Kazuaki Taira. Boundary Value Problems and Markov Processes

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8.3 Proof of Part (i) of Theorem 1.3 119

Proof. (i) First, we recall that

u ∈ W

2,p

(D) for all 1 <p<∞.

Hence we have the assertion

(x, η)u ∈ W

2,p

(D).

(ii) Secondly, it is easy to verify (see Figure 8.4) that the function ϕ

(x, η)u,

x ∈

D, satisﬁes the boundary condition

L(ϕ

(x, η)u)=0 on∂D.

B(x;´)

B(x

;´)

Fig. 8.4.

Indeed, this is obvious if we have the condition

supp (ϕ

(x, η)u) ⊂ B(x

,η),x

∈ D.

Moreover, if we have the condition

supp (ϕ

(x, η)u) ⊂ B(x

,η) ∩ D, x

∈ ∂D,

then it follows that

∂

∂x

(ϕ

(x, η))





(0) · θ





− x





=0,

since θ



(0) = 0. This proves that

∂

∂n

(ϕ

(x, η)) = 0 on ∂D.

120 8 Proof of Theorem 1.3, Part (i)

Therefore, we have the assertion

L(ϕ

(x, η)u)=μ(x



)

∂

∂n

(ϕ

(x, η)u)+γ(x



)ϕ

(x, η)u

= ϕ

(x, η)(Lu)+μ(x



)



∂

∂n

(ϕ

(x, η))



=0 on∂D,

since Lu =0on∂D.

Summing up, we have proved that

(x, η)u ∈D(A

) for all 1 <p<∞.

The proof of Claim 8.5 is complete. 

(3) Now we take a positive number p such that

N<p<∞.

Then, by Claim 8.5 we can apply inequality (8.4) to the function ϕ

(x, η)u,

u ∈D(A), to obtain that

|λ|

1/2

|u|



)

+ |λ|·|u|

C( G



)

≤|λ|

1/2

|ϕ

(x, η)u|



)

+ |λ|·|ϕ

(x, η)u|

C( G



)

= |λ|

1/2

|ϕ

(x, η)u|

(D)

+ |λ|·|ϕ

(x, η)u|

C( D)

≤ C|λ|

N/2p

(A − λ)(ϕ

(x, η)u)

(D)

= C|λ|

N/2p

(A − λ)(ϕ

(x, η)u)



)

, 0 <η<η

, (8.9)

sincewehavetheassertions



(x, η)=1 onG



supp (ϕ

(x, η)u) ⊂ G



However, we have the formula

(A −λ)(ϕ

(x, η)u)=ϕ

(x, η)((A − λ)u)+[A, ϕ

(x, η)]u, (8.10)

where [A, ϕ

(x, η)] is the commutator of A and ϕ

(x, η) deﬁned by the formula

[A, ϕ

(x, η)]u = A(ϕ

(x, η)u) − ϕ

(x, η)Au



i,j=1

(x)

∂ϕ

∂x

∂u

∂x

⎛

⎝



i,j=1

(x)

∂

∂x



i=1

(x)

∂ϕ

∂x

⎞

⎠

u. (8.11)

8.3 Proof of Part (i) of Theorem 1.3 121

Here we need the following elementary inequality:

Claim 8.6. We have, for all v ∈ C



), j =0, 1, 2,

v

j,p



)

≤|G



1/p

v



)

where |G



| denotes the measure of G



Proof. It suﬃces to note that we have, for all w ∈ C(







|w(x)|

dx ≤|G



||w|



)

This proves Claim 8.6. 

Sincewehave(seeFigure8.3),forsomepositiveconstantc,



|≤|B(x

,η)|≤cη

it follows from an application of Claim 8.6 that

ϕ

(x, η)((A − λ)u)



)

≤ c

1/p

N/p

|(A − λ)u|

C( G



)

, 0 <η<η

. (8.12)

Furthermore, we remark that

(x, η)| = O



−|α|



as η ↓ 0.

Hence it follows from an application of Claim 8.6 that



∂ϕ

∂x

∂u

∂x





)

≤

|u|

1,p,G



≤ Cη

−1+N/p

|u|



)

, 0 <η<η

, (8.13)



∂

∂x





)

≤

|u|



)

≤ Cη

−2+N/p

|u|

C( G



)

, 0 <η<η

, (8.14)



∂ϕ

∂x





)

≤

|u|



)

≤ Cη

−1+N/p

|u|

C( G



)

, 0 <η<η

. (8.15)

By using inequalities (8.13), (8.14) and (8.15), we obtain from formula (8.11)

that

[A, ϕ

(x, η)]u



)

≤ C



−1+N/p

|u|



)

+ η

−2+N/p

|u|

C( G



)

+ η

−1+N/p

|u|

C( G



)



≤ C



−1+N/p

|u|

(D)

+ η

−2+N/p

|u|

C( D)



, 0 <η<η

. (8.16)

In view of formula (8.10), it follows from inequalities (8.12) and (8.16) that

(A − λ)(ϕ

(x, η)u)



)

≤ϕ

(x, η)((A − λ)u)



)

+ [A, ϕ

(x, η)]u



)

≤ Cη

N/p



|(A − λ)u|

C( G



)

+ η

−1

|u|

(D)

+ η

−2

|u|

C( D)



0 <η<η

. (8.17)

122 8 Proof of Theorem 1.3, Part (i)

Therefore, by combining inequalities (8.9) and (8.17) we obtain that

|λ|

1/2

|u|



)

+ |λ|·|u|

C( G



)

≤ C|λ|

N/2p

(A − λ)(ϕ

(x, η)u)



)

≤ C|λ|

N/2p

N/p



|(A − λ)u|

C( G



)

+ η

−1

|u|



)

+ η

−2

|u|

C( G



)



≤ C|λ|

N/2p

N/p



|(A − λ)u|

C( D)

+ η

−1

|u|

(D)

+ η

−2

|u|

C( D)



0 <η<η

. (8.18)

We remark (see Figure 8.5) that the closure

D = D ∪ ∂D can be covered

by a ﬁnite number of sets of the forms

B(x



,η/2) ∩ D, x



∈ ∂D,

and

B(x

,η/2),x

∈ D.

B(x

;´=2)

B(x



;´=2)



Fig. 8.5.

Hence, by taking the supremum of inequality (8.18) over x ∈ D we ﬁnd

that

|λ|

1/2

|u|

(D)

+ |λ|·|u|

C( D)

≤ C|λ|

N/2p

N/p



|(A − λ)u|

C( D)

+ η

−1

|u|

(D)

+ η

−2

|u|

C( D)



0 <η<η

. (8.19)

(4) We now choose the localization parameter η.Welet

η =

|λ|

1/2

8.3 Proof of Part (i) of Theorem 1.3 123

where K is a positive constant (to be chosen later) satisfying the condition

0 <η=

|λ|

1/2

K<η

that is,

0 <K<|λ|

1/2

Then it follows from inequality (8.19) that

|λ|

1/2

|u|

(D)

+ |λ|·|u|

C( D)

≤ Cη

N/p

|(A − λ)u|

C( D)



Cη

N/p−1

−1+N/p



|λ|

1/2

·|u|

(D)



Cη

N/p−2

−2+N/p



|λ|·|u|

C( D)

for all u ∈D(A). (8.20)

However, since the exponents −1+N/p and −2+N/p are negative for N<

p<∞, we can choose the constant K so large that

Cη

N/p−1

−1+N/p

< 1,

and

Cη

N/p−2

−2+N/p

< 1.

Then the desired inequality (8.8) follows from inequality (8.20).

The proof of Lemma 8.4 is complete. 

Step (II): The next lemma, together with Lemma 8.4, proves that the

resolvent set of A contains the set

Σ(ε)=



λ = r

iθ

: r ≥ r(ε), −π + ε ≤ θ ≤ π − ε



that is, the resolvent (A − λI)

−1

exists for all λ ∈ Σ(ε).

Lemma 8.7. If λ ∈ Σ(ε), then, for any f ∈ C

(D \M), there exists a unique

function u ∈D(A) such that (A − λI)u = f .

Proof. Since we have the assertion

f ∈ C

(D \ M ) ⊂ L

(D) for all 1 <p<∞,

it follows from an application of Theorem 1.2 that if λ ∈ Σ(ε)thereexistsa

unique function u ∈ W

2,p

(D) such that

(A −λ)u = f in D, (8.21)

and

Lu = μ(x



)

∂u

∂n

+ γ(x



)u =0 on∂D. (8.22)

124 8 Proof of Theorem 1.3, Part (i)

However, part (ii) of Theorem 8.1 asserts that

u ∈ W

2,p

(D) ⊂ C

2−N/p

(D) ⊂ C

(D)ifN<p<∞.

Hence we have, by formula (8.22) and condition (B),

u =0onM = {x



∈ ∂D : μ(x



)=0},

so that

u ∈ C

(D \ M ).

Furthermore, in view of formula (8.21) it follows that

Au = f + λu ∈ C

(D \ M ).

Summing up, we have proved that



u ∈D(A),

(A − λI)u = f.

Now the proof of part (i) of Theorem 1.3 is complete. 

Proof of Theorem 1.3, Part (ii)

In this chapter we prove Theorem 1.4 and part (ii) of Theorem 1.3. This

chapter is the heart of the subject. General existence theorems for Feller

semigroups are formulated in terms of elliptic boundary value problems with

spectral parameter (Theorem 9.12). First, we study Feller semigroups with

reﬂecting barrier (Theorem 9.14) and then, by using these Feller semigroups

we construct Feller semigroups corresponding to such a diﬀusion phenomenon

that either absorption or reﬂection phenomenon occurs at each point of the

boundary (Theorem 9.18). Our proof is based on the generation theorems of

Feller semigroups discussed in Section 2.2.

9.1 General Existence Theorem for Feller Semigroups

The purpose of this section is to give a general existence theorem for Feller

semigroups in terms of boundary value problems (Theorem 9.12), following

Taira [Ta2, Section 9.6] (cf. [BCP], [SU], [Ta3]).

Let D be a bounded domain of Euclidean space R

, with smooth bound-

ary ∂D;itsclosure

D = D ∪ ∂D is an N-dimensional, compact smooth man-

ifold with boundary. We let

A =



i,j=1

(x)

∂

∂x



i=1

(x)

∂

∂x

+ c(x)

be a second-order, elliptic diﬀerential operator with real coeﬃcients such that:

(1) a

∈ C

∞

(D)anda

(x)=a

(x) for all x ∈ D and 1 ≤ i, j ≤ N ,and

there exists a positive constant a

such that



i,j=1

(x)ξ

≥ a

|ξ|

for all (x, ξ) ∈ D × R

K. Taira, Boundary Value Problems and Markov Processes, 125

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

 Springer-Verlag Berlin Heidelberg 2009

126 9 Proof of Theorem 1.3, Part (ii)

(2) b

∈ C

∞

(D) for all 1 ≤ i ≤ N .

(3) c ∈ C

∞

(D)andc(x) ≤ 0onD.

The diﬀerential operator A describes analytically a strong Markov process

with continuous paths in the interior D such as Brownian motion (see

Figure 1.4). The functions a

(x), b

(x)andc(x) are called the diﬀusion coef-

ﬁcients, the drift coeﬃcients and the termination coeﬃcient, respectively.

Let L be a ﬁrst-order boundary condition such that

Lu = μ(x



)

∂u

∂n

+ γ(x



)u,

where:

(4) μ ∈ C

∞

(∂D)andμ(x



) ≥ 0on∂D.

(5) γ ∈ C

∞

(∂D)andγ(x



) ≤ 0on∂D.

(6) n =(n

,...,n

) is the unit interior normal to the boundary ∂D (see

Figure 1.1).

The boundary condition L is called a ﬁrst-order Ventcel’ boundary condition

(cf. [We]). Its terms μ(x



)(∂u)/(∂n)andγ(x



)u are supposed to correspond

to the reﬂection and absorption phenomena, respectively (seeFigure1.5).

We are interested in the following problem:

Problem. Given analytic data (A, L), can we construct a Feller semigroup

}

t≥0

on D whose inﬁnitesimal generator A is characterized by (A, L)?

First, we consider the following Dirichlet problem: Given functions f(x)

and ϕ(x



) deﬁned in D and on ∂D, respectively, ﬁnd a function u(x)inD

such that



Au = f in D,

u = ϕ on ∂D.

(9.1)

The next theorem summarizes the basic facts about the Dirichlet problem

in the framework of H¨older spaces (cf. [GT]):

Theorem 9.1. (i) (Existence and Uniqueness)Iff ∈ C

(D) with 0 <θ<1

and if ϕ ∈ C(∂D), then problem (9.1) has a unique solution u(x) in C(

D) ∩

2+θ

(D).

(ii) ( Interior Regularity)Ifu ∈ C

(D) and Au = f ∈ C

k+θ

(D) for some

non-negative integer k, then it follows that u ∈ C

k+2+θ

(D).

(iii) (Global Regularity)Iff ∈ C

k+θ

(D) and ϕ ∈ C

k+2+θ

(∂D) for some

non-negative integer k, then a solution u ∈ C(

D) ∩ C

(D) of problem (9.1)

belongs to the space C

k+2+θ

(D).

Next we consider the following Dirichlet problem with spectral parameter:

For given functions f(x)andϕ(x



) deﬁned in D and on ∂D, respectively, ﬁnd

a function u(x)inD such that

9.1 General Existence Theorem for Feller Semigroups 127



(α − A)u = f in D,

u = ϕ on ∂D,

(9.2)

where α is a positive parameter.

By applying Theorem 9.1 with A := A − α, we obtain that problem (9.2)

has a unique solution u(x)inC

2+θ

(D) for any f ∈ C

(D)andanyϕ ∈

2+θ

(∂D)with0<θ<1. Therefore, we can introduce linear operators

: C

(D) −→ C

2+θ

(D),

and

: C

2+θ

(∂D) −→ C

2+θ

(D)

as follows.

(a) For any f ∈ C

(D), the function G

f ∈ C

2+θ

(D) is the unique solution

of the problem



(α − A)G

f = f in D,

f =0 on∂D.

(9.3)

(b) For any ϕ ∈ C

2+θ

(∂D), the function H

ϕ ∈ C

2+θ

(D) is the unique solu-

tion of the problem



(α − A)H

ϕ =0 inD,

ϕ = ϕ on ∂D.

(9.4)

The operator G

is called the Green operator and the operator H

is called

the harmonic operator, respectively.

Then we have the following result:

Lemma 9.2. The operator G

, α>0, considered from C(D) into itself, is

non-negative and continuous (bounded) with norm



∞

=max

x∈D

1(x).

Proof. Let f (x) be an arbitrary function in C

(D) such that f (x) ≥ 0on

D. Then, by applying the weak maximum principle (see Theorem A.1) with

A := A − α to the function −G

f we obtain from formula (9.3) that

f ≥ 0onD.

This proves the non-negativity of G

Since G

is non-negative, we have, for all f ∈ C

(D),

−G

f

∞

≤ G

f ≤ G

f

∞

on D.

This implies the continuity of G

with norm

G

 = G

1

∞

The proof of Lemma 9.2 is complete. 

128 9 Proof of Theorem 1.3, Part (ii)

Similarly, we have the following result:

Lemma 9.3. The operator H

, α>0, considered from C(∂D) into C(D),is

non-negative and continuous (bounded) with norm

H

 = H

1

∞

=max

x∈D

1(x).

More precisely, we have the following fundamental results:

Theorem 9.4. (i) (a) The operator G

, α>0, can be uniquely extended to

a non-negative, bounded linear operator on C(

D) into itself, denoted again by

,withnorm



∞

≤

. (9.5)

(b) For any f ∈ C(

D), we have the assertion

f =0 on ∂D.

f − G

f +(α − β)G

f =0,f∈ C(D). (9.6)

(d) For any f ∈ C(

D), we have the assertion

lim

α→+∞

αG

f(x)=f(x) for all x ∈ D. (9.7)

Furthermore, if f(x



)=0on ∂D, then this convergence is uniform in x ∈ D,

that is, we have the assertion

lim

α→+∞

αG

f = f in C(D). (9.8)

(e) The operator G

maps C

k+θ

(D) into C

k+2+θ

(D) for any non-negative

integer k.

(ii) (a



) The operator H

, α>0, can be uniquely extended to a non-

negative, bounded linear operator on C(∂D) into C(

D), denoted again by H

with norm H

 =1.



) For any ϕ ∈ C(∂D), we have the assertion

ϕ = ϕ on ∂D.



) For all α, β>0, we have the equation

ϕ − H

ϕ +(α − β)G

ϕ =0,ϕ∈ C(∂D). (9.9)



) The operator H

maps C

k+2+θ

(∂D) into C

k+2+θ

(D) for any non-negative

integer k.