Kazuaki Taira. Boundary Value Problems and Markov Processes
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5
ProofofTheorem1.1
This chapter is devoted to the proof of Theorem 1.1. The idea of our proof is
stated as follows. First, we reduce the study of the boundary value problem
'
(A −λ)u = f in D,
Lu = μ(x
)
∂u
∂n
+ γ(x
)u = ϕ on ∂D
(1.1)
to that of a first-order pseudo-differential operator T (λ)=LP(λ)onthe
boundary ∂D, just as in Section 4.3. Then we prove that conditions (A) and
(B) are sufficient for the validity of the aprioriestimate
u
2,p
≤ C(λ)
f
p
+ |ϕ|
2−1/p,p
+ u
p
. (1.2)
More precisely, we construct a parametrix S(λ)forT (λ)intheH¨ormander
class L
0
1,1/2
(∂D) (Lemma 5.2), and apply the Besov-space boundedness theo-
rem (Theorem 3.15) to S(λ) to obtain the desired estimate (1.2) (Lemma 5.1).
5.1 Boundary Value Problem with Spectral Parameter
Let D be a bounded domain of Euclidean space R
N
with smooth boundary
∂D.Itsclosure
D = D ∪ ∂D is an N -dimensional, compact smooth manifold
with boundary. We may assume that
D is the closure of a relatively compact
open subset D of an N -dimensional, compact smooth manifold
"
D without
boundary in which D has a smooth boundary ∂D. This manifold
"
D is the
double of D (see Figure 5.1).
We let
A =
N
i,j=1
a
ij
(x)
∂
2
∂x
i
∂x
j
+
N
i=1
b
i
(x)
∂
∂x
i
+ c(x)
K. Taira, Boundary Value Problems and Markov Processes,87
Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6
5,
c
Springer-Verlag Berlin Heidelberg 2009
88 5 Proof of Theorem 1.1
D
∂D
D
Fig. 5.1.
be a second-order, elliptic differential operator with real coefficients such that:
(1) a
ij
∈ C
∞
(
"
D)anda
ij
(x)=a
ji
(x) for all x ∈
"
D,1≤ i, j ≤ N,andthere
exists a positive constant a
0
such that
N
i,j=1
a
ij
(x)ξ
i
ξ
j
≥ a
0
|ξ|
2
on T
∗
(
"
D),
where T
∗
(
"
D) is the cotangent bundle of the double
"
D.
(2) b
i
∈ C
∞
(
"
D) for all 1 ≤ i ≤ N .
(3) c ∈ C
∞
(
"
D)andc(x) ≤ 0inD.
In this chapter we consider the elliptic boundary value problem with spec-
tral parameter
(A −λ)u = f in D,
Lu = μ(x
)
∂u
∂n
+ γ(x
)u = ϕ on ∂D.
(1.1)
Here we recall that:
(4) λ is a complex parameter.
(5) μ(x
)andγ(x
) are real-valued, smooth functions on the boundary ∂D.
(6) n =(n
1
,n
2
,...,n
N
) is the unit interior normal to the boundary ∂D (see
Figure 1.1).
The purpose of this chapter is to prove Theorem 1.1. More precisely, we prove
the aprioriestimate
u
2,p
≤ C(λ)
(A − λ)u
p
+ |Lu|
2−1/p,p
+ u
p
, (1.2)
provided that the following two conditions (A) and (B) are satisfied:
(A) μ(x
) ≥ 0on∂D.
(B) γ(x
) < 0onM := {x
∈ ∂D : μ(x
)=0}.
5.2 Proof of Estimate (1.2) 89
5.2 Proof of Estimate (1.2)
The proof of Estimate (1.2) is divided into three steps.
Step I: It suffices to show that estimate (1.2) holds true for some λ
0
> 0,
sincewehave,forallλ ∈ C,
(A −λ
0
)u =(A − λ)u +(λ − λ
0
)u.
We take a positive constant λ
0
so large that the function c(x)−λ
0
satisfies
the condition
c(x) − λ
0
< 0 on the double
"
D of D. (5.1)
This condition (5.1) implies that condition (4.1) is satisfied for the operator
A − λ
0
. Therefore, by applying Theorems 4.4 and 4.3 to the operator A − λ
0
we can obtain the following two fundamental results:
(a) The Dirichlet problem
(A −λ
0
)w =0 inD,
w = ϕ on ∂D
has a unique solution w ∈ H
t,p
(D) for any function ϕ ∈ B
t−1/p,p
(∂D)
with t ∈ R.
(b) The Poisson operator
P (λ
0
):B
t−1/p,p
(∂D) −→ H
t,p
(D),
defined by w = P (λ
0
)ϕ, is an isomorphism of the space B
t−1/p,p
(∂D)onto
the null space N(A − λ
0
,t,p)={u ∈ H
t,p
(D):(A − λ
0
)u =0inD} for
all t ∈ R; and its inverse is the trace operator γ
0
on the boundary ∂D.
We let
T (λ
0
): C
∞
(∂D) −→ C
∞
(∂D)
ϕ −→ LP (λ
0
)ϕ.
Then we have the formula
T (λ
0
)=μ(x
)Π(λ
0
)+γ(x
),
where
Π(λ
0
)ϕ =
∂
∂n
(P (λ
0
)ϕ)
∂D
.
It is known that the operator Π(λ
0
) is a classical pseudo-differential operator
of first order on the boundary ∂D and that its complete symbol is given by
the following formula (cf. [Ta2, Section 10.2]):
p
1
(x
,ξ
)+
√
−1 q
1
(x
,ξ
)
+
p
0
(x
,ξ
)+
√
−1 q
0
(x
,ξ
)
+ terms of order ≤−1 depending on λ
0
,
90 5 Proof of Theorem 1.1
where
p
1
(x
,ξ
) < 0
on the bundle T
∗
(∂D) \{0} of non-zero cotangent vectors. (5.2)
For example, if A is the usual Laplacian
Δ=
∂
2
∂x
2
1
+
∂
2
∂x
2
2
+ ...+
∂
2
∂x
2
N
,
then we have the formula
p
1
(x
,ξ
) = minus the length |ξ
| of ξ
with respect to the Riemannian
metric of ∂D induced by the natural metric of R
N
.
Therefore, we obtain that the operator T (λ
0
)=μ(x
)Π(λ
0
)+γ(x
)isa
classical pseudo-differential operator of first order on the boundary ∂D and
further that its complete symbol t(x
,ξ
; λ
0
) is given by the following formula:
t(x
,ξ
; λ
0
)=μ(x
)
p
1
(x
,ξ
)+
√
−1 q
1
(x
,ξ
)
+
[γ(x
)+μ(x
)p
0
(x
,ξ
)] +
√
−1 μ(x
)q
0
(x
,ξ
)
+ terms of order ≤−1 depending on λ
0
. (5.3)
Then, by arguing just as in Section 4.3 we can prove that the question of
the validity of aprioriestimates and the question of regularity for solutions
of problem (1.1) for λ = λ
0
are reduced to the corresponding questions for
the operator T (λ
0
) (cf. Theorems 4.9 and 4.10).
Step II: Therefore, in order to prove estimate (1.2) for λ = λ
0
it suffices
to show the following:
Lemma 5.1. Assume that conditions (A) and (B) are satisfied:
(A) μ(x
) ≥ 0 on ∂D.
(B) γ(x
) < 0 on M = {x
∈ ∂D : μ(x
)=0}.
Then we have, for all s ∈ R,
ϕ ∈D
(∂D),T(λ
0
)ϕ ∈ B
s,p
(∂D)=⇒ ϕ ∈ B
s,p
(∂D). (5.4)
Furthermore, for any t<s, there exists a positive constant C
s,t
such that
|ϕ|
s,p
≤ C
s,t
(|T (λ
0
)ϕ|
s,p
+ |ϕ|
t,p
) . (5.5)
Proof. (a) The proof of Lemma 5.1 is based on the following lemma (cf. [Ka,
Theorem 3.1]):
5.2 Proof of Estimate (1.2) 91
Lemma 5.2. Assume that conditions (A) and (B) are satisfied. Then, for
each point x
of ∂D, we can find a neighborhood U(x
) of x
such that:
For any compact K ⊂ U(x
) and any multi-indices α, β, there exist positive
constants C
K,α,β
and C
K
such that we have, for all x
∈ K and all |ξ
|≥C
K
,
D
α
ξ
D
β
x
t(x
,ξ
; λ
0
)
≤ C
K,α,β
|t(x
,ξ
; λ
0
)|(1 + |ξ|)
−|α|+(1/2)|β|
, (5.6a)
|t(x
,ξ
; λ
0
)|
−1
≤ C
K
. (5.6b)
Granting Lemma 5.2 for the moment, we shall prove Lemma 5.1.
(b) First, we cover ∂D by a finite number of local charts {(U
j
,χ
j
)}
m
j=1
in each of which inequalities (5.6a) and (5.6b) hold true. Since the operator
T (λ
0
) satisfies conditions (3.7a) and (3.7b) of Theorem 3.16 with μ := 0,
ρ := 1 and δ =1/2, it follows from an application of the same theorem that
there exists a parametrix S(λ
0
)intheclassL
0
1,1/2
(U
j
)forT (λ
0
):
'
T (λ
0
)S(λ
0
) ≡ I mod L
−∞
(U
j
),
S(λ
0
)T (λ
0
) ≡ I mod L
−∞
(U
j
).
Let {ϕ
j
}
m
j=1
be a partition of unity subordinate to the covering {U
j
}
m
j=1
,and
choose a function ψ
j
(x
) ∈ C
∞
0
(U
j
) such that ψ
j
(x
) = 1 on supp ϕ
j
,sothat
ϕ
j
(x
)ψ
j
(x
)=ϕ
j
(x
).
Now we may assume that ϕ ∈ B
t,p
(∂D)forsomet<sand that T (λ
0
)ϕ ∈
B
s,p
(∂D). We remark that the operator T (λ
0
)canbewritteninthefollowing
form:
T (λ
0
)=
m
j=1
ϕ
j
T (λ
0
)ψ
j
+
m
j=1
ϕ
j
T (λ
0
)(1 − ψ
j
).
However, by applying Theorems 3.12 and 3.8 to our situation we obtain that
the second terms ϕ
j
T (λ
0
)(1 −ψ
j
)areinL
−∞
(∂D). Indeed, it suffices to note
that
ϕ
j
(x
)(1− ψ
j
(x
)) = ϕ
j
(x
) − ϕ
j
(x
)=0.
Hence we are reduced to the study of the first terms ϕ
j
T (λ
0
)ψ
j
. This implies
that we have only to prove the following local version of assertions (5.4) and
(5.5):
ψ
j
ϕ ∈ B
t,p
(U
j
),T(λ
0
)ψ
j
ϕ ∈ B
s,p
(U
j
)=⇒ ψ
j
ϕ ∈ B
s,p
(U
j
). (5.7)
|ψ
j
ϕ|
s,p
≤ C
s,t
|T (λ
0
)ψ
j
ϕ|
2
s,p
+ |ψ
j
ϕ|
2
t,p
. (5.8)
However, by applying the Besov-space boundedness theorem (Theorem 3.15)
to our situation we obtain that the parametrix S(λ
0
)mapsB
σ,p
loc
(U
j
)contin-
uously into itself for all σ ∈ R. This proves the desired assertions (5.7) and
(5.8), since we have the assertion
ψ
j
ϕ ≡ S(λ
0
)(T (λ
0
)ψ
j
ϕ)modC
−∞
(U
j
).
92 5 Proof of Theorem 1.1
Lemma 5.1 is proved, apart from the proof of Lemma 5.2.
Step III: Proof of Lemma 5.2
The proof of Lemma 5.2 is divided into five steps.
Step III-1: First, we verify condition (5.6b).
By assertions (5.3) and (5.2), we can find positive constants c
0
and c
1
such
that we have, for all sufficiently large |ξ
|,
|t(x
,ξ
; λ
0
)|≥μ(x
) |p
1
(x
,ξ
)+p
0
(x
,ξ
)|−γ(x
)
≥
c
0
μ(x
)|ξ
|−
1
2
γ(x
)if0≤ μ(x
) ≤ c
1
,
c
0
2
μ(x
)|ξ
|−γ(x
)ifc
1
≤ μ(x
) ≤ 1,
since γ(x
) < 0onM = {x
∈ ∂D : μ(x
)=0}. Hence we have, for all
sufficiently large |ξ
|,
|t(x
,ξ
; λ
0
)|≥C (μ(x
)|ξ
| +1). (5.9)
Here in the following the letter C denotes a generic positive constant.
Inequality (5.9) implies the desired condition (5.6b):
|t(x
,ξ
; λ
0
)|≥C. (5.10)
Step III-2: Next we verify condition (5.6a) for |α| =1and|β| =0.
Since we have, for all sufficiently large |ξ
|,
D
α
ξ
t(x
,ξ
; λ
0
)
≤ C
μ(x
)+|ξ
|
−1
,
it follows from inequality (5.9) that
D
α
ξ
t(x
,ξ
; λ
0
)
≤ C(1 + |ξ
|)
−1
(μ(x
)|ξ
| +1)
≤ C(1 + |ξ
|)
−1
|t(x
,ξ
; λ
0
)|.
This inequality proves the desired condition (5.6a) for |α| =1and|β| =0.
Step III-3: We verify condition (5.6a) for |β| =1and|α| =0.Todothis,
we need the following elementary lemma on non-negative functions.
Lemma 5.3. Let f(x) be a non-negative, C
2
function on R such that we have,
for some positive constant c,
sup
x∈R
|f
(x)|≤c. (5.11)
Then we have the inequality
|f
(x)|≤
)
2c
)
f(x) on R. (5.12)
5.2 Proof of Estimate (1.2) 93
Proof. In view of Taylor’s formula, it follows that
0 ≤ f(y)=f(x)+f
(x)(y − x)+
f
(ξ)
2
(y − x)
2
,
where ξ is between x and y. Thus, by letting z = x−y we obtain from estimate
(5.11) that
0 ≤ f(x)+f
(x)z +
f
(ξ)
2
z
2
≤ f(x)+f
(x)z +
c
2
z
2
for all z ∈ R.
This implies the desired inequality (5.12) if we take the discriminant of the
quadratic equation.
Step III-4: Since we have, for all sufficiently large |ξ
|,
D
β
x
t(x
,ξ
; λ
0
)
≤ C
D
β
x
μ(x
)
·|ξ
| + μ(x
)|ξ
|+1
,
it follows from an application of Lemma 5.3 and inequalities (5.9) and (5.10)
that
D
β
x
t(x
,ξ
; λ
0
)
≤ C
*
)
μ(x
) |ξ
|+1
+(μ(x
)|ξ
|+1)
+
≤ C
*
|ξ
|
1/2
(μ(x
)|ξ
| +1)
1/2
+(μ(x
)|ξ
|+1)
+
≤ C |t(x
,ξ
; λ
0
)|
|ξ
|
1/2
|t(x
,ξ
; λ
0
)|
−1/2
+1
≤ C |t(x
,ξ
; λ
0
)|(1 + |ξ
|)
1/2
.
This inequality proves the desired condition (5.6a) for |β| =1and|α| =0.
Step III-5: Similarly, we can verify condition (5.6a) for the general case:
|α| + |β| = k, k ∈ N.
Now the proof of Lemma 5.1 and hence that of Theorem 1.1 is complete.
6
APrioriEstimates
This Chapter 6 and the next Chapter 7 are devoted to the proof of
Theorem 1.2. In this chapter we study the operator A
p
, and prove apri-
ori estimates for the operator A
p
− λI (Theorem 6.3) which will play a
fundamental role in the next chapter. In the proof we make good use of
Agmon’s method (Proposition 6.4). This is a technique of treating a spec-
tral parameter λ as a second-order, elliptic differential operator of an extra
variable and relating the old problem to a new problem with the additional
variable.
Recall that the operator A
p
is a unbounded linear operator from L
p
(D)
into itself defined by the following formulas:
(a) The domain of definition D(A
p
)ofA
p
is the space
D(A
p
)=
u ∈ H
2,p
(D)=W
2,p
(D):Lu = μ(x
)
∂u
∂n
+ γ(x
)u =0
.
(1.3)
(b) A
p
u = Au, u ∈D(A
p
).
We remark that the operator A
p
is densely defined, since the domain D(A
p
)
contains the space C
∞
0
(D).
First, we have the following:
Lemma 6.1. Assume that conditions (A) and (B) are satisfied:
(A) μ(x
) ≥ 0 on ∂D.
(B) γ(x
) < 0 on M = {x
∈ ∂D : μ(x
)=0}.
Then we have the aprioriestimate
u
2,p
≤ C (Au
p
+ u
p
) ,u∈D(A
p
). (6.1)
Proof. The aprioriestimate (6.1) follows immediately from estimate (1.2) of
Theorem 1.1 with ϕ := 0.
K. Taira, Boundary Value Problems and Markov Processes,95
Lecture Notes in Mathematics 1499, DOI 10.1007/978-3-642-01677-6
6,
c
Springer-Verlag Berlin Heidelberg 2009
96 6 A Priori Estimates
Corollary 6.2. The operator A
p
is a closed operator.
Proof. Let {u
j
} be an arbitrary sequence in the domain D(A
p
) such that:
u
j
−→ u in L
p
(D),
Au
j
−→ v in L
p
(D).
Then, by applying estimate (6.1) to the sequence {u
j
} we find that {u
j
} is a
Cauchy sequence in the space W
2,p
(D). This proves that
u ∈ W
2,p
(D),
and that
u
j
−→ u in W
2,p
(D).
Hence we have the formula
Au = lim
j→∞
Au
j
= v in L
p
(D),
and also, by Proposition 4.7 with B := L,
Lu = lim
j→∞
Lu
j
=0 inB
1−1/p,p
(∂D).
Summing up, we have proved that u ∈D(A
p
)andA
p
u = v.
The proof of Corollary 6.2 is complete.
The next theorem is an essential step in the proof of Theorem 1.2 (cf. the
proof of Theorem 7.1 in Chapter 7):
Theorem 6.3. Assume that conditions (A) and (B) are satisfied. Then, for
every −π<θ<π, there exists a positive constant R(θ) depending on θ such
that if λ = r
2
e
iθ
and |λ| = r
2
≥ R(θ), we have, for all u ∈ W
2,p
(D) satisfying
Lu =0on ∂D (i. e., u ∈D(A
p
)),
|u|
2,p
+ |λ|
1/2
·|u|
1,p
+ |λ|·u
p
≤ C(θ) (A − λ)u
p
, (6.2)
with a positive constant C(θ) depending on θ.Here|·|
j,p
, j =1, 2,isthe
seminorm on the space W
2,p
(D) defined by the formula
|u|
j,p
=
⎛
⎝
D
|α|=j
|D
α
u(x)|
p
dx
⎞
⎠
1/p
.
Proof. The proof of Theorem 6.3 is divided into two steps.
Step I: We shall make use of a method essentially due to Agmon (cf. [Ag],
[Fu], [LM], [Ta1]).
We introduce an auxiliary variable y of the unit circle
S = R/2πZ,
6 A Priori Estimates 97
and replace the complex parameter λ by the second-order differential operator
−e
iθ
∂
2
∂y
2
, −π<θ<π.
Namely, we replace the operator A − λ by the operator
,
Λ(θ)=A + e
iθ
∂
2
∂y
2
, −π<θ<π,
and consider instead of the problem with spectral parameter
(A −λ)u = f in D,
Lu = μ(x
)
∂u
∂n
+ γ(x
)u =0 on∂D
(1.1)
the following boundary value problem:
'
,
Λ(θ),u =
A + e
iθ
∂
2
∂y
2
,u =
,
f in D × S,
L,u = μ(x
)
∂,u
∂n
+ γ(x
),u =0 on∂D × S.
(6.3)
We remark that the operator
,
Λ(θ)iselliptic for −π<θ<π.
Then we have the following result, analogous to Lemma 6.1:
Proposition 6.4. Assume that conditions (A) and (B) are satisfied. Then we
have, for all ,u ∈ W
2,p
(D ×S) satisfying L,u =0on ∂D × S,
,u
2,p
≤
,
C(θ)
,
Λ(θ),u
p
+ ,u
p
, (6.4)
with a positive constant
,
C(θ) depending on θ.
Proof. We reduce the study of problem (6.3) to that of a pseudo-differential
operator on the boundary, just as in problem (1.1).
We can prove that Theorems 4.3 and 4.4 remain valid for the operator
,
Λ(θ)=A + e
iθ
∂
2
/∂y
2
, −π<θ<π:
(,a) The Dirichlet problem
,
Λ(θ) ,w =0 inD × S,
,w = ,ϕ on ∂D × S
has a unique solution ,w ∈ H
t,p
(D×S) for any function ,ϕ ∈ B
t−1/p,p
(∂D×
S)witht ∈ R.
(
,
b) The Poisson operator
,
P (θ):B
t−1/p,p
(∂D × S) −→ H
t,p
(D ×S),
defined by ,w =
,
P (θ) ,ϕ, is an isomorphism of the space B
t−1/p,p
(∂D ×
S) onto the null space N(
,
Λ(θ),t,p)={,u ∈ H
t,p
(D × S):
,
Λ(θ),u =
0inD × S} for all t ∈ R; and its inverse is the trace operator on the
boundary ∂D × S.