Kazuaki Taira. Boundary Value Problems and Markov Processes

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108 7 Proof of Theorem 1.2

7.1.1 Proof of Proposition 7.2

The proof of Proposition 7.2 is divided into three steps.

Step 1: First, we study the null spaces N



(θ)



and N (T

(λ



)) when



= 

iθ

with  ∈ Z:



(θ)





˜ϕ ∈ B

2−1/p,p

(∂D × S):

T (θ)˜ϕ =0



N (T

(λ



)) =



ϕ ∈ B

2−1/p,p

(∂D):T (λ



)ϕ =0



Since the pseudo-diﬀerential operators

T (θ)andT (λ



)arebothhypoelliptic,

it follows that



(θ)





˜ϕ ∈ C

∞

(∂D × S):

T (θ)˜ϕ =0



N (T

(λ



)) = {ϕ ∈ C

∞

(∂D):T (λ



)ϕ =0}.

Therefore, we can apply [Ta2, Proposition 8.4.6] to obtain the following most

important relationship between the null spaces N



(θ)



and N (T

(λ



))

when λ



= 

iθ

with  ∈ Z:

Lemma 7.4. The following two conditions are equivalent:

(1) dim N



(θ)



< ∞.

(2) There exists a ﬁnite subset I of Z such that



dim N



(

iθ

)



< ∞ if  ∈ I,

dim N



(

iθ

)



=0 if  ∈ I.

Moreover, in this case we have the formulas



(θ)



∈I



(

iθ

)



⊗ e

iy

dim N



(θ)





∈I

dim N



(

iθ

)



Step 2: Secondly, we study the ranges R



(θ)



and R(T

(λ



)) when



= 

iθ

with  ∈ Z. To do this, we consider the adjoint operators

(θ)

∗

and T

(λ



)

∗

(θ)andT

(λ



), respectively.

The next lemma allows us to give a characterization of the adjoint oper-

ators

(θ)

∗

and T

(λ



)

∗

in terms of pseudo-diﬀerential operators (cf. [Ta2,

Lemma 8.4.8]):

Lemma 7.5. Let M be a compact smooth manifold without boundary. If T is

a classical pseudo-diﬀerential operator of order m on M , we deﬁne a densely

deﬁned, closed linear operator

7.1 Proof of Theorem 1.2, Part (i) 109

T : B

s,p

(M) −→ B

s−m+1,p

(M)(s ∈ R)

as follows.

(a) The domain D(T ) of T is the space

D(T )=



ϕ ∈ B

s,p

(M):Tϕ∈ B

s−m+1,p

(M)



(b) T ϕ = Tϕ, ϕ ∈D(T ).

Then the adjoint operator T

∗

of T is characterized as follows:

∗

) of T

∗

is contained in the space



ψ ∈ B

−s+m−1,p



(M):T

∗

ψ ∈ B

−s,p



(M)



where p



= p/(p − 1) and T

∗

∈ L

(M) is the adjoint of T .

(d) T

∗

ψ = T

∗

ψ, ψ ∈D(T

∗

Proof. Let ψ be an arbitrary element of D(T

∗

) ⊂ B

−s+m−1,p



(M), and let

{ψ

} be a sequence in C

∞

(M) such that ψ

→ ψ in B

−s+m−1,p



(M). Then

we have, for all ϕ ∈ C

∞

(M) ⊂D(T ),

∗

ψ, ϕ)=(ψ, T ϕ)

=(ψ, Tϕ)

= lim

j→∞

(ψ

,Tϕ)

= lim

j→∞

∗

,ϕ)

=(T

∗

ψ, ϕ) .

This proves that

∗

ψ = T

∗

ψ ∈ B

−s,p



(M).

The proof of Lemma 7.5 is complete. 

We remark that the pseudo-diﬀerential operators T (λ)

∗

and

T (θ)

∗

also

satisfy conditions (3.7a) and (3.7b) of Theorem 3.16 with μ := 0, ρ := 1 and

δ := 1/2; hence they are hypoelliptic.

Therefore, by applying Lemma 7.5 to the operators

T (θ)andT (λ



)we

obtain the following:

Lemma 7.6. The null spaces N



(θ)

∗



and N (T

(λ



)

∗

) are characterized

respectively as follows:



(θ)

∗





ψ ∈ C

∞

(∂D × S):

T (θ)

∗

ψ =0



N (T

(λ



)

∗

)={ψ ∈ C

∞

(∂D):T (λ



)

∗

ψ =0}.

110 7 Proof of Theorem 1.2

By using Lemma 7.6, we ﬁnd that Lemma 7.4 remains valid for the adjoint

operators

(θ)

∗

and T

(λ



)

∗

(cf. [Ta2, Lemma 8.4.10]):

Lemma 7.7. The following two conditions are equivalent:

(1) dim N



(θ)

∗



< ∞.

(2) There exists a ﬁnite subset J of Z such that

dim N



(

iθ

)

∗



< ∞ if  ∈ J,

dim N



(

iθ

)

∗



=0 if  ∈ J.

Moreover, in this case we have the formula

dim N



(θ)

∗





∈J

dim N



(

iθ

)

∗



Hence, by combining Lemma 7.7 and the closed range theorem ([Yo,

Chapter VII, Section 5, Theorem]) we obtain the most important relation-

ship between codim R



(θ)



and codim R(T

(λ



)) when λ



= 

iθ

,  ∈ Z

(cf. [Ta2, Proposition 8.4.11]):

Lemma 7.8. The following two conditions are equivalent:

(1) codim R



(θ)



< ∞.

(2) There exists a ﬁnite subset J of Z such that



codim R



(

iθ

)



< ∞ if  ∈ J,

codim N



(

iθ

)



=0 if  ∈ J.

Moreover, in this case we have the formula

codim R



(θ)





∈J

codim R



(

iθ

)

∗



Step 3: Proposition 7.2 is an immediate consequence of Lemmas 7.4 and

7.8, with K := I ∪J.

The proof of Proposition 7.2 is now complete. 

Summing up, we have proved Theorem 7.1 and hence part (i) of

Theorem 1.2. 

7.2 Proof of Theorem 1.2, Part (ii)

Part (ii) of Theorem 1.2 may be proved as follows. Theorem 7.1 asserts that,

for suﬃciently large μ

> 0, the operator A

− μ

I satisﬁes condition (2.1)

(see Figure 7.1). Thus, by applying Theorem 2.2 (and Remark 2.1) to the

operator A

− μ

I we obtain part (ii) of Theorem 1.2:

7.2 Proof of Theorem 1.2, Part (ii) 111

Fig. 7.1.

Theorem 7.9. Assume that conditions (A) and (B) are satisﬁed. Then the

operator A

generates a semigroup U

on L

(D) which is analytic in the sector

= {z = t + is : z =0, |arg z| <π/2 − ε}

for any 0 <ε<π/2, and enjoys the following three properties:

(a) The operators A

and

are bounded operators on L

(D) for each

z ∈ Δ

, and satisfy the relation

= A

for all z ∈ Δ

(b) For each 0 <ε<π/2, there exist positive constants



(ε),



(ε) and μ

such that

U

≤



(ε)e

·Re z

for all z ∈ Δ

A

≤



(ε)

|z|

·Re z

for all z ∈ Δ

∈ L

(D), we have, as z → 0,z ∈ Δ

−→ u

in L

(D).

The proof of Theorem 1.2 is now complete. 

Proof of Theorem 1.3, Part (i)

This Chapter 8 and the next Chapter 9 are devoted to the proof of Theorem 1.3

and Theorem 1.4. In this chapter we prove part (i) of Theorem 1.3. In the

proof we make use of Sobolev’s imbedding theorems (Theorems 8.1 and 8.2)

and a λ-dependent localization argument due to Masuda [Ma] (cf. Lemma 8.4)

in order to adjust estimate



− λI)

−1



≤

(ε)

|λ|

for all λ ∈ Σ

(ε)(1.4)

to obtain the desired estimate

(A − λI)

−1

≤

c(ε)

|λ|

for all λ ∈ Σ(ε). (1.6)

Here we recall that

D(A



u ∈ H

2,p

(D)=W

2,p

(D):Lu = μ(x



)

∂u

∂n

+ γ(x



)u =0



. (1.3)

D(A )=



u ∈ C

(D \ M ):Au ∈ C

(D \ M),Lu=0



. (1.5)

8.1 The Space C

(D \ M)

First, we consider a one-point compactiﬁcation K

∂

= K ∪{∂} of the space

K =

D \ M .

We say that two points x and y of

D are equivalent modulo M if x = y or

x, y ∈ M ;wethenwritex ∼ y.Itiseasytoverifythatthisrelation∼ enjoys

the so-called equivalence laws. We denote by

D/M the totality of equivalence

classes modulo M.Ontheset

D/M we deﬁne the quotient topology induced

by the projection

K. Taira, Boundary Value Problems and Markov Processes, 113

Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6

 Springer-Verlag Berlin Heidelberg 2009

114 8 Proof of Theorem 1.3, Part (i)

q : D −→ D/M.

Namely, a subset O of

D/M is deﬁned to be open if and only if the inverse

image q

−1

(O)ofO is open in D. It is easy to see that the topological space

D/M is a one-point compactiﬁcation of the space D \ M and that the point

at inﬁnity ∂ corresponds to the set M (see Figure 8.1):



∂

:= D/M,

∂ := M.

∂

D/M

∂D

Fig. 8.1.

Furthermore, we obtain the following two assertions:

(i) If ˜u is a continuous function deﬁned on K

∂

, then the function ˜u ◦ q is

continuous on

D and constant on M.

(ii) Conversely, if u is a continuous function deﬁned on

D and constant on M,

then it deﬁnes a continuous function ˜u on K

∂

In other words, we have the following isomorphism:

C(K

∂

)

∼



u ∈ C(D):u(x) is constant on M



. (8.1)

Now we introduce a closed subspace of C(K

∂

) as in Subsection 2.2.1:

(K)={u ∈ C(K

∂

):u(∂)=0}.

Then we have, by assertion (8.1),

(K)

∼

(D \ M )=



u ∈ C(D):u(x)=0onM



. (8.2)

8.2 Sobolev’s Imbedding Theorems

It is the imbedding characteristics of Sobolev spaces of L

type that render

these spaces so useful in the study of partial diﬀerential equations. We need

the following imbedding properties of Sobolev spaces:

8.3 Proof of Part (i) of Theorem 1.3 115

Theorem 8.1 (Sobolev). Let D be a bounded domain in the Euclidean space

with boundary ∂D of class C

. Then we have the following two assertions:

(i) If 1 ≤ p<N, we have the continuous injection

2,p

(D) ⊂ W

1,q

(D) for

−

≤

(ii) If N/2 <p<∞, p = N, we have the continuous injection

2,p

(D) ⊂ C

(D) for 0 <ν≤ 2 −

Theorem 8.2 (Gagliardo–Nirenberg). Let D be a bounded domain in R

with boundary of class C

,and1 ≤ p, r ≤∞. Then we have the following

assertions:

(i) If p = N and if

+ θ



−



+(1− θ)

for

≤ θ ≤ 1,

then we have, for all u ∈ W

2,p

(D) ∩ L

(D),

u

1,q

≤ C

u

2,p

u

1−θ

with a positive constant C

= C

(D, p, r, θ).

(ii) If N/2 <p<∞, p = N and if

0 ≤ ν<θ



2 −



− (1 − θ)

then we have, for all u ∈ W

2,p

(D) ∩ L

(D),

u

(D)

≤ C

u

2,p

u

1−θ

, (8.3)

with a positive constant C

= C

(D, p, r, θ).

For a proof of Theorem 8.1, see Adams–Fournier [AF, Theorem 5.4] and

for a proof of Theorem 8.2, see Friedman [Fr1, Part I, Theorem 10.1], and also

Taira [Ta4].

8.3 Proof of Part (i) of Theorem 1.3

The proof is carried out in a chain of auxiliary lemmas.

Step (I): We begin with a version of estimate (7.1):

Lemma 8.3. Let N<p<∞. Assume that the following conditions (A) and

(B) are satisﬁed:

116 8 Proof of Theorem 1.3, Part (i)

(A) μ(x



) ≥ 0 on ∂D.

(B) γ(x



) < 0 on M = {x



∈ ∂D : μ(x



)=0}.

Then, for every ε>0, there exists a positive constant r

(ε) such that if

λ = r

iθ

with r ≥ r

(ε) and −π + ε ≤ θ ≤ π −ε, we have, for all u ∈D(A

|λ|

1/2

|u|

(D)

+ |λ|·|u|

C( D)

≤ C

(ε)|λ|

N/2p

(A − λ)u

, (8.4)

with a positive constant C

(ε).Here

D(A



u ∈ H

2,p

(D)=W

2,p

(D):Lu = μ(x



)

∂u

∂n

+ γ(x



)u =0



Proof. First, by applying Theorem 8.2 with p := r>N, θ := N/p and ν := 0

we obtain from the Gagliardo–Nirenberg inequality (8.3) that

|u|

C( D)

≤ C|u|

N/p

1,p

u

1−N/p

. (8.5)

Here and in the following the letter C denotes a generic positive constant

depending on p and ε, but independent of u and λ.

Combining inequality (7.1) with inequality (8.5), we ﬁnd that

|u|

C( D)

≤ C



|λ|

−1/2

(A − λ)u



N/p



|λ|

−1

(A − λ)u



1−N/p

= C|λ|

−1+N/2p

(A − λ)u

so that

|λ|·|u|

C( D)

≤ C|λ|

N/2p

(A − λ)u

for all u ∈D(A

). (8.6)

Similarly, by applying inequality (8.5) to the functions D

u ∈ W

1,p

(D),

1 ≤ i ≤ n, we obtain that

C( D)

≤ C|D

N/p

1,p

D

u

1−N/p

≤ C|u|

N/p

2,p

|u|

1−N/p

1,p

≤ C ((A − λ)u

)

N/p



|λ|

−1/2

(A − λ)u



1−N/p

= C|λ|

−1/2+N/2p

(A − λ)u

This proves that

|λ|

1/2

|u|

(D)

≤ C|λ|

N/2p

(A − λ)u

for all u ∈D(A

). (8.7)

Therefore, the desired inequality (8.4) follows by combining inequalities

(8.6) and (8.7). 

8.3 Proof of Part (i) of Theorem 1.3 117

The next lemma proves estimate (1.6):

Lemma 8.4. Assume that conditions (A) and (B) are satisﬁed. Then, for

every ε>0, there exists a positive constant r(ε) such that if λ = r

iθ

with

r ≥ r(ε) and −π + ε ≤ θ ≤ π − ε, we have, for all u ∈D(A),

|λ|

1/2

|u|

(D)

+ |λ|·|u|

C( D)

≤ c(ε)|(A − λ)u|

C( D)

, (8.8)

with a positive constant c(ε).Here

D(A )=



u ∈ C

(D \ M ):Au ∈ C

(D \ M),Lu=0



Proof. We shall make use of a λ-dependent localization argument due to

Masuda [Ma] in order to adjust the term (A − λ)u

in inequality (8.4)

to obtain inequality (8.8).

First, we remark that

A ⊂ A

for all 1 <p<∞.

Indeed, since we have, for any u ∈D(A),

u ∈ C(

D) ⊂ L

(D),Au∈ C(D) ⊂ L

(D)andLu =0,

it follows from an application of Theorem 4.9 and Lemma 5.1 that

u ∈ W

2,p

(D).

(1) Let x

be an arbitrary point of the closure D = D ∪ ∂D.

If x



is a boundary point and if χ is a smooth coordinate transformation

such that χ maps B(x

,η

) ∩ D into B(0,δ) ∩ R

and ﬂattens a part of the

boundary ∂D into the plane x

= 0 (see Figure 8.2), then we let

= B(x



,η

) ∩ D,



= B(x



,η) ∩ D, 0 <η<η



= B(x



,η/2) ∩ D, 0 <η<η

Here B(x, η) denotes the open ball of radius η about x.

Similarly, if x

is an interior point and if χ is a smooth coordinate trans-

formation such that χ maps B(x

,η

)intoB(0,δ), then we let (see Figure 8.3)

= B(x

,η



= B(x

,η), 0 <η<η



= B(x

,η/2), 0 <η<η

(2) Now we take a function θ(t)inC

∞

(R) such that θ(t) equals one near

the origin, and deﬁne

ϕ(x)=θ(|x



) θ(x

),x=(x



118 8 Proof of Theorem 1.3, Part (i)



=(x

;:::;x

N−1

)

∂D

B(x

, η

)

B (0;±)

Fig. 8.2.

∂D

= B(x

;´

)





Fig. 8.3.

Here we may assume that the function ϕ(x) is chosen so that



supp ϕ ⊂ B(0, 1),

ϕ(x)=1onB(0, 1/2).

We introduce a localizing function

(x, η) ≡ ϕ



x − x



= θ





− x







− t



=(x



,t) ∈ D.

We remark that



supp ϕ

⊂ B(x

,η),

(x, η)=1onB(x

,η/2).

Then we have the following:

Claim 8.5. If u ∈D(A), then it follows that ϕ

(x, η)u ∈D(A

) for all 1 <

p<∞.