Kazuaki Taira. Boundary Value Problems and Markov Processes

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3.3 Pseudo-Diﬀerential Operators 75

A classical pseudo-diﬀerential operator A ∈ L

(M)issaidtobeelliptic

of order m if its homogeneous principal symbol σ

(x, ξ) does not vanish on

the bundle T

∗

(M) \{0} of non-zero cotangent vectors.

Then we have the following assertion:

(6) An operator A ∈ L

(M) is elliptic if and only if there exists a parametrix

B ∈ L

−m

(M)forA:



AB ≡ I mod L

−∞

(M),

BA ≡ I mod L

−∞

(M).

Let Ω be an open subset of R

. A properly supported pseudo-diﬀerential

operator A on Ω is said to be hypoelliptic if it satisﬁes the condition

sing supp u = sing supp Au, u ∈D



(Ω).

For example, Theorem 3.13 asserts that elliptic operators are hypoelliptic. We

remark that this notion may be transferred to manifolds.

The following criterion for hypoellipticity is due to H¨ormander (cf. [Ho2,

Theorem 4.2]):

Theorem 3.16. Let A = p(x, D) ∈ L

ρ,δ

(Ω) be properly supported. Assume

that, for any compact K ⊂ Ω and any multi-indices α, β, there exist positive

constants C

K,α,β

, C

and a real number μ such that we have, for all x ∈ K

and |ξ|≥C



p(x, ξ)



≤ C

K,α,β

|p(x, ξ)|(1 + |ξ|)

−ρ|α|+δ|β|

, (3.7a)

|p(x, ξ)|

−1

≤ C

(1 + |ξ|)

. (3.7b)

Then there exists a parametrix B ∈ L

ρ,δ

(Ω) for A.

Approach to Elliptic Boundary

Value Problems

In this chapter we study elliptic boundary value problems in the framework of

-spaces, by using the L

theory of pseudo-diﬀerential operators. For more

thorough treatments of this subject, the reader might refer to H¨ormander

[Ho1], Seeley [Se2], Taylor [Ty, Chapter XI] and also Taira [Ta2, Chapter 8]

-version).

4.1 The Dirichlet Problem

In this section we shall consider the Dirichlet problem in the framework of

Sobolev spaces of L

type. This is a generalization of the classical potential

approach to the Dirichlet problem.

Let Ω be a bounded domain of Euclidean space R

with smooth boundary

Γ = ∂Ω.Itsclosure

Ω = Ω ∪Γ is an n-dimensional, compact smooth manifold

with boundary. We may assume that

Ω is the closure of a relatively compact

open subset Ω of an n-dimensional, compact smooth manifold M without

boundary in which Ω has a smooth boundary Γ . This manifold M is the

double of Ω (see Figure 4.1).

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

∞

(M)anda

(x)=a

(x) for all x ∈ M ,1≤ i, j ≤ n,andthere

exists a positive constant a

such that



i,j=1

(x)ξ

≥ a

|ξ|

on T

∗

(M).

Here T

∗

(M) is the cotangent bundle of the double M.

K. Taira, Boundary Value Problems and Markov Processes,77

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

 Springer-Verlag Berlin Heidelberg 2009

78 4 L

Approach to Elliptic Boundary Value Problems

Γ=∂ Ω

Fig. 4.1.

(2) b

∈ C

∞

(M) for all 1 ≤ i ≤ n.

(3) c ∈ C

∞

(M)andc(x) ≤ 0inM .

Furthermore, for simplicity, we assume that:

The function c(x)doesnotvanishidentically on the double M. (4.1)

The next theorem states the existence of a volume potential for A,which

plays the same role for A as the Newtonian potential plays for the Laplacian

(cf. [Se1, Theorem 1] and [Ta2, Theorem 8.2.1]):

Theorem 4.1. (i) The operator A : C

∞

(M) → C

∞

(M) is bijective, and its

inverse Q is a classical elliptic pseudo-diﬀerential operator of order −2 on M.

(ii) The operators A and Q extend respectively to isomorphisms

A : H

s,p

(M) −→ H

s−2,p

(M),

Q : H

s−2,p

(M) −→ H

s,p

(M)

for all s ∈ R, which are still inverses of each other.

Next we construct a surface potential for A, which is a generalization of

the classical Poisson kernel for the Laplacian.

We let

Kv = Q(v ⊗ δ)|

,v∈ C

∞

(Γ ).

Here v(x



) ⊗ δ(t) is a distribution on M deﬁned by the formula

v ⊗ δ, ϕ · μ = v, ϕ(·, 0) · ω,ϕ(x



,t) ∈ C

∞

(M),

where μ is a strictly positive density on M and ω is a strictly positive density

on Γ = {t =0}, respectively.

Then we have the following (cf. [Ta2, Theorem 8.2.2]):

4.1 The Dirichlet Problem 79

Theorem 4.2. (i) The operator K is a classical elliptic pseudo-diﬀerential

operator of order −1 on Γ .

(ii) The operator K : C

∞

(Γ ) → C

∞

(Γ ) is bijective, and its inverse L is a

classical elliptic pseudo-diﬀerential operator of ﬁrst order on Γ .Furthermore,

the operators K and L extend respectively to isomorphisms

K : B

σ,p

(Γ ) −→ B

σ+1,p

(Γ ),

L : B

σ+1,p

(Γ ) −→ B

σ,p

(Γ )

for all σ ∈ R, which are still inverses of each other.

Now we let

Pϕ = Q(Lϕ ⊗ δ)|

,ϕ∈ C

∞

(Γ ).

Then the operator P maps C

∞

(Γ ) continuously into C

∞

(Ω), and extends to

a continuous linear operator

P : B

s−1/p,p

(Γ ) −→ H

s,p

(Ω)

for all s ∈ R.Furthermore,wehave,forallϕ ∈ B

s−1/p,p

(Γ ),



AP ϕ = AQ(Lϕ ⊗ δ)|

=(Lϕ ⊗ δ)|

=0 inΩ,

Pϕ|

= KLϕ = ϕ on Γ.

The operator P is called the Poisson operator.

We let

N(A, s, p)={u ∈ H

s,p

(Ω):Au =0inΩ},s∈ R.

Since the injection H

s,p

(Ω) →D



(Ω) is continuous, it follows that the null

space N(A, s, p) is a closed subspace of H

s,p

(Ω); hence it is a Banach space.

Then we have the following fundamental result (cf. [Se2, Theorems 5

and 6]):

Theorem 4.3. The Poisson operator P maps the Besov space B

s−1/p,p

(Γ )

isomorphically onto the null space N (A, s, p) for all s ∈ R.

By combining Theorems 4.1 and 4.3, we can obtain the following existence

and uniqueness theorem for the Dirichlet problem (cf. [ADN]):

Theorem 4.4. Let s ≥ 2. The Dirichlet problem



Au = f in Ω,

u = ϕ on Γ

(4.2)

has a unique solution u(x) in H

s,p

(Ω) for any f ∈ H

s−2,p

(Ω) and any ϕ ∈

s−1/p,p

(Γ ).

Furthermore, we can prove the following existence and uniqueness theorem

for the Neumann problem (cf. [ADN]):

80 4 L

Approach to Elliptic Boundary Value Problems

Theorem 4.5. Let s ≥ 2. The Neumann problem



Au = f in Ω,

∂u

∂n

= ϕ on Γ

(4.3)

has a unique solution u(x) in H

s,p

(Ω) for any f ∈ H

s−2,p

(Ω) and any ϕ ∈

s−1−1/p,p

(Γ ).Heren is the unit interior normal to the boundary Γ .

By Theorem 4.5, we can introduce a linear operator

: H

s−2,p

(Ω) −→ H

s,p

(Ω)

as follows: For any f ∈ H

s−2,p

(Ω), the function G

f ∈ H

s,p

(Ω) is the unique

solution of the problem



Au = f in Ω,

∂u

∂n

=0 onΓ.

The operator G

is called the Green operator for the Neumann problem.

4.2 Formulation of a Boundary Value Problem

If u ∈ H

2,p

(Ω)=W

2,p

(Ω), we can deﬁne its traces γ

u and γ

u respectively

by the formulas



u = u|

u =

∂u

∂n



and let

γu = {γ

u, γ

u}.

Then we have the following (cf. [St1]):

Theorem 4.6 (the trace theorem). The trace map

γ : H

2,p

(Ω) −→ B

2−1/p,p

(Γ ) ⊕ B

1−1/p,p

(Γ )

is continuous and surjective for all 1 <p<∞.

We deﬁne a boundary condition

Bu := a(x



)

∂u

∂n

+ b(x





= a(x



)γ

u + b(x



)γ

u, u ∈ H

2,p

(Ω),

where a(x



)andb(x



) are real-valued, smooth functions on Γ .

Then we have the following:

Proposition 4.7. The mapping

B : H

2,p

(Ω) −→ B

1−1/p,p

(Γ )

is continuous for all 1 <p<∞.

Now we can formulate our boundary value problem for (A, B) as follows:

Given functions f ∈ L

(Ω)andϕ ∈ B

2−1/p,p

(Γ ), ﬁnd a function u ∈ H

2,p

(Ω)

such that



Au = f in Ω,

Bu = ϕ on Γ.

(4.4)

4.3 Reduction to the Boundary 81

4.3 Reduction to the Boundary

In this section, by using the operators P and G

we shall show that problem

(4.4) can be reduced to the study of a pseudo-diﬀerential operator on the

boundary.

First, we remark that every function u(x)inH

2,p

(Ω) can be written in

the following form:

u(x)=v(x)+w(x), (4.5)

where



v = G

(Au) ∈ H

2,p

(Ω),

w = u − v ∈ N (A, 2,p)=



z ∈ H

2,p

(Ω):Az =0inΩ



Since the operator G

: L

(Ω) → H

2,p

(Ω) is continuous, it follows that the

decomposition (4.5) is continuous; more precisely, we have the inequalities

v

2,p

≤ CAu

≤ Cu

2,p

;

w

2,p

≤u

2,p

+ v

2,p

≤ Cu

2,p

Here the letter C denotes a generic positive constant.

Now we assume that u ∈ H

2,p

(Ω) is a solution of the boundary value

problem (4.4)



Au = f in Ω,

Bu = ϕ on Γ.

Then, by virtue of the decomposition (4.5) of u(x)thisisequivalenttosaying

that w ∈ H

2,p

(Ω) is a solution of the boundary value problem



Aw =0 inΩ,

Bw = ϕ − Bv on Γ.

(4.6)

Here v(x)=G

f ∈ H

2,p

(Ω)andw(x)=u(x) − v(x). However, Theorem 4.3

asserts that the spaces N (A, 2,p)andB

2−1/p,p

(Γ ) are isomorphic in such a

way that:

N(A, 2,p)

−→ B

2−1/p,p

(Γ ).

N(A, 2,p) ←−

2−1/p,p

(Γ ).

Therefore, we ﬁnd that w ∈ H

2,p

(Ω) is a solution of problem (4.6) if and only

if ψ(x



) ∈ B

2−1/p,p

(Γ ) is a solution of the equation

BPψ = ϕ − Bv on Γ. (4.7)

Here ψ = γ

w,orequivalently,w = Pψ.

Summing up, we obtain the following:

82 4 L

Approach to Elliptic Boundary Value Problems

Proposition 4.8. For functions f ∈ L

(Ω) and ϕ ∈ B

2−1/p,p

(Γ ),thereexists

asolutionu ∈ H

2,p

(Ω) of problem (4.4) if and only if there exists a solution

ψ ∈ B

2−1/p,p

(Γ ) of equation (4.7). Furthermore, the solutions u(x) and ψ(x



)

are related as follows:

u = G

f + Pψ.

We remark that equation (4.7) is a generalization of the classical Fredholm

integral equation.

We let

T : C

∞

(Γ ) −→ C

∞

(Γ )

ϕ −→ BPϕ.

Then we have, by condition (4.2),

T = a(x



)Π + b(x



where

Πϕ = γ

Pϕ =

∂

∂n

(Pϕ)



,ϕ∈ C

∞

(Γ ).

It is known (cf. [Ho1], [Se2]) that the operator Π is a classical pseudo-

diﬀerential operator of ﬁrst order on Γ ; hence the operator T is a classical

pseudo-diﬀerential operator of ﬁrst order on the boundary Γ .

Consequently, Proposition 4.8 asserts that problem (4.4) can be reduced to

the study of the ﬁrst-order pseudo-diﬀerential operator T on the boundary Γ .

We shall formulate this fact more precisely in terms of functional analysis.

First, we remark that the operator T : C

∞

(Γ ) → C

∞

(Γ ) extends to a

continuous linear operator

T : B

s,p

(Γ ) −→ B

s−1,p

(Γ ),s∈ R.

Then we have the formula

Tϕ= BPϕ, ϕ ∈ B

2−1/p,p

(Γ ),

since the Poisson operator

P : B

2−1/p,p

(Γ ) −→ N(A, 2,p)

and the boundary operator

B : H

2,p

(Ω) −→ B

1−1/p,p

(Γ )

are both continuous.

We associate with problem (4.4) a linear operator

A : H

2,p

(Ω) −→ L

(Ω) ⊕ B

2−1/p,p

(Γ )

as follows.

4.3 Reduction to the Boundary 83

(a) The domain D(A)ofA is the space

D(A)=



u ∈ H

2,p

(Ω)=W

2,p

(Ω):Bu ∈ B

2−1/p,p

(Γ )



(b) Au = {Au, Bu}, u ∈D(A).

Note that the space B

2−1/p,p

(Γ ) is a right boundary space associated with

the Dirichlet condition: a(x



) ≡ 0andb(x



) ≡ 1onΓ .

Since the operators A : H

2,p

(Ω) → L

(Ω)andB : H

2,p

(Ω) → B

1−1/p,p

(Γ )

are both continuous, it follows that A is a closed operator. Furthermore, the

operator A is densely deﬁned, since the domain D(A) contains the space

∞

(Ω).

Similarly, we associate with equation (4.7) a linear operator

T : B

2−1/p,p

(Γ ) −→ B

2−1/p,p

(Γ )

as follows.

(α) The domain D(T )ofT is the space

D(T )=



ϕ ∈ B

2−1/p,p

(Γ ):Tϕ ∈ B

2−1/p,p

(Γ )



(β) T ϕ = Tϕ, ϕ ∈D(T ).

Then the operator T is a densely deﬁned, closed operator, since the opera-

tor T : B

2−1/p,p

(Γ ) → B

1−1/p,p

(Γ ) is continuous and since the domain D(T )

contains the space C

∞

(Γ ).

The next theorem states that A has regularity property if and only if T

has.

Theorem 4.9. The following two conditions are equivalent:

u ∈ L

(Ω),Au∈ L

(Ω) and Bu ∈ B

2−1/p,p

(Γ )=⇒ u ∈ H

2,p

(Ω). (4.8)

ϕ ∈ B

−1/p,p

(Γ ) and Tϕ∈ B

2−1/p,p

(Γ )=⇒ ϕ ∈ B

2−1/p,p

(Γ ). (4.9)

Proof. (i) First, we show that assertion (4.8) implies assertion (4.9). To do

this, assume that

ϕ ∈ B

−1/p,p

(Γ )andTϕ∈ B

2−1/p,p

(Γ ).

Then, by letting u = Pϕ we obtain that

u ∈ L

(Ω),Au=0 and Bu = Tϕ∈ B

2−1/p,p

(Γ ).

Hence it follows from condition (4.8) that

u ∈ H

2,p

(Ω),

84 4 L

Approach to Elliptic Boundary Value Problems

so that, by Theorem 4.6,

ϕ = γ

u ∈ B

2−1/p,p

(Γ ).

(ii) Conversely, we show that assertion (4.9) implies estimate (4.8). To do

this, assume that

u ∈ L

(Ω),Au∈ L

(Ω)andBu ∈ B

2−1/p,p

(Γ ).

Then the distribution u(x) can be decomposed as follows:

u(x)=v(x)+w(x),

where



v = G

(Au) ∈ H

2,p

(Ω),

w = u − v ∈ N (A, 0,p)={z ∈ L

(Ω):Az =0inΩ}.

Moreover, Theorem 4.3 asserts that the distribution w(x) can be written in

the form

w = Pϕ, ϕ = γ

w ∈ B

−1/p,p

(Γ ).

Hence we have, by Theorem 4.6,

Tϕ= BPϕ = Bw = Bu − Bv = Bu − b(x



)γ

v ∈ B

2−1/p,p

(Γ ),

since γ

v = 0. Thus it follows from condition (4.9) that

ϕ ∈ B

2−1/p,p

(Γ ),

so that again, by Theorem 4.3,

w = Pϕ ∈ H

2,p

(Ω).

This proves that

u = v + w ∈ H

2,p

(Ω).

The proof of Theorem 4.9 is complete. 

The next theorem states that aprioriestimates for A are entirely equiv-

alent to corresponding aprioriestimates for T .

Theorem 4.10. The following two estimates are equivalent:

u

2,p

≤ C



Au

+ |Bu|

2−1/p,p

+ u



,u∈D(A). (4.10)

|ϕ|

2−1/p,p

≤ C



|T ϕ|

2−1/p,p

+ |ϕ|

−1/p,p



,ϕ∈D(T ). (4.11)

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

4.3 Reduction to the Boundary 85

Proof. (i) First, we show that estimate (4.10) implies estimate (4.11).

By taking u = Pϕ with ϕ ∈D(T ) in estimate (4.10), we obtain that

Pϕ

2,p

≤ C



|T ϕ|

2−1/p,p

+ Pϕ



. (4.12)

However, Theorem 4.3 asserts that the Poisson operator P maps the Besov

space B

s−1/p,p

(Γ ) isomorphically onto the null space N(A, s, p) for all s ∈ R.

Thus the desired estimate (4.11) follows from estimate (4.12).

(ii) Conversely, we show that estimate (4.11) implies estimate (4.10).

To do this, we express a function u ∈D(A)intheform

u(x)=v(x)+w(x),

where



v = G

(Au) ∈ H

2,p

(Ω),

w = u − v ∈ N (A, 2,p)=



z ∈ H

2,p

(Ω):Az =0inΩ



Then we have, by Theorem 4.5 with s := 2,

v

2,p

= G

(Au)

2,p

≤ CAu

. (4.13)

Furthermore, by applying estimate (4.11) to the distribution γ

w we obtain

that

|γ

2−1/p,p

≤ C



|T (γ

w)|

2−1/p,p

+ |γ

−1/p,p



= C



|Bw|

2−1/p,p

+ |γ

−1/p,p



≤ C



|Bu|

2−1/p,p

+ |Bv|

2−1/p,p

+ |γ

−1/p,p



In view of Theorem 4.3, this proves that

w

2,p

≤ C



|Bu|

2−1/p,p

+ |Bv|

2−1/p,p

+ w



≤ C



|Bu|

2−1/p,p

+ |Bv|

2−1/p,p

+ u

+ v



≤ C



|Bu|

2−1/p,p

+ |Bv|

2−1/p,p

+ u

+ v

2,p



. (4.14)

However, since γ

v = 0, it follows from an application of Theorem 4.6 that

|Bv|

2−1/p,p

= |b(x



)γ

2−1/p,p

≤ Cv

2,p

. (4.15)

Thus, by carrying estimates (4.13) and (4.15) into estimate (4.14) we obtain

that

w

2,p

≤ C



Au

+ |Bu|

2−1/p,p

+ u



. (4.16)

Therefore, the desired estimate (4.10) follows from estimates (4.13) and

(4.16), since u(x)=v(x)+w(x).

The proof of Theorem 4.10 is complete. 