Benzoni-Gavage S., Serre D. Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications
Подождите немного. Документ загружается.
Energy estimates 245
On the other hand, since each U
j
, j ≥ 1isdiffeomorphictoahalf-space,
Theorem 9.4 shows that
γ e
−γt
ϕ
j
u
2
L
2
(Ω×R)
+ e
−γt
ϕ
j
u
2
L
2
(∂Ω×R)
1
γ
e
−γt
L(ϕ
j
u)
2
L
2
(Ω×R)
+ e
−γt
Bϕ
j
u
2
L
2
(∂Ω×R)
1
γ
e
−γt
Lu
2
L
2
(Ω×R)
+
1
γ
e
−γt
u
2
L
2
(Ω×R)
+ e
−γt
Bu
2
L
2
(∂Ω×R)
.
Summing all the inequalities obtained for j ∈{0,...,J},weget
γ e
−γt
u
2
L
2
(Ω×R)
+ e
−γt
u
2
L
2
(∂Ω×R)
1
γ
e
−γt
Lu
2
L
2
(Ω×R)
+
1
γ
e
−γt
u
2
L
2
(Ω×R)
+ e
−γt
Bu
2
L
2
(∂Ω×R)
,
hence the announced estimate for γ large enough.
9.1.5 Less-smooth coefficients
Preparing the way for Chapter 11 on non-linear problems, we consider here initial
boundary value problems in which, similarly as in Theorem 2.4 for the Cauchy
problem, the matrices A
j
and B depend not directly on (x, t) but on v(x, t),
with v Lipschitz continuous, and A
j
and B are C
∞
of w ∈ R
n
(or w in an open
subset of R
n
). For this purpose we are going to modify the rest of our notations
accordingly.
To simplify the presentation, we go back to the basic domain Ω = {x; x
d
> 0}.
Notations Introducing contractible open subsets of R
n
,sayW and W
0
⊂ W,
intended to contain, respectively, v(
¯
Ω) and v(∂Ω), we redefine in this section
X
1
:= {X =(w, η, τ) ∈ W × R
d−1
× C
+
, |τ |
2
+ η
2
=1},
X
0
1
:= {X =(w, η, τ) ∈ W
0
× R
d−1
× C
+
, |τ |
2
+ η
2
=1}.
For all X =(w, η, τ) ∈ X
1
such that Re τ>0, E
−
(X) denotes the stable sub-
space of
A(X):=−(A
d
(w))
−1
( τI
n
+ iA(w, η,0) ) ,
where A(w, ξ):=
d
j=1
ξ
j
A
j
(w) for all (w, ξ) ∈ W × R
d
,andE
−
is extended
by continuity to all points of X
1
. For a given Lipschitz continuous v, L
v
stands
for the variable-coefficients differential operator
L
v
:= ∂
t
+
j
A
j
v
∂
j
, where A
j
v
(x, t):=A
j
(v(x, t)) ,
and B
v
stands for the variable-coefficients boundary matrix defined by
B
v
(y, t)=B(v(y, 0,t))
246 Variable-coefficients initial boundary value problems
for all (y,t) ∈ R
d−1
× R . Finally, for any ω>0 we denote by V
ω
the set of
Lipschitz continuous
v : R
d−1
× R
+
× R → W
such that v is constant outside a compact subset, v
|x
d
=0
takes its values in W
0
and
v
W
1,∞
(R
d−1
×R
+
×R))
≤ ω.
Theorem 9.6 Given C
∞
matrix-valued mappings A
j
(j =1,...,d)andB on
W and W
0
, respectively, our main assumptions are that
r
the operator
∂
t
+
j
A
j
(w) ∂
j
is constantly hyperbolic, that is, the matrices A(w, ξ) are diagonalizable
with real eigenvalues of constant multiplicities on W × (R
d
\{0});
r
the matrix A
d
(w) is non-singular for all w ∈ W
0
;
r
the matrix B(w) is for all w ∈ W
0
of maximal rank equal to the number of
positive eigenvalues of A
d
(w) (counted with multiplicity),
r
the uniform Kreiss–Lopatinski˘ı condition holds:
(UKL) for all X =(w, η, τ) ∈ X
0
1
, there exists C>0 so that
V ≤C B(w) V for all V ∈ E
−
(X) .
Then there exists c = c(ω) > 0 and γ
0
= γ
0
(ω) > 0 such that for all γ ≥ γ
0
,for
all v ∈ V
ω
, for all u ∈ D(R
d−1
× R
+
× R), u
γ
(x, t):=e
−γt
u(x, t) satisfies
γ u
γ
2
L
2
(R
d−1
×R
+
×R)
+ (u
γ
)
|x
d
=0
2
L
2
(R
d−1
×R)
≤ c
1
γ
(γ + L
v
)u
γ
2
L
2
(R
d−1
×R
+
×R)
+ (B
v
u
γ
)
|x
d
=0
2
L
2
(R
d−1
×R)
.
Proof We first rewrite the equation L
v
u = f as
∂
d
u
γ
− P
γ
v
u
γ
=(A
d
v
)
−1
f
γ
,P
γ
v
:= −(A
d
v
)
−1
( ∂
t
+ γ +
d−1
j=1
A
j
v
∂
j
) .
Here the notation (A
d
v
)
−1
stands for
(x, t) → (A
d
v
(x, t))
−1
=(A
d
(v(x, t))
−1
.
Since the matrix (A
d
v
(x, t))
−1
is uniformly bounded by a constant depending
only on ω for v ∈ V
ω
, the problem amounts to proving that for all v ∈ V
ω
and
Energy estimates 247
all u ∈ D (R
d−1
× R
+
× R)
γ u
2
L
2
(R
d−1
×R
+
×R)
+ u
|x
d
=0
2
L
2
(R
d−1
×R)
1
γ
f
2
L
2
(R
d−1
×R
+
×R)
+ g
2
L
2
(R
d−1
×R)
(9.1.22)
with f = ∂
d
u − P
γ
v
u and g = B
v
u
|x
d
=0
. In fact, it suffices to prove (9.1.22)
for the paralinearized expressions f = ∂
d
u − T
γ
A
v
u,whereA
v
(x, t, η, τ):=
A(v(x, t),η,τ), and g = T
γ
B
v
u
|x
d
=0
. Indeed, the error estimates in Theorem C.20
show that
P
γ
v
u − T
γ
A
v
u
L
2
≤ C(ω) u
L
2
,γB
v
u − T
γ
B
v
u
L
2
≤ C(ω) u
L
2
.
Consequently, if we have the estimate
γ u
2
L
2
(R
d−1
×R
+
×R)
+ u
|x
d
=0
2
L
2
(R
d−1
×R)
1
γ
∂
d
u − T
γ
A
v
u
2
L
2
(R
d−1
×R
+
×R)
+ T
γ
B
v
u
|x
d
=0
2
L
2
(R
d−1
×R)
we also have, for γ large enough
γ u
2
L
2
(R
d−1
×R
+
×R)
+ u
|x
d
=0
2
L
2
(R
d−1
×R)
1
γ
∂
d
u − P
γ
v
u
2
L
2
(R
d−1
×R
+
×R)
+ B
v
u
|x
d
=0
2
L
2
(R
d−1
×R)
.
The proof of the estimate for the para-linearized problem
∂
d
u − T
γ
A
v
u = f, T
γ
B
v
u
|x
d
=0
= g
of course relies on Kreiss symmetrizers. But we must overcome the fact that
the sharp G˚arding inequality is not true for Lipschitz coefficients. This is
why we introduce after Chazarain and Piriou [31] a refined version of Kreiss
symmetrizers, which will allow us to use the standard G˚arding inequality.
Definition 9.4 Given C
∞
functions
A : X → M
n
(C)
X =(w, τ,η) →A(X)
and B : W
0
→ M
p×n
(R)
w → B(w) ,
a refined Kreiss symmetrizer for A and B at some point X
∈ X
1
is a matrix-
valued function r in some neighbourhood V of X
in X
1
, which is associated with
another matrix-valued function T , both being C
∞
, and such that
i) the matrix r(X) is Hermitian and T (X) is invertible for all X ∈ V ,
ii) the matrix Re ( r(X) T (X)
−1
A(X)T (X)) is block-diagonal, with blocks
h
0
(X) and h
1
(X) such that h
0
(X)/γ is C
∞
and
Re ( h
0
(X)) ≥ CγI
p
, Re ( h
1
(X)) ≥ CI
n−p
, (9.1.23)
for some C>0 independent of X ∈ V
248 Variable-coefficients initial boundary value problems
iii) and additionally, if X ∈ X
0
, there exist α>0 and β>0 independent of
X ∈ V so that
r(X) ≥ αI
n
− β (B(w)T(X))
∗
B(w)T (X) . (9.1.24)
End of the proof of Theorem 9.6. Our assumptions allow us to construct
local Kreiss symmetrizers r, which, moreover, satisfy the refined property in ii)
(see Section 9.1.3 for some hints and [31], pp. 381–390 for details): we shall point
out where this property is so important; besides this ‘technical’ point, the proof
of Theorem 9.6 follows a standard strategy, which we describe now.
For any v ∈ V
ω
, we may consider the mapping r
v
defined by r
v
(y, x
d
,t,η,τ)=
r(v(y, x
d
,t),η,τ) and extend it to a global symmetrizer R
v
, homogeneous degree
0in(η, τ) as in the proof of Theorem 9.1. The resulting matrix-valued function
R
v
(x
d
):(y, t,η, δ, γ) →R
v
(y, x
d
,t,η,τ = γ + iδ)
may be viewed, for all x
d
, as a symbol in the variables (y, t) (with associated
frequencies (η, δ)) and parameter γ, which belongs to Γ
0
1
, the order 0 coming from
the homogeneity degree 0 in (η, τ) and the regularity index is 1 coming from the
fact that v, hence also A
v
and B
v
, are Lipschitz in (y,t). By construction, R
v
satisfies inequalities
R
v
(y, 0,t,η,τ) ≥ αI
n
− βB
v
(y, t)
T
B
v
(y, t), (9.1.25)
Re ( R
v
(y, x
d
,t,η,τ) A
v
(y, x
d
,t,η,τ)) ≥ CγI
n
, (9.1.26)
for some constants α>0, β>0andC>0 depending only on the Lipschitz
bound ω for v. (We have not used the refined property in ii) yet.) Note also that
R
v
is Lipschitz continuous in x
d
. We attempt a para-differential version of the
proof of Theorem 9.1/Proposition 9.1 by considering the family of operators
R
γ
v
(x
d
):=
1
2
( T
γ
R
v
(x
d
)
+(T
γ
R
v
(x
d
)
)
∗
) .
By construction, R
γ
v
(x
d
) is a self-adjoint, bounded operator on L
2
(R
d
, dy dt),
whose norm is bounded uniformly in x
d
and γ. This is true also for dR
γ
v
/dx
d
.
Additionally, by Theorem C.21 and Remark C.2,
R
γ
v
(x
d
) − T
γ
R
v
(x
d
)
B(L
2
)
1
γ
.
Hence, the inequality in (9.1.25) together with the error estimates in Theorems
C.20 and C.22 and the G˚arding inequality in Theorem C.23 imply
R
γ
v
(0) u(0) ,u(0) + β Re T
γ
B
v
T
B
v
u(0) ,u(0) ≥
α
2
u
2
L
2
(R
d
,dy dt)
for γ large enough. (Here ·, · denotes the inner product on L
2
(R
d
, dy dt).)
Assume for now that we also have
Re
R
γ
v
T
γ
A
v
u, u dx
d
≥ γ
C
2
u
2
L
2
(R
d
×R
+
,dy dt dx
d
)
(9.1.27)
Energy estimates 249
(which can not be deduced from (9.1.26), as for the G˚arding inequality in
Theorem C.23 to apply to the degree 1 symbol R
γ
v
A
v
it would require λ
γ,1
instead of γ in the right-hand side of (9.1.26)): then it is easy to complete the
proof of Theorem 9.6. Indeed, integrating in x
d
the equality
d
dx
d
R
γ
v
u, u =
dR
γ
v
dx
d
u, u +2ReR
γ
v
(∂
d
u − T
γ
A
v
u) ,u
+2Re R
γ
v
T
γ
A
v
u,u,
we get
α
2
u(0)
2
L
2
(R
d
)
− β Re T
γ
B
v
T
B
v
u(0) ,u(0)
≤ ( C
2
+ γ (εC
1
− C)) u
2
L
2
(R
d
×R
+
)
+
C
1
4εγ
f
2
L
2
(R
d
×R
+
)
.
In this inequality, the constants C
1
and C
2
come from bounds for R
γ
v
(x
d
)and
dR
γ
v
/dx
d
, ε>0 is arbitrary, and we have set f = ∂
d
u − T
γ
A
v
u. Now, choosing
ε = C/(2C
1
), we obtain
α
2
u(0)
2
L
2
(R
d
)
+ γ
C
4
u
2
L
2
(R
d
×R
+
)
≤ β Re T
γ
B
v
T
B
v
u(0) ,u(0)
+
C
2
1
2Cγ
f
2
L
2
(R
d
×R
+
)
for all γ ≥ 4C
2
/C. Finally, using again Theorems C.21, C.22 and Remark C.2,
we arrive at
α
4
u(0)
2
L
2
(R
d
)
+ γ
C
4
u
2
L
2
(R
d
×R
+
)
≤ β T
γ
B
v
u(0)
2
L
2
(R
d
)
+
C
2
1
2Cγ
f
2
L
2
(R
d
×R
+
)
for γ large enough.
Proof of (9.1.27) This is where we are to make use of ii). Indeed, going back
to the construction of the global symmetrizer R
v
(see the proof of Theorem 9.1),
we see it is of the form
R
v
(x, t, η, τ)=
j
P
j
(X)
∗
r
j
(X) P
j
(X) ,
where the sum is finite, X =(v(x, t),η,τ = γ + iδ), and P
j
(X)(=
ϕ(X)
1/2
T
j
(X)
−1
with the notations of Theorem 9.1), r
j
(X) are homogeneous
degree 0 in (η, τ), with
j
P
∗
j
P
j
uniformly bounded by below. Additionally, the
250 Variable-coefficients initial boundary value problems
refined property in ii) for the r
j
shows that
R
v
(x, t, η, τ) A(X)=
j
P
j
(X)
∗
γh
0,j
(X) 0
0 h
1,j
(X)
P
j
(X) ,
with h
0,j
homogeneous degree 0, h
1,j
homogeneous degree 1 in (η, τ), satisfying
lower bounds (see (9.1.23))
Re (h
0,j
(X)) ≥ C
j
I
p
j
, Re (h
1,j
(X)) ≥ C
j
λ
γ,1
(η, δ) I
n−p
j
.
Both these bounds are elligible for the G˚arding inequality (Theorem C.23), hence
Re T
γ
h
0,j
u
0,j
,u
0,j
≥
C
j
4
u
0,j
2
L
2
,
i.e. Re γT
γ
h
0,j
u
0,j
,u
0,j
≥γ
C
j
4
u
0,j
2
L
2
for all u
0,j
with values in C
p
j
(upper block) and
Re T
γ
h
1,j
u
1,j
,u
1,j
≥
C
j
4
u
1,j
2
H
1/2
≥ γ
C
j
4
u
1,j
2
L
2
for all u
1,j
with values in C
n−p
j
(lower block). Now the conclusion will follow
from standard error estimates. Indeed,
Re R
γ
v
T
γ
A
v
u, u≥
j
2
γT
γ
h
0,j
0
0 T
γ
h
1,j
T
γ
P
j
u, T
γ
P
j
u
3
− C
u
2
L
2
≥
j
γ
C
j
4
T
γ
P
j
u
2
L
2
− C
u
2
L
2
by Theorems C.21, C.22 first and the inequalities obtained above, hence
Re R
γ
v
T
γ
A
v
u, u≥( Cγ − C
) u
2
L
2
once more by Theorems C.21, C.22, Remark C.2 and the G˚arding inequality
(Theorem C.23, applied to the degree 0 symbol
j
P
∗
j
P
j
), which proves (9.1.27)
for large enough γ ≥ 2C
/C.
It is possible to go further than the L
2
estimates of Theorem 9.6 and
derive Sobolev estimates, provided that v enjoys some additional, though limited
regularity.
Theorem 9.7 In the framework of Theorem 9.6, assume, moreover, that W and
W
0
contain zero and v is compactly supported, with v ∈ H
m
(R
d−1
× R
+
× R) and
v
|x
d
=0
∈ H
m
(R
d−1
× R) for some integer m>(d +1)/2+1 and
v
H
m
(R
d−1
×R
+
×R))
≤ µ, v
|x
d
=0
H
m
(R
d−1
×R))
≤ µ.
Energy estimates 251
Then there exists γ
m
= γ
m
(ω, µ) ≥ 1 and C
m
= C
m
(ω, µ) > 0 such that, for all
γ ≥ γ
m
, for all u ∈ D(R
d−1
× R
+
× R)
γ u
2
H
m
γ
(R
d−1
×R
+
×R)
+ u
|x
d
=0
2
H
m
γ
(R
d
)
≤ C
m
1
γ
L
v
u
2
H
m
γ
(R
d−1
×R
+
×R)
+ B
v
u
|x
d
=0
2
H
m
γ
(R
d
)
.
Proof The idea is to prove first an estimate in L
2
(R
+
; H
m
γ
(R
d
)) and then infer
the estimate in H
m
γ
(R
d−1
× R
+
× R) by differentiation with respect to x
d
.
1) Tangential derivatives The L
2
(R
+
; H
m
γ
(R
d
)) estimate will be deduced
from the result of Theorem 9.6 (regarded as a L
2
(R
+
; H
0
γ
(R
d
)) estimate) applied
to derivatives of u in the (y, t) directions. In order to estimate the ‘error terms’
we will also need the non-linear estimate in Theorem C.12, as well as the product
and commutator estimates in Theorem C.10 and Corollary C.2, or more precisely
their ‘parameter version’ (the proof of which is left as an exercise) stated in the
following.
Lemma 9.3
r
For al l q, r, s with r + s>0 and q ≤ min(r, s), q<r+ s − d/2, there exists
C>0 so that for all a ∈ H
r
and all u ∈ H
s
γ
,
au
H
q
γ
≤ C a
H
r
u
H
s
γ
.
r
If m is an integer greater than d/2+1 and α is a d-uple of length |α|∈
[1,m], there exists C>0 such for all γ ≥ 1, that for all a in H
m
and all
u ∈ H
|α|−1
γ
,
e
−γt
[ ∂
α
,a] u
L
2
≤ C a
H
m
u
H
|α|−1
γ
.
So, let us take a d-uple α =(α
0
,α
) of length |α|≤m (with m>(d +1)/2+1),
and denote
u
α
:= ∂
α
u = ∂
α
0
t
∂
α
y
u
for u ∈ D (R
d−1
× R
+
× R). The L
2
(R
+
; H
0
γ
(R
d
)) estimate (Theorem 9.6)
applied to u
α
reads
γ e
−γt
u
α
2
L
2
(R
d−1
×R
+
×R)
+ e
−γt
(u
α
)
|x
d
=0
2
L
2
(R
d−1
×R)
≤ c
1
γ
e
−γt
L
v
u
α
2
L
2
(R
d−1
×R
+
×R)
+ e
−γt
(B
v
u
α
)
|x
d
=0
2
L
2
(R
d−1
×R)
.
(9.1.28)
Our main task is to estimate the right-hand side in terms of f := L
v
u, g :=
B
v
u
|x
d
=0
, and ‘error terms’ depending on u, and check the error terms can be
252 Variable-coefficients initial boundary value problems
absorbed in the left-hand side. Let us denote f
α
:= L
v
u
α
and g
α
:= (B
v
u
α
)
|x
d
=0
and begin with the easier case of g
α
. By definition we have
g
α
= ∂
α
g +[B
v
,∂
α
]u
|x
d
=0
.
The estimate of the first part is trivial, and the estimate of the commutator will
obviously rely on Lemma 9.3. A slight difficulty arises here because, unlike v, B
v
is not H
m
in general (recall that B(w) is of constant, generally non-zero rank
for w ∈ W
0
!). However, since v is H
m
and compactly supported,
B
v
= B
0
+ C
v
, with C
v
:= B
v
− B
0
∈ H
m
by Theorem C.12, and
C
v
H
m
(R
d
)
≤ c
m
(v
L
∞
) v
|x
d
=0
H
m
(R
d
)
.
Therefore
e
−γt
[B
v
,∂
α
]u
|x
d
=0
L
2
(R
d
)
= e
−γt
[C
v
,∂
α
]u
|x
d
=0
L
2
(R
d
)
≤ c
m,0
(ω, µ) u
|x
d
=0
H
|α|−1
γ
(R
d
)
by Lemma 9.3, which implies
e
−γt
[B
v
,∂
α
]u
|x
d
=0
L
2
(R
d
)
≤
1
γ
c
m,0
(ω, µ) u
|x
d
=0
H
|α|
γ
(R
d
)
.
(It is the factor 1/γ that will enable us to absorb this contribution in the left-hand
side). This yields a constant c
m,0
= c
m,0
(ω, µ) such that
|α|≤m
γ
2(m−|α|)
e
−γt
g
α
2
L
2
(R
d
)
≤ c
m,0
(ω, µ)
g
2
H
m
γ
(R
d
)
+
1
γ
2
u
|x
d
=0
2
H
m
γ
(R
d
)
.
The estimate of f
α
is a little trickier, because the coefficient of ∂
d
in L
v
is the
nonconstant matrix-valued function A
−1
d
. For this reason we write f
α
in the
following twisted way:
f
α
= A
d
v
∂
α
((A
d
v
)
−1
f )+A
d
v
[(A
d
v
)
−1
L
v
,∂
α
]u.
On the one hand, we have by definition of the H
m
γ
norm,
γ
m−|α|
e
−γt
A
d
v
∂
α
((A
d
v
)
−1
f )
L
2
(R
d−1
× R
+
×R)
≤A
d
v
L
∞
(A
d
v
)
−1
f
L
2
(R
+
;H
m
γ
(R
d
))
,
and by Lemma 9.3 and Theorem C.12,
(A
d
v
)
−1
f
L
2
(R
+
;H
m
γ
(R
d
))
≤
(A
d
0
)
−1
+ a
m
v
L
2
(R
+
;H
m
(R
d
))
f
L
2
(R
+
;H
m
γ
(R
d
))
Energy estimates 253
with a
m
depending continuously on v
L
∞
. On the other hand, the commutator
reads
[(A
d
v
)
−1
L
v
,∂
α
]u =[(A
d
v
)
−1
,∂
α
] ∂
t
u +
d−1
j=1
[(A
d
v
)
−1
A
j
v
,∂
α
] ∂
j
u.
Therefore, applying again Lemma 9.3 and Theorem C.12 (using the same trick
as for B to deal with the fact that the matrices (A
d
(0))
−1
A
j
(0) are non-zero),
we find a constant c
m,1
depending only on v
L
∞
and v
L
2
(R
+
;H
m
(R
d
))
such that
e
−γt
[(A
d
v
)
−1
L
v
,∂
α
]u
L
2
(R
d−1
× R
+
× R)
≤ c
m,1
(∂
t
u, ∂
1
u,...,∂
d−1
u)
L
2
(R
+
;H
|α|−1
γ
(R
d
))
,
hence
e
−γt
[(A
d
v
)
−1
L
v
,∂
α
]u
L
2
(R
d−1
× R
+
× R)
≤ c
m,1
u
L
2
(R
+
;H
|α|
γ
(R
d
))
.
So, finally, we find a constant c
1,m
= c
1,m
(ω, µ) such that
|α|≤m
γ
2(m−|α|)
e
−γt
f
α
2
L
2
(R
d−1
× R
+
× R)
≤ c
1,m
f
2
L
2
(R
+
;H
m
γ
(R
d
))
+ u
2
L
2
(R
+
;H
m
γ
(R
d
))
.
Consequently, summing on α the inequality (9.1.28) and using the estimates
of the right-hand side obtained here above we find that
γ u
2
L
2
(R
+
;H
m
γ
(R
d
))
+ u
|x
d
=0
2
H
m
γ
(R
d
)
≤ c
1
γ
f
2
L
2
(R
+
;H
m
γ
(R
d
))
+ g
2
H
m
γ
(R
d
)
+
1
γ
u
2
L
2
(R
+
;H
m
γ
(R
d
))
+
1
γ
2
u
|x
d
=0
2
H
m
γ
(R
d
)
,
with c = c max(c
0,m
, c
1,m
), hence
γ u
2
L
2
(R
+
;H
m
γ
(R
d
))
+ u
|x
d
=0
2
H
m
γ
(R
d
)
≤ 2c
1
γ
L
v
u
2
L
2
(R
+
;H
m
γ
(R
d
))
+ B
v
u
|x
d
=0
2
H
m
γ
(R
d
)
(9.1.29)
for γ ≥
√
2c.
2) Normal derivatives Once we have (9.1.29), we readily infer a bound for
the L
2
norm of e
−γt
∂
d
u by writing
∂
d
u =(A
d
v
)
−1
f − ∂
t
u +
d−1
j=1
A
j
v
∂
j
u
. (9.1.30)
254 Variable-coefficients initial boundary value problems
More generally, for any differentiation operator ∂
α
in the (y,t) direction only,
the d-uple α being of length |α|≤m − 1, writing
∂
d
∂
α
u =(A
d
v
)
−1
f
α
− ∂
t
∂
α
u +
d−1
j=1
A
j
v
∂
j
∂
α
u
,
and using the estimate of f
α
:= L
v
∂
α
u obtained in the first part, namely
γ
2(m−|α|)
e
−γt
f
α
2
L
2
(R
d−1
× R
+
× R)
f
2
L
2
(R
+
;H
m
γ
(R
d
))
+ u
2
L
2
(R
+
;H
|α|
γ
(R
d
))
,
(9.1.31)
together with the L
2
(R
+
; H
m
γ
(R
d
)) estimate (9.1.29) we find that
γ e
−γt
∂
d
u
α
2
L
2
(R
d−1
×R
+
×R)
1
γ
L
v
u
2
L
2
(R
+
;H
m
γ
(R
d
))
+ B
v
u
|x
d
=0
2
H
m
γ
(R
d
)
.
Finally, we can show by induction on k, differentiation of (9.1.30) and repeated
use of Lemma 9.3 (in dimension d + 1 instead of d), the estimate
γ e
−γt
∂
k
d
u
α
2
L
2
(R
d−1
×R
+
R)
1
γ
L
v
u
2
H
m
γ
(R
d−1
×R
+
×R))
+ B
v
u
|x
d
=0
2
H
m
γ
(R
d
)
.
for all integer k and all d-uple such that k + |α|≤m. (The details are left
to the reader.) Summing all these inequalities with (9.1.29) we obtain the
H
m
γ
(R
d−1
× R
+
× R) estimate announced. 2
Remark 9.10 In the proof of Theorem 9.7 here above, we have used, in a
crucial way, the inequality
u
H
k−1
γ
(R
d
)
1
γ
u
H
k
γ
(R
d
)
,
which either can be viewed as a straightforward consequence of the equivalence
u
H
k
γ
(R
d
)
e
−γt
u
H
k
γ
(R
d
)
(independently of γ) and the inequality
v
H
m−1
γ
(R
d
)
≤
1
γ
v
H
m
γ
(R
d
)
,
or can be proved directly by using the definition
u
H
m−1
γ
(R
d
)
=
|α|≤m−1
γ
2(m−1−|α|)
e
−γt
∂
α
u
L
2
(R
d
)
and the inequality
e
−γt
w
L
2
(R
d
)
≤
1
γ
e
−γt
∂
t
w
L
2
(R
d
)
(9.1.32)
for w ∈ H
1
γ
(R
d
)=e
γt
L
2
(R
d
). The latter can be proved in a completely
elementary way. Indeed, by integration by parts we have γ e
−γt
w
2
L
2
=
Re e
−γt
w, e
−γt
∂
t
w, which implies (9.1.32) merely by the Cauchy–Schwarz
inequality. We point out this here because, when R
d
is replaced by R
d
× [0,T]
(as we shall do later), the inequality (9.1.32) is no longer valid.