can be made arbitrarily small uniformly in t by choosing U to be a small enough neighborhood of y. Combining the above three lines, we obtain (9.4).

O

REMARK 9.3. As we see from the last part of the proof, (9.4), in fact, holds for arbitrary f E L°° (M) provided f is continuous at the point y.

The next lemma is needed for the proof of main result of this section Theorem 9.5.

LEMMA 9.4. Let u (t, x) be a non-negative smooth function on R+ x M such that u (t, ) - ay as t -4 0. Then, for any open set f C= M and any

f E Cb (f),

L. as t--30.

u (t, x) f (x) dµ (x) -->

j f (y), l 0,

if y E 0,

ifyEM\S2,

(9.7)

9.1. FUNDAMENTAL SOLUTIONS

245

REMARK. Extend f to M by setting f = 0 outside Q. Then in the both cases in (9.7), f is continuous at y. Hence, (9.7) follows from Lemma 9.2 and Remark 9.3, provided u satisfies (9.2). However, we need (9.7) without the hypothesis (9.2), which requires a more elaborate argument.

PROOF. If f E D (S2) then, by hypothesis u (t, ) which implies (9.7).

6y, we have (9.4),

Representing f E Cb (Il) as f = f+ - f_, it suffices to prove (9.7) for non-negative f . By scaling f, we can assume without loss of generality that

O< f <1. Let us first prove (9.7) for y E M\n. Let b E D (M) be a cutoff function of SZ in M \ {y} (see Fig. 9.1).

FIGURE 9.1. Functions f and

.

Then f
fu(t,.)d(y)=0as t

0,

Z

which implies

f u (t, f dp

0.

Assume now y E S2 and set f (y) = a. By the continuity of f at y, for any E > 0 there exists an open neighborhood U C= SZ of y such that

a-E< f
so that

(a-E)cp< f <(a+E)z/5 in U.

y FIGURE 9.2. Functions f, cp, 0.

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

246

have

Therefore,

fd<(a+E)fu (t,dµ.

fu (a - e)Ju U

Passing to the limit as tf-+ 0, we obtain lim sup t--+O

JU

u (t, )fdµ < (a + E) V) (y) = a +

(9.8)

and

liminf J u (t, ) fdµ > (a - e) cp (y) = a - E. t->o

Let us show that

(9.9)

U

f

f

f dµ-+0.

(9.10)

Indeed, let V C U be an open neighborhood of y. Since P \ V is a relatively

compact open set and y 0 sZ \ V, we obtain by the first part of the proof that 0,

whence (9.10) follows.

Finally, combining together (9.8), (9.9), (9.10) and letting E -3 0, we obtain fdµ-*a= f (y) ast--30. The next theorem provides a characterization of the heat kernel, which can serve as an alternative definition. THEOREM 9.5. For any y E M, the heat kernel pt (x, y) is the minimal non-negative fundamental solution of the heat equation at y. PROOF. Let u (t, x) be another non-negative fundamental solution at y, and fix s > 0. The function t, x H u (t + s, x) satisfies the heat equation in R+ x M and, hence, u (t + s, x) can be considered as a non-negative solution to the Cauchy problem in R+ x M with the initial function f (x) = u (s, x). Since u is a smooth function, we have f E L210c (M) and 2

u(t+s,) L-` f ast-->0. all t > 0 and x E M, By Theorem 8.1, we conclude that, for all (9.11) u (t + s, x) > Pt f (x) = fMPt (x, z) u (s, z) dµ (z) . Fix now t > 0, x E M and choose an open set SZ C M containing y. Then

pt (x, ) E Cb (S2) and, by Lemma 9.4, in pt (x, z) a (s, z) dµ (z) --+ pt (x, y) ass ---* 0.

247

9.1. FUNDAMENTAL SOLUTIONS

Hence, letting s -4 0 in (9.11), we obtain u (t, x) > pt (x, y), which was to be proved.

COROLLARY 9.6. Let (M, g, µ) be a stochastically complete weighted manifold. If u (t, x) is a regular fundamental solution at a point y E M,

then u (t, x) - pt (x, y). PROOF. By Theorem 9.5, we have u (t, x) > pt (x, y), which implies

1 > f u (t, x) dp (x) > J pt (x, y) dp (x) = 1, M

M

where in the last part we have used the stochastic completeness of M. We conclude that all the inequalities above are actually equalities, which is only possible when u (t, x) = pt (x, y). The next theorem helps establishing the identity of a fundamental solution and the heat kernel using the "boundary condition", which may be useful in the case of stochastic incompleteness. THEOREM 9.7. Let u (t, x) be a non-negative fundamental solution to the

heat equation at y E M. If u (t, x) 0 as x -f oc where the convergence is uniform in t E (0,T) for any T > 0, then u (t, x) = pt (x, y). REMARK 9.8. The hypothesis that u (t, x) is non-negative can be relaxed

to the assumption that lim sup u (tk, xk) > 0, k-oo

(9.12)

for any sequence (tk, xk) such that tk -+ 0 and xk -4 x E M. Indeed, by the maximum principle of Exercise 8.6, (9.12) together with the other hypotheses implies u > 0. PROOF. By Theorem 9.5, we have u (t, x) > pt (x, y) so that we only need to prove the opposite inequality. Fix some s > 0 and notice that the function v (t, x) = u (t + s, x) solves the heat equation with the initial function f (x) = v (0, x) = u (s, x). Since v (t, x) = 0 as x --> oo, we obtain by Corollary 8.2 that v (t, x) = Ptf (x) that is, u (t + s, x) = Ptf (x) = fM pt (x, ) u (s, ) dp.

Let {Slk}k1 be a compact exhaustion sequence in M. For any k, we have u (t + s, x)

- in

pt (x, ) u (s, ) dp =

SZ k

fM\Slk

pt (x, ) u (s, ) dg

sup a (s, ) fmn\Qk pt (x, ) dµ. M\Slk

Since the total integral of the heat kernel is bounded by 1 and sup u (s, z) -+ 0 as k --+ oo, sup sE(O,T) zEM\Slk

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

248

we see that, for any E > 0 there is k so big that, for all s c (0, T),

u (t + s, x) -

pt (x, ) u (s, ) d < a.

Letting here s -* 0 and applying Lemma 9.4 in 92k with function f = pt (x, ), we obtain

U(t,x)-Pt(x,y) <E. Since E > 0 is arbitrary, we conclude u (t, x) < pt (x, y), which was to be proved.

EXAMPLE 9.9. By Lemma 1.1, the Gauss-Weierstrass function

Pt (x, y) _ (47rt )n/2 exp -'x 4tyl

(9.13)

is a regular fundamental solution of the heat equation in R. Since Rn is stochastically complete, we conclude by Corollary 9.6 that pt (x, y) is the heat kernel in R n. Alternatively, this can be concluded by Theorem 9.7 because pt (x, y) --3 0 as x -3 oo uniformly in t (cf. Exercise 1.5). Yet another proof of the fact that pt (x, y) is the heat kernel in ' was given in Example 4.12.

EXAMPLE 9.10. For any weighted manifold (M, g, /2) and an open set 92 C M, denote as before by pt (x, y) the heat kernel in (92, g, it). Extend pt (x, y) to all x, y c M by setting it to 0 if either x or y is outside Q. Let us show that if 01 and 922 are two disjoint open sets and 92 = Q1 U 522 then

Pt = pt' + pt2

.

(9.14)

Indeed, if y c 921 then pt ' (x, y) is a fundamental solution not only in 921 but also in 92 because, being identical zero in i2i it satisfies the heat equation also in 922. Therefore, by the minimality of pt, we conclude pt (x, y) < pt"' (X, Y) On the other hand, since 921 C 92, we have the opposite inequality by Exercise 7.40. Hence, -

Pt (x, y) = At" (x, y) , which implies (9.14) because p0t2 (x, y) = 0. Similarly, (9.14) holds if y E 922.

The identity (9.14) implies that pt (x, y) = 0 if the points x, y belong to different connected components of M (assuming, of course, that M is disconnected). Indeed, if x is contained in a connected component 121 and 922 := M \ 01 then y E 922 and, hence, pt' (x, y) = 0 for i = 1, 2, whence Pt (x, y) = A" (x, y) +p2 (x, y) = 0.

9.2. Some examples We give here some examples of application of the techniques developed in the previous sections.

9.2. SOME EXAMPLES

249

9.2.1. Heat kernels on products. Let (X, gx, µx) and (Y, gy, µy) be two weighted manifold and (M, g, t) is their direct (see Section 3.8). Denote by pX and pt the heat kernels on X and Y, respectively. THEOREM 9.11. Assume that (M, g, p) is stochastically complete. Then the heat kernel pt on M satisfies the identity

pt ((x, y) , W, y')) = pt (x, x') pt (y, y')

,

(9.15)

for alit >0 andx,x' E X, y,y' E Y. The statement is true without the hypothesis of stochastic completeness, but the proof of that requires a different argument (see Exercise 7.41).

PROOF. Denote by OX and Ay the Laplace operators on X and Y, respectively. Then the Laplace operator 0,,, on M is given by

Duu = Oxu + Dyu, where u (x, y) is a smooth function on M, and OX acts on the variable x, Ay acts on the variable y (see Section 3.8). Fix (x', y') E M and prove that the function

u(t, (X, Y)) =pt (x,x')pt (y,y') is a regular fundamental solution at (x', y'), which will imply (9.15) by Corollary 9.6. Indeed, the heat equation for u is verified as follows:

at

at pt (x,

x') pt (y, y') + pt, (x, x')

pt (y, y')

oxpX (x, x') pt (y,) + Pt (x, x') Oypt (y, y') (Ax + Dy) u = AMU.

The integral of u is evaluated by

f

M

u (t, ) dp =

f MpX (., x') dµx JMpt (', y') d/ty < 1.

To check the condition (9.3) of Lemma 9.2, it suffices to take the set U C M

in the form U = V x W where V C X and W C Y. If (x', y') E U then x' E V and y' E W, which implies

fu u

(t,) dp =

pX

x') dlux fW pt (', y') dµy -+ 1

V

as t -4 0. Hence, by Lemma 9.2, u is a regular fundamental solution at (x', y').

11

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

250

9.2.2. Heat kernels and isometries. We use here the notion of isometry of weighted manifolds introduced in Section 3.12. THEOREM 9.12. Let J : M -* M be an isometry of a weighted manifold (M, g, .t). Then the heat kernel of M is J-invariant, that is, for all t > 0 and X, y E M,

Pt (Jx, Jy) = Pt (x, y) See also Exercise 7.24 for an alternative proof.

(9.16)

PROOF. Let us first show that the function u (t, x) = pt (Jx, Jy) is a fundamental solution at y. Indeed, by Lemma 3.27, for any smooth function

f on M,

(0µf) (Jx) = Dµ (f (Jx)) Applying this for f = pt au

Jy), we obtain

a

at Pt (Jx, Jy) = (D&Pt) (Jx, Jy) = Dµu, so that u solves the heat equation. By Lemma 3.27, we have the identity at

f (x) dµ (x) , fm for any integrable function f. Hence, for any co E D (M),

f M f (Jx) dp (x) =

(9.17)

J u(t,x),p(x)dµ(x) = J pt(Jx,Jy)cp(x)dp(x) M

M

= fMPt (x, Jy) w (J-1x) da (x)

.

Since pt (x, Jy) is a fundamental solution at Jy, the last integral converges

as t-+0to

(gyp o J-1)

(Jy) = co (y)

,

which proves that u (t, x) by. Hence, u (t, x) is a non-negative fundamental solution at y, which implies by Theorem 9.5

u(t,x) ?Pt(x,y), that is,

Pt(Jx,Jy) : Pt(x,y) Applying the same argument to J-1 instead of J, we obtain the opposite inequality, which finishes the proof.

EXAMPLE 9.13. By Exercise 3.46, for any four points x, y, x', y' E H7' such that d (x', y') = d (x, y),

there exists a Riemannian isometry J : El -4 H1 such that Jx' = x and Jy' = y. By Theorem 9.12, we conclude

Pt(x,y) =Pt(x,y)

9.2. SOME EXAMPLES

251

Hence, pt (x, y), as a function of x, y, depends only on the distance d (x, y).

9.2.3. Heat kernel on model manifolds. Let (M, g, p) be a weighed model as in Sections 3.10 and 8.4.3. That is, M is either a ball B,0 = {IxI < ro} in R7L or M = R7b (in this case ro = oo), and the metric g and the density function of p depend only on the polar radius. Let S (r) be the area function of (M, g, p) and pt (x, y) be the heat kernel. Let (M, g, µ) be another weighted model based on the same smooth manifold M, and let S (r) and pt (x, y) be its area function and heat kernel, respectively.

THEOREM 9.14. If S (r) - S (r) then pt (x, 0) = pt (y, 0) for all x, y E M such that jxj = !yL.

Note that the area function S (r) does not fully identify the structure of the weighted model unless the latter is a Riemannian model. Nevertheless, A (x, 0) is completely determined by this function. PROOF. Let us first show that pt (x, 0) = pt (y, 0) if I xI = lyI I. Indeed,

there is a rotation J of Il87L such that Jx = Jy and JO = 0. Since j is an isometry of (M, g, p), we obtain by Theorem 9.12 that pt is J-invariant, which implies the claim. By Lemma 9.2, the fact that a smooth non-negative function u (t, x) on I[8+ x M is a regular fundamental solution at 0, is equivalent to the conditions 8u

8t fM

u (t, x) dp (x) < 1,

(9.18)

fu(t,x)dP(x)1 ast0, E

for all 0 < e < ro. The heat kernel pt (x, 0) is a regular fundamental solution on (M, g, p) at the point 0, and it depends only on t and r = jxj so that we can write pt (x, 0) = u (t, r). Using the fact that u does not depend on the polar angle, we obtain from (3.93)

82u S' (r) 8u Auu-are+S(r)8r

For 0 < E < ro, we have by (3.86), (3.88), (3.91)

Judp = n f 6 fn 1 it (t, r) S (r) dOdr =

Be

f

it (t, r) S (r) dr.

252

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

Hence, we obtain the following equivalent form of (9.18):

au _ 02u

S' (r) au

ar2 + S (r) 8r'

at ro

u (t, r) S (r) dr < 1,

J

(9.19)

o

fu(t,r)S(r)dr1 ast0. Since by hypothesis S (r) = S (r), the conditions (9.19) are satisfied also with S replaced by S, which means that u (t, r) is a regular fundamental solution at 0 also on the manifold (M, R, ;u). By Theorem 9.5, we conclude that u (t, I xl) > pt (x, 0), that is, pt (x, 0) ? At (x, 0)

The opposite inequality follows in the same way by switching pt and pt, which finishes the proof.

9.2.4. Heat kernel and change of measure. Let (M, g, h) be a weighted manifold. Any smooth positive function h on M determines a new measure µ on M by dµ = h2dp, (9.20) and, hence, a new weighted manifold (M, g, µ). Denote by Pt and pt respectively the heat semigroup and the heat kernel on (M, g, µ). THEOREM 9.15. Let h be a smooth positive function on M that satisfies the equation

&,h +ah=0,

(9.21)

where a is a real constant. Then the following identities holds C1 _

o (0t, + aid) o h,

Pt=eat1 oPtoh, at pt (x7 y)

pt ( x1 2J ) = e h (x) h (y)

(9.22) (9.23)

(

9 24) .

for all t > 0 and x, y E M. In (9.22) and (9.23), h and h are regarded as multiplication operators, the domain of the operators in (9.22) is CO° (M), and the domain of the operators in (9.23) is L2 (M, µ). The change of measure (9.20) satisfying (9.21) and the associated change of operator (9.22) are referred to as Doob's h-transform.

253

9.2. SOME EXAMPLES

PROOF. By the definition of the weighted Laplace operator (see Section 3.6), we obtain, for any smooth function f on M,

Of=

2

diva,(h2V f) = diva,(V f) + h2 (Vh2, Vf)g

+2(Oh,Vf)g

AI.f

(9.25)

On the other hand, using the equation (9.21) and the product rule for t (cf. Exercise 3.8) and (9.21), we obtain

1 A,. (hf) = h (h0,,,f + 2(Vh, Vf)g + f 0jh) +2(Vh,Vf)g+

OIf

f0ph

= Oj,f - af. Hence, we have the identity

O-f = A,, (hf) + af,

(9.26)

which is equivalent to (9.22). Next, fix a point y E M, set

u (t, x) = eat ht)

h (y)

and show that u (t, x) is a fundamental solution on (M, g, i) at point y. Using (9.26), we obtain

au (x, y) - au + eat ata pt h (x) h (y) at eat

au +

h (x) h (y)

= au + h (x) 1 `

Oot (x, y) h (x) eat pt (x, y) h (x) h (y)

= au + h A,, (hu) = au + ATu - au = D;,u, so that u solves the heat equation on (M, g'µ)For any function co E D (M), we have fM

u (t, x) cp (x) dµ (x) = f

r

eat pt (x' y) h (x) cp (x) dy (x) h (y)

Cet

h (y) Pt

(y).

Since h (x) cp (x) ED (M), we have Pt (h(P) (y) -4 hcp (y) as t -+ 0,

(9.27)

254

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

whence it follows that fM u (t, x) o (x) dl (x)

and, hence, u (t, )

jo(y)

Sy on (M, g, fit).

Therefore, u is a non-negative fundamental solution on (M, g, E,c) at point

y and, by Theorem 9.5, we conclude that u (t, x) > pt (x, y), that is, eat pt (x, y) h (x) h (y)

> At (x, y)

(9.28)

To prove the opposite inequality, observe that the function h := h satisfies the equation

ah = 0, which follows from (9.26) because hh = 1 and

Aj,h =

cah = ah.

Switching the roles of /,t and /c, replacing a by -a and h by h, we obtain by the above argument e

-«t pt (x, y) h (x) h (y)

> Pt (x, Y),

which is exactly the opposite inequality in (9.28). Finally, using (9.20) and (9.24), we obtain, for any f e L2 (M, /2), Pt f

Im A (x, y) f (y) dµ (y)

f

eat pt (x) y) h (x) h (y)

M

f (y) h2 (y) dµ (y)

1

eat - Pt (f h) ,

whence (9.23) follows. EXAMPLE 9.16. The heat kernel in (RTh, gRn, p) with the Lebesgue mea-

sure µ is given by

A (x, y) _

)n/2 exp - I x 4t

(9.29)

(4-7rt

Let h be any positive smooth function on Rn that determines a new measure µ on RTh by dµ = h2 dµ. Then we have 0A = d2 and

0- =

d2 dx2

c

+2

d

h dx

(cf. (9.25)). The equation (9.21) becomes

h"+ah=0,

() 9.30

9.2. SOME EXAMPLES

255

which is satisfied, for example, if h (x) = cosh,(3x and a we have by (9.30)

In this case,

2

Du = d +2,Qcoth,3xd-. 22

By Theorem 9.15, we obtain eat pt (x, y) h (x) h (y)

pt (x) y)

-

1

1

(41rt)"2 cosh Ox cosh y

exp

lx-yl`

-2t

4t

EXAMPLE 9.17. Using the notation of the previous Example, observe that the heat kernel (9.29) is a regular fundamental solution also on (M, gRn, p)

where M := R' \ {0}. If n > 2 then by Exercise 8.11, M is stochastically complete, which implies by Corollary 9.6 that pt is the heat kernel also in M. Assuming n > 2, consider the following function h (x) = 1 x

12-n

,

that is harmonic in M (cf. Exercise 3.24). Hence, (9.21) holds for this function with a = 0. Defining measure µ by dE = h2d1.c, we obtain by Theorem 9.15 that the heat kernel in (M, g, µ) is given by

ll-2 n

pt (x, y) _ (4

lyln-2

exp

Ix _ 12 4t y

EXAMPLE 9.18. Consider in R' measure u is given by dp = ex2dx,

where dx is the Lebesgue measure. Then, by (9.30) with h = e2x2, 2

a = dx2 + 2x dx .

(9.31)

We claim that the heat kernel pt (x, y) of (R, gR, µ) is given by the explicit formula:

(2xYe2t_x2 _ pt (x) y) =

-

(9.32) t J1 e 4t which is a modification of the Mehler kernel (cf. Exercise 11.18). It is a matter of a routine (but hideous) computation to verify that the function ,

1

(27r sink 2t)

(9.32) does solve the heat equation and satisfy the conditions of Lemma 9.2, which implies that is it a regular fundamental solution. It is easy to see that x_4tyl 2

Pt (x, y) <-

exp

1

-t

,

which implies that pt (x) y) -+ 0 as x - oc uniformly in t (cf. Exercise 1.5). Hence, we conclude by Theorem 9.7 that pt (x, y) is indeed the heat kernel.

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

256

Alternatively, this follows from Corollary 9.6 provided we know that the manifold (R, gR, p) is stochastically complete. To prove the latter, observe that (R, gR, p) fits the description of "a model with two ends" of Exercise 8.16. Its area function is S (r) = ere, and the volume function

V (r) =

ex2dx

J [O,rl

is even and satisfies the condition

+°°V(r)dr-oo, S (r)

because

2r as r -4 +oo.

S

Hence, by Exercise 8.16, (R, gg, ,u) is stochastically complete. Alternatively,

this conclusion follows also from Theorem 11.8, which will be proved in Chapter 11, because 1R is geodesically complete and, for large r, V (r) < exp (Cr2). EXAMPLE 9.19. Continuing the previous example, it easily follows from (9.31) that function e-x2

h (x) =

satisfies the equation

&h+2h=0.

Clearly, the change of measure dµ = h2dµ is equivalent to e-,2

d=

dx.

By Theorem 9.15 and (9.32), we obtain that the heat kernel pt of (R, gR, µ) is given by

pt (x) y) =

e2t

x

h (x) h (y) = pt (x, y) eXP (x2 + 2xye-2t

_1

(27r sinh2t) "2

exp

y2 + 2t)

- (x2 + y2) e-4t + 1 - e-4t

t

9.2.5. Heat kernel in 1H13. As was shown in Example 9.13, the heat kernel pt (x, y) in the hyperbolic space IEI[' is a function of r = d (x, y) and t. The following formulas for pt (x, y) are known: if n = 2m + 1 then

(-1) m

1

Pt (x1 y) _ (27r)' (47rt)1/2

and if n = 2m + 2 then _ (-1)m'_/ V Pt (x' y)

(29r)m (4irt)

e

2mf1 2 t 4

a

(sinhr 8r

am ( sinh rOr )

e

_m2t_r2 4t

,

(9.33)

Se- 4t ds

1

r

(cosh s - cosh r)2 (9.34)

9.2. SOME EXAMPLES

257

In particular, the heat kernel in 12 is given by _1t e

pt ( )

4

se-4tds

I

1,

(cosh s - cosh r) 2

9.35 (

)

and the heat kernel in 1H13 is expressed by a particularly simple formula 1

Pt (X, y) =

r

(4,,t) 3/2 sink r

r2 exp - 4t t

.

(

(9.36)

Of course, once the formula is known, one can prove it by checking that it is a regular fundamental solution, and then applying Corollary 9.6, because H is stochastically complete. We will give here a non-computational proof of (9.36), which to some extend also explains why the heat kernel has this form. Rename y by o and let (r, 0) be the polar coordinates (r, 0) in i3 \ {o}. By means of the polar coordinates, 13 can be identified with ii and considered as a model (see Sections 3.10 and 9.2.3). The area function of 1H13 is given by S (r) = 4ir sinh2 r,

and the Laplacian in the polar coordinates is as follows: 8 a2 + 2 coth r OHs = + 1 OS2 Ore

Or

sinh2 r

.

(9.37)

Denote by p the Riemannian measure of IE113. For a smooth positive function h on IEL3, depending only on r, consider the weighted model (]H[3, /.t) where dµ = h2dp. The area function of (IH[3, µ) is given by

S (r) = h2 (r) S (r) . Choose function h as follows:

r sinhr'

so that S (r) = Orr2 is equal to the area function of 1183. By a miraculous coincidence, function h happens to satisfy in H3 \ {o} the equation Ooh + h = 0, (9.38) which follows from (9.37) by a straightforward computation. The function h extends by continuity to the origin o by setting h (o) = 1. In fact, the extended function is smooth' in TEL3, which can be seen, for example, by representing h in another coordinate system (cf. Exercise 3.23). Denoting by pt the heat kernel of (JHI3, E:t), we obtain by Theorem 9.15

that pt (x, y) =

etpt (x, y) h (x) h (y)

(9.39)

'It is true for any weighted manifold of dimension n > 2 that any bounded solution to (9.38) in a punctured neighborhood of a point extends smoothly to this point, but we do not prove this result here.

9. HEAT KERNEL AS A FUNDAMENTAL SOLUTION

258

Since the area functions of the weighted models ()E3[3,µ) and R3 are the same,

we conclude by Theorem 9.14 that their heat kernels at the origin are the same, that is 2

pt(x,o)=

13/2exp(-r4t)

(47rt)

Combining with (9.39), we obtain r2

pt(x,o) =e-tpt(x,o)h(x)h(o) _ (47rt)3/2 sinh r exp ( 4t - t which was to be proved.

Exercises. 9.1. Let t be a measure in R" defined by

dµ= where dx is the Lebesgue measure and c is a constant vector from '. Prove that the heat kernel of (1R', gR' t) is given by ,

pt (x, y) _

1

(47rt)

exp (_.c (x + y) - Icl2 t -

Ix - y12

4t

(9.40)

9.2. (Heat kernel in half-space) Let

M={(x...... xn) ERn:xn>0}. Prove that the heat kernel of M with the canonical Euclidean metric and the Lebesgue measure is given by

pt (x, y) _ (4 In/2 (exp

(12)

at

where y is the reflection of y

_

z

- exp (- Lx4tyl ))

(9.41)

the hyperplane x" = 0, that is,

y = (y1, ... , 9.3. (Heat kernel in Weyl's chamber) Let

yn-1 -yn)

.

<x2 <... <xn}.

M={(x1,...,xn) E E

Prove that the heat kernel of M with the canonical Euclidean metric and the Lebesgue measure is given by lln

pt (x, y) = det where

(9.42)

(xt' yj

is the heat kernel in

9.4. Let (M, g, µ) be a weighted manifold, and let h be a smooth positive function on M satisfying the equation

-D,,h+0h=0,

(9.43)

where 4' is a smooth function on M. Define measure µ on M by dµ = h2dµ. (a) Prove that, for any f E C°° (M),

A,,f - Of = h0;j

(h-1 f)

.

(9.44)

o.

(9.45)

(b) Prove that, for any f E D (M),

fM (vj2 + 4'f2) d

9.3. ETERNAL SOLUTIONS

259

9.5. Applying (9.45) in 1R'\{0} with suitable functions h and ', prove the Hardy inequality:

for any f ED(R"\{0}), z

fR ID.fI dx >

fzdx.

(n - 2)2

(9.46)

fl' IxIz

4

9.3. Eternal solutions In this section, we consider solutions to the heat equation defined for all t E (-oo, +oo), which, hence, are called eternal solutions. Let u (t, x) be a regular fundamental solution at a point y E M. Let us extend u (t, x) tot < 0 by setting u (t, x) - 0. Since for any t E R, fM

u (t, x) da (x) < 1,

(9.47)

we see that u (t, x) E L10C (R x M). In particular, u (t, x) can be regarded as a distribution on R x M. THEOREM 9.20. Let u (t, x) be a regular fundamental solution of the heat

equation at y E M, extended to t < 0 by setting u (t, x) - 0. Then u (t, x) satisfies in R x M the following equation au

at - L p.u = S(o,y)

(9.48)

Here S(9,y) is the delta function at the point (0, y) on the manifold R x M, defined by (S(o,y), cp) =