Finite Propagation Speed for First Order Systems and Huygens' Principle for Hyperbolic Equations

0.2 MB·English

by  Alan McIntosh

FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS
AND HUYGENS’ PRINCIPLE FOR HYPERBOLIC EQUATIONS
ALAN MCINTOSH AND ANDREW J. MORRIS
Abstract. We prove that strongly continuous groups generated by first order
systems on Riemannian manifolds have finite propagation speed. Our procedure
provides a new direct proof for self-adjoint systems, and allows an extension to
2
operatorsonmetricmeasurespaces. Asanapplication,wepresentanewapproach
1
to the weak Huygens’ principle for second order hyperbolic equations.
0
2

n
a
J
1. Introduction
5
2 For a self-adjoint first order differential operator D acting on a space L2( ),
V
] where is a vector bundle over a complete Riemannian manifold M, it is known
P V
that the unitary operators eitD act with finite propagation speed. Indeed, if the
A
principal symbol satisfies
.
h
t (1.1) σ (x,ξ) κ ξ x M, ξ T∗M
a | D | ≤ | | ∀ ∈ ∀ ∈ x
m
for some positive number κ, and if sppt(u) K M, then
[ ⊂ ⊂

2 sppt(eitDu) Kκ|t| := x M ; dist(x,K) κ t t R,
⊂ { ∈ ≤ | |} ∀ ∈
v
8 that is, eitD has finite propagation speed κ κ. See, for instance, Proposi-
D
1 ≤
tions 10.2.11 and 10.3.1 in [13].
8
1 Noting that σD(x, η(x))u(x) = [D,ηI]u(x) for all bounded real-valued C1 func-
∇
. tions η on M, we see that condition (1.1) implies that the commutator [ηI,D] is a
1
0 multiplication operator which satisfies the bound
2
1 (1.2) [ηI,D]u κ η u u Dom(D) Dom(Dη),
2 ∞ 2
: k k ≤ k∇ k k k ∀ ∈ ⊂
v
i where Dom(D) denotes the domain of D and [ηI,D]u = ηDu D(ηu).
X −
On stating the result in terms of commutators, we can remove the differentiability
r
a assumptions altogether and consider operators D defined on metric measure spaces
instead. Our aim is to also weaken the self-adjointness condition on D to the
requirement that iD generates a C0 group (eitD)t∈R with
(1.3) eitDu ceω|t| u t R, u L2( )
2 2
k k ≤ k k ∀ ∈ ∀ ∈ V
for some c 1 and ω 0. (When D is self-adjoint, condition (1.3) holds with
≥ ≥
c = 1 and ω = 0.) At the same time we could replace the L2 space by an Lp space
if we wished. This requires a new proof of finite propagation speed. Let us state the
result here for Riemannian manifolds.
Date: 24 January 2012.
2010 Mathematics Subject Classification. Primary: 35F35, 35L20; Secondary: 47D06.
Key words and phrases. Finite propagation speed, first order systems, C0 groups, Huygens’
principle, hyperbolic equations.
1
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 2
Theorem 1.1. Let D be a first order differential operator which acts on a space
L2( ), where is a complex vector bundle with a Hermitian metric, over a separable
V V
Riemannianmanifold M. Suppose that iD generates a C0 group (eitD)t∈R satisfying
(1.3) and that the commutators of D with bounded real-valued C∞ functions η on M
satisfy (1.2). Then the group (eitD)t∈R has finite propagation speed κD cκ.
≤
In particular, the operator D acting on L2( ) could denote a first order system
acting on L2(Rn, CN) for some positive integerVs n and N, or on L2(Ω, CN) where
Ω is an open subset of Rn. We remark that the constants in (1.2) and (1.3) could
be with respect to another norm on L2( ) equivalent to the standard one. We shall
V
return to this point.
In Section 3, we prove our main result, Theorem 3.1, which is a generalisation
of Theorem 1.1 to metric measure spaces. The proof utilises a higher-commutator
technique introduced by McIntosh and Nahmod in Section 2 of [16], and used to
derive off-diagonal estimates, otherwise known as Davies–Gaffney estimates, by Ax-
elsson, Keith and McIntosh in Proposition 5.2 of [5], and by Carbonaro, McIntosh
and Morris in Lemma 5.3 of [8]. The proof also simplifies the argument based on
energy estimates that is known for self-adjoint operators.
A weak Huygens’ principle for second order hyperbolic equations on Rn is proved
as an application in Section 5. Homogeneous and inhomogeneous hyperbolic equa-
tions are treated separately. The homogeneous version in Theorem 5.2 only requires
the finite propagation speed result for self-adjoint first order systems. The inhomo-
geneous version in Theorem 5.6, however, requires the generality of Theorem 3.1.
These results are achieved by introducing a first order elliptic system BD, where
D is a first order constant coefficient system, and B is a multiplication operator,
such that the second order hyperbolic equation contains a component of the system
(BD)2. This approach is motivated by the work of Auscher, McIntosh and Nah-
mod [4], and of Axelsson, Keith and McIntosh [5, 6], in which the solution of the
Kato square root problem for second order elliptic operators is reduced to proving
quadratic estimates for related first order elliptic systems. See also the survey by
Auscher, Axelsson and McIntosh [3].
2. Notation
Henceforth, M denotesametric measure spacewithametric d(x,y) anda σ-finite
Borel measure µ. In particular, M could be a Riemannian manifold as in the Intro-
duction, or an open subset of Rn with Euclidean distance and Lebesgue measure.
If K,K˜ M and x M, then d(x,K) := inf d(x,y); y K and d(K,K˜) :=
⊂ ∈ { ∈ }
inf d(x,y); x K,y K˜ . We define, for τ > 0, K := x M ; d(x,K) τ .
τ
{ ∈ ∈ } { ∈ ≤ }
By Lip(M) we mean the space of all bounded real-valued functions η on M with
finite Lipschitz norm
η = sup |η(x)−η(y)| .
k kLip d(x,y)
x6=y
Whenever K M and α > 0, the real-valued function η defined by
K,α
⊂
(2.1) η (x) := max 1 αd(x,K), 0
K,α
{ − }
belongs to Lip(M) with η α, and sppt(η ) K .
K,α Lip K,α 1/α
k k ≤ ⊂
A vector bundle over M refers to a complex vector bundle π : M equipped
V V →
with a Hermitian metric , that depends continuously on x M. The examples
x
to be presented in Sectiohn· ·4i will be trivial bundles M Cm∈with inner product
×
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 3
ζ,ξ = ζ ξ . For every vector bundle , there are naturally defined Banach
h ix j j j V
spaces Lp( ), 1 p , of measurable sections. In particular, L2( ) denotes the
PV ≤ ≤ ∞ V
Hilbert space of square integrable sections of with the inner product (u,v) :=
u(x),v(x) dµ(x). In the case of the triviaVl bundle M Cm, these are denoted
aMs uhsual by Lpi(xM,Cm). ×
R
The Banach algebra of all bounded linear operators on a Banach space is
denoted by ( ). Given A L∞(M, (Cm)), the same symbol A is also useXd to
denote the mLuXltiplication ope∈rator on LLp(M,Cm) defined by u Au. Note that
7→
Au A u . Multiplication operators on Lp( ) are defined in the natural
p ∞ p
k k ≤ k k k k V
way. For any function η Lip(M), the multiplication operator ηI : Lp( ) Lp( )
∈ V → V
is defined by (ηI)u(x) := η(x)u(x) for all u Lp( ) and µ-almost all x M. This
∈ V ∈
is a multiplication operator by virtue of the facts that η is bounded and continuous,
and µ is a Borel measure. The commutator [A,T] of a multiplication operator A
with a (possibly unbounded) operator T in Lp( ) with domain Dom(T) is defined
by [A,T]u = ATu TAu provided u Dom(T)V Dom(TA).
− ∈ ∩
Given an operator D in Lp( ) (1 p < ), we say that iD generates a C
0
group (V(t))t∈R provided t VV(t) is≤a stron∞gly continuous mapping from R to
7→
(Lp( )) with V(s + t) = V(s)V(t), V(0) = I (the identity map on Lp( )) and
LdV(t)Vu = iDu for all u Dom(D) = u Lp( ); dV(t)u exists inVLp( ) .
dt |t=0 ∈ { ∈ V dt |t=0 V }
We write V(t) = eitD. Such a group automatically has dense domain Dom(D),
and satisfies an estimate of the form eitD ceω|t| for some c 1 and ω 0.
k k ≤ ≥ ≥
An introduction to the theory of strongly continuous groups can be found in, for
instance, [14] or [11]. We remark that, when D is self-adjoint in L2( ), Stone’s
V
Theorem guarantees that the operators eitD are unitary, so iD generates a C
0
group with c = 1 and ω = 0.
The group (eitD)t∈R is said to have finite propagation speed when there exists a
finite constant κ 0, such that for all u Lp( ) satisfying sppt(u) K M, and
all t R, it holds≥that sppt(eitDu) K ∈. ThVe propagation speed κ⊂ is⊂defined to
κ|t| D
∈ ⊂
be the least such κ.
3. The Main Result
The following theorem is the main result of the paper. Theorem 1.1 is proved as
a special case at the end of the section.
Theorem3.1. Let denotea complexvectorbundle overa metric measurespace M
and let 1 p < .VSuppose that D : Dom(D) Lp( ) Lp( ) is a linear opera-
≤ ∞ ⊂ V → V
tor with the following properties:
(1) there exist finite constants c 1 and ω 0 such that iD generates a C
0
≥ ≥
group (eitD)t∈R in Lp( ) with eitDu p ceω|t| u p t R, u Lp( );
V k k ≤ k k ∀ ∈ ∀ ∈ V
(2) there exists a finite constant κ > 0 such that for all η Lip(M), one has
ηu Dom(D) and [ηI,D]u κ η u and [ηI,[∈ηI,D]]u = 0 for all
p Lip p
u ∈Dom(D). k k ≤ k k k k
∈
Then the group (eitD)t∈R has finite propagation speed κD cκ.
≤
Remark 3.2. As the commutator [ηI,D] is bounded on the dense domain Dom(D),
it extends uniquely toanoperator(denoted by thesame symbol) [ηI,D] (Lp( ))
∈ L V
with the same bound.
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 4
Remark 3.3. The theorem remains true when the norm on Lp( ) is replaced by
V
another equivalent norm. This is used later in Case II of Section 4.
Remark 3.4. The results remain true in Bochner spaces L2( ) when the fibres of
V V
are infinite dimensional Hilbert spaces.
For completeness, we prove a known formula for the commutator [ηI,eitD].
Lemma 3.5. Under the hypotheses of Theorem 3.1, the following holds:
1
[ηI,eitD]u = it eistD[ηI,D]ei(1−s)tDuds t R, u Lp( ).
∀ ∈ ∀ ∈ V
Z0
Proof. It suffices to verify the expression when u Dom(D). The property that
(eitD)t∈R is a C0 group then guarantees that eitD∈u Dom(D) with derivative
d(eitDu) = iDeitDu = ieitDDu for all t R. The pr∈operty that ηu Dom(D)
dt ∈ ∈
then implies that eistD(ηI)ei(1−s)tDu is differentiable with respect to s, with
d ′ ′
eistD(ηI)ei(1−s)tDu = ei(·)tD(ηI)ei(1−s)tDu (s)+eistD (ηI)ei(1−·)tDu (s)
ds
(cid:16) (cid:17) = (cid:16)iteistD[ηI,D]ei(1−s)(cid:17)tDu (cid:16) (cid:17)
−
for all s R. This version of the chain rule can be found in, for instance,
∈
Lemma B.16 in [11]. Using the fundamental theorem of calculus, we then have
1 d 1
[ηI,eitD]u = eistD(ηI)ei(1−s)tDu ds = it eistD[ηI,D]ei(1−s)tDuds
− ds
Z0 (cid:16) (cid:17) Z0
(cid:3)
as required.
Proof of Theorem 3.1. Given t R and u Lp( ) with sppt(u) K M, our
∈ ∈ V ⊂ ⊂
aim is to prove that sppt(eitDu) K . To do this, it suffices to prove that
cκ|t|
(eitDu,v) = 0 for all v Lp′( ) wi⊂th d(sppt(v),K) > cκ t (where p′ = p ). Let
∈ V | | p−1
us fix K,t,u,v, and choose α > 0 such that cκ t < 1/α < d(sppt(v),K). On
| |
defining the cut-off function η := η Lip(M) as in (2.1), we have ηu = u,
K,α
∈
ηv = 0 and cκ t η cκ t α < 1.
Lip
| |k k ≤ | |
To simplify the computations, set δ : (Lp( )) (Lp( )) to be the derivation
L V → L V
defined by δ(S) = [ηI,S] for all S (Lp( )), and adopt the convention that
∈ L V
δ0(S) := S. We see that
(3.1) (eitDu,v) = (eitDηnu,v) = (δ(eitD)η(n−1)u,v) =...= ( 1)n(δn(eitD)u,v)
− −
for all n N. The derivation formula δ(ST) = δ(S)T +Sδ(T) is readily verified for
∈
any S,T (Lp( )). Using Lemma 3.5 and the fact, given by property (2) of D,
∈ L V
that δ([ηI,D]) = [ηI,[ηI,D]] = 0, we then obtain
1 m
(3.2) δm+1(eitD)u = it m δm−k(eistD)[ηI,D]δk(ei(1−s)tD)uds
k
Z0 k=0
X(cid:0) (cid:1)
for all m N := N 0 , where the binomial coefficient m := m! . We now
∈ 0 ∪{ } k k!(m−k)!
prove by induction that
(cid:0) (cid:1)
(3.3) δn(eitD) (c t [ηI,D] )nceω|t|
k k ≤ | |k k
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 5
for all n N . For n = 0, this is given by property (1) of D. Now let m N and
0
∈ ∈
suppose that (3.3) holds for all integers n m. We then use (3.2) to obtain
≤
1 m
δm+1(eitD) t m δm−k(eistD) [ηI,D] δk(ei(1−s)tD) ds
k k ≤ | | k k kk kk k
Z0 k=0
X(cid:0) (cid:1)
1 m
(c t [ηI,D] )m+1ceω|t| m sm−k(1 s)kds
≤ | |k k k −
Z0 k=0
X(cid:0) (cid:1)
1
= (c t [ηI,D] )m+1ceω|t| (s+(1 s))mds
| |k k −
Z0
= (c t [ηI,D] )m+1ceω|t| .
| |k k
This proves (3.3) for all n N .
0
∈
Therefore, using the estimate (3.3) in (3.1), together with property (2), we obtain
(eitDu,v) (cκ t η Lip)nceω|t| u p v p′ (cκ t α)nceω|t| u p v p′
| | ≤ | |k k k k k k ≤ | | k k k k
for all n N . We have cκ t α < 1, so (eitDu,v) = 0 as required. (cid:3)
0
∈ | |
Remark 3.6. In fact we have proved the stronger statement that
sppt(eitDu) K˜ (D) K ,
t cκ|t|
⊂ ⊂
˜
where K (D) = sppt(η); η Lip(M), η 1 on K, c t [ηI,D] < 1 .
t ∞
For example, i∩f{M = Rn an∈d ∂ does no≡t appear in|D|k, then tkhere is}no propa-
∂x1
gation in the x direction. See the recent paper of Cowling and Martini [9] for some
1
related results.
We conclude the section by proving Theorem 1.1.
Proof of Theorem 1.1. From what was explained in the introduction, together with
the fact that [ηI,[ηI,D]] = 0 when [ηI,D] is a multiplication operator, Lemma 3.5
also holds under the hypotheses of Theorem 1.1. We then consider ǫ > 0 and
repeat the proof of Theorem 3.1, replacing the cut-off function η Lip(M) with
K,α
∈
a [0,1]-valued function η˜ C∞(M) satisfying η˜ = 1 on K, sppt(η˜) K , and
1/α
∈ ⊂
η˜ (1 + ǫ) η . These approximations exist because the Riemannian
∞ K,α Lip
k∇ k ≤ k k
manifold M is separable (see, for instance, Corollary 3 in [7]). The result follows
because ǫ > 0 can be chosen arbitrarily. (cid:3)
Remark 3.7. The only reason that the Riemannian manifold M was required to be
separable in Theorem 1.1, was so that we could construct smooth approximations to
Lipschitz functions in the proof above. Indeed, Theorem 3.1 provides an analogous
result without requiring separability.
4. Some Special Cases
A typical example of a first order system is the Hodge–Dirac operator D = δ+δ∗
actingon L2(Λ(M)) when M isacompleteRiemannianmanifold. Forthisoperator,
the group (eitD)t∈R has finite propagation speed 1. We shall consider the case when
the manifold is Rn or an open subset thereof, and restrict attention to the leading
components of the Hodge–Dirac operator (the components acting between scalar-
valued functions and vector fields), and perturbations thereof. For this purpose,
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 6
when Ω is an open subset of Rn, we require the Sobolev space W1,2(Ω) consisting
of all f in L2(Ω) with generalised derivatives satisfying
f 2 := f 2 + f 2 < ,
k kW1,2(Ω) k kL2(Ω) k∇ kL2(Ω) ∞
where f = (∂ f) .
j j=1,...,n
∇
Case I. Let M = Rn (n N), and let D denote the self-adjoint operator
∈
W1,2(Rn) L2(Rn) L2(Rn)
0 div
D = − :
(cid:20) ∇ 0 (cid:21) Dom⊕(div) ⊂ L2(R⊕n,Cn) → L2(R⊕n,Cn),
where : f (∂ f) has domain W1,2(Rn), and div = ∗ : (u ) ∂ u
∇ 7→ j j −∇ j j 7→ j j j
has domain u L2(Rn,Cn); divu L2(Rn) .
{ ∈ ∈ } P
It follows from known results that (eitD)t∈R has finite propagation speed 1. It is
also a consequence of Theorem 3.1 with c = 1 and ω = 0 because D is self-adjoint,
and with κ = 1 because (using η = η ) we have
∞ Lip
k∇ k k k
f (∂ η)u f
[ηI,D] = j j j η η Lip(Rn).
(u ) (∂ η) f ≤ k kLip (u ) ∀ ∈
(cid:13) (cid:20) j j (cid:21)(cid:13)2 (cid:13)(cid:20) P− j j (cid:21)(cid:13)2 (cid:13)(cid:20) j j (cid:21)(cid:13)2
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
C(cid:13)ase II. Let D d(cid:13)enote(cid:13)theoperatorinC(cid:13)aseI,andc(cid:13)onsiderthe(cid:13)perturbedoperator
BD with thesame domain, where B L∞(Rn, (C1+n)) satisfies B(x)ζ,ζ λ ζ 2
fora.e. x Rn andall ζ C1+n, fors∈ome λ > 0L. Themultiplicatiohnoperatior≥B |is|a
strictlypo∈sitiveself-adjoi∈ntoperatorin L2(Rn,C1+n), since (Bu,u) λ u 2. Hence
≥ k k
B1/2 = B 1/2 = B 1/2 and B−1, B−1/2 both exist as bounded operators.
∞
k Usinkg tkheske factsk, wke find that BD is self-adjoint in L2(Rn,C1+n) under the
inner product (u,v) := (B−1u,v), whose associated norm u = B−1/2u is
B B
equivalent to u . So iBD generates a C0 group (eitBD)t∈R ink Lk2(Rn,kC1+n) wkith
k k
eitBDu B 1/2 eitBDu B 1/2 u λ−1/2 B 1/2 u .
∞ B ∞ B ∞
k k ≤ k k k k ≤ k k k k ≤ k k k k
The commutator [ηI,BD] = B[ηI,D] is a multiplication operator which satisfies
[ηI,BD]u B ∞ η Lip u , so by Theorem 3.1, the group (eitBD)t∈R has finite
k k ≤ k k k k k k
propagation speed κ λ−1/2 B 3/2.
BD ∞
≤ k k
Actually, thiscanbeimproved. AsnotedinRemark3.3, theequivalentnorm u
B
on L2(Rn,C1+n) can be used in the proof of Theorem 3.1. The operator BD iskseklf-
adjoint in this norm, and [ηI,BD]u = B−1/2B[ηI,D]u B η u ,
B ∞ Lip B
k k k k ≤ k k k k k k
so we conclude that (eitBD)t∈R has finite propagation speed κBD B ∞.
≤ k k
Case III. Now we allow inhomogeneous terms, and consider an operator of the
form BD acting on an open subset Ω Rn. Suppose that V is a closed subspace of
⊂
W1,2(Ω) which contains C∞ (the C∞ functions with compact support), and which
c
has the property that ηf V for all η Lip(Ω) and f V . For example, the
∈ ∈ ∈
space W1,2(Ω) itself has this property, as does W1,2(Ω) (the closure of C∞(Ω) in
0 c
W1,2(Ω)). (This last statement follows from the facts that, given η Lip(Ω), then
∈
ηf W1,2(Ω) for all f C∞(Ω) and ηI (W1,2(Ω)).)
D∈efin0e : V L2(∈Ω) c L2(Ω,Cn) b∈yL f = (∂ f) , and set div = ∗.
V V j j V V
That is, d∇iv u =⊂divu for→all u Dom(div∇ ) = u L2(Ω,Cn); divu −L∇2(Ω)
V V
and ( f ,u) = (f ,divu) f V∈ . In particular{, w∈e have ηu Dom(d∈iv ) for
V
all η −L∇ip(Ω) and u Dom∀(div∈ ).} ∈
V
∈ ∈
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 7
Define the self-adjoint operator
V L2(Ω) L2(Ω)
0 I div
V
− ⊕ ⊕ ⊕
D = I 0 0 : L2(Ω) L2(Ω) L2(Ω)
  ⊂ →
0 0
V
 ∇  Dom⊕(divV) L2(Ω⊕,Cn) L2(Ω⊕,Cn)
and the multiplication operator
a 0 0
B = 0 A (A ) L∞(Ω, (C2+n))
00 0k
  ∈ L
0 (A ) (A )
j0 jk
 
with a(x) λ > 0 and n A (x)ζ ζ λ ζ 2 for a.e. x Ω and all ζ Cn,
≥ j,k=1 jk k j ≥ | | ∈ ∈
for some λ > 0.
P
Theoperator BD satisfies(1.2)asinCaseII.If A,andhence B,werepositiveself-
adjoint, then iBD would generate a C group as before, but we have only assumed
0
this for the matrix-valued function (A ) with j,k 1,...,n . We remedy this by
jk
∈ { }
writing
0 a adiv
BD = A + A ∂ 0 − 0 = B˜D +C ,
 00 k 0k k 
(A + A ∂ ) 0 0
j0 Pk jk k
 
a 0 0 0 0 0
P
where B˜ = 0 ReA +α (A ) , C = iImA α 0 0 , and α > 0
00 0k 00
   − 
0 (A ) (A ) (A A ) 0 0
0j jk j0 0j
−
is chosen large enough so that B˜(x)ζ,ζ λ ζ 2 for a.e. x Ω and all ζ
h i ≥ 2| | ∈ ∈
C2+n. As in Case II, we see that B˜D is self-adjoint in L2(Ω,C2+n) under the inner
product (u,v) := (B˜−1u,v), so iB˜D generatesa C groupwith eitB˜Du u .
0 k kB˜ ≤ k kB˜
Lemma 4.1 below then allows us to deduce that iBD = iB˜D +iC generates a C
0
group (eitBD)t∈R with
eitBDu B˜ 1/2 eit(B˜D+C)u B˜ 1/2eω˜|t| u c˜eω˜|t| u
k k ≤ k k∞ k kB˜ ≤ k k∞ k kB˜ ≤ k k
for some finite c˜ 1 and ω˜ 0. By Theorem 3.1, we conclude that (eitBD)t∈R has
≥ ≥
finite propagation speed.
Lemma 4.1. Let be a Banach space, and suppose that T : Dom(T) is
X ⊂ X → X
a linear operator that generates a C0 group (etT)t∈R in satisfying etT ceω|t|
for some c 1 and ω 0. If B ( ), then the sum TX+B on Dom(kT) gke≤nerates
≥ ≥ ∈ L X
a C0 group (et(T+B))t∈R on satisfying et(T+B) ce(ω+ckBk)|t|.
X k k ≤
Lemma 4.1 is a well known result based on the work of Phillips in [17]. The proof
for semigroups in Theorem III.1.3 of [11] can be extended to give the above result.
5. Weak Huygens’ Principle
In this section, we apply Theorem 3.1 to prove a weak Hugyens’ principle for
second order hyperbolic equations. For motivation, we start with the wave equation
on Rn. A homogeneous equation with bounded measurable coefficients is treated
next, followed by an inhomogeneous version on a domain Ω Rn.
⊂
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 8
L2(Rn)
0 div
In Case I, we considered the operator D = − in , which
(cid:20) ∇ 0 (cid:21) L2(R⊕n,Cn)
is self-adjoint, and noted that iD generates a C0 group (eitD)t∈R with finite propa-
gation speed 1. Consequently, the cosine family cos(tD) = 1(eitD +e−itD) (t 0)
2 ≥
also has finite propagation speed 1, where this is defined in the obvious way. Note
that the cosine operators, being even functions of D, satisfy cos(tD) = cos(t√D2),
∆ 0
where D2 = − and ∆ = div denotes the Laplacian operator with
0 div ∇
(cid:20) −∇ (cid:21)
domain Dom(∆) = f W1,2(Rn); f Dom(div) . On restricting attention to
{ ∈ ∇ ∈ }
the first component, we deduce that the cosine family (cos(t√ ∆)) has finite
t≥0
−
propagation speed 1. This is at the heart of the weak Huygens’ principle for the
wave equation:
Theorem 5.1. If f W1,2(Rn), g L2(Rn) with sppt(f) sppt(g) K Rn,
∈ ∈ ∪ ⊂ ⊂
then the solution
t
F(t) = cos(t√ ∆)f + cos(s√ ∆)gds
− −
Z0
of the Cauchy problem
∂2 F(t) ∆F(t) = 0 (t > 0)
∂t2 −
limF(t) = f
t→0
lim ∂ F(t) = g
∂t
t→0
has support sppt(F(t)) K .
t
⊂
This result is very well known. The solution F belongs to C1(R+,L2(Rn))
C0(R+,W1,2(Rn)). There is a considerable literature on the wave equation, so w∩e
shall not proceed further with statements of uniqueness, energy estimates, etc.
We turn now to the corresponding result for homogeneous equations with L∞
coefficients. Let L = adivA = a n ∂ A ∂ , where a L∞(Rn) and A =
− ∇ − j,k=1 j jk k ∈
(A ) L∞(Rn, (Cn)) with a(x) λ > 0 and A(x)ζ,ζ λ ζ 2 for a.e. x Rn
jk
and al∈l ζ Cn.L Here L : Dom(L≥) PL2(Rn) h L2(Rni)≥wit|h|Dom(L) = ∈f
W1,2(Rn);∈A f Dom(div) . ⊂ → { ∈
∇ ∈ }
Theorem 5.2. If f W1,2(Rn), g L2(Rn) with sppt(f) sppt(g) K Rn,
∈ ∈ ∪ ⊂ ⊂
then the solution
t
F(t) = cos(t√L)f + cos(s√L)gds C1(R+,L2(Rn)) C0(R+,W1,2(Rn))
∈ ∩
Z0
of the Cauchy problem
∂2 F(t)+LF(t) = 0 (t > 0)
∂t2
limF(0) = f
t→0
lim ∂ F(0) = g
∂t
t→0
has support sppt(F(t)) K , where α := ( a A )1/2.
αt ∞ ∞
⊂ k k k k
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 9
a 0 1/2
Proof. Apply Case II with B = β and β = kak∞ . Then
0 βA kAk∞
(cid:20) (cid:21)
(cid:16) (cid:17)
a 0 0 div 0 a div
BD = β − = −β
0 βA 0 βA 0
(cid:20) (cid:21)(cid:20) ∇ (cid:21) (cid:20) ∇ (cid:21)
L 0
and (BD)2 = with L as above and L˜ = A adiv. From Case II, the
0 L˜ − ∇
(cid:20) (cid:21)
C0 group (eitBD)t∈R has finite propagation speed κBD B ∞ = α. On defining
≤ k k
cos(tBD) = 1(eitBD +e−itBD), it is clear that the cosine family
2
cos(t√L) 0
(cos(tBD)) = (cos(t (BD)2)) =
t≥0 t≥0 " 0 cos(t L˜) #!
t≥0
p
has the same bound on its propagation speed. It follows that thpe first component
(cos(t√L)) , acting on L2(Rn), has the same bound α on its propagation speed,
t≥0
(cid:3)
as required.
Remark 5.3. The operators cos(tBD) agree with those defined in the standard func-
tional calculus of the self-adjoint operator BD acting on L2(Rn,C1+n) with inner
product (u,v) = (B−1u,v). Their properties are those of cosine functions as pre-
B
sented, for example, in [1, 2]. Note that the use of the square root sign is purely
a symbolism to express the fact that the cosine operators cos(tBD) are diagonal.
The function cos(t√z) is an analytic function of z.
Remark 5.4. When a = 1, then L is the self-adjoint operator in L2(Rn) associated
with the sesquilinear form J : W1,2(Rn) W1,2(Rn) C defined by
A
× →
n
J (f,g) = A (x)(∂ f(x))∂ g(x)dx
A jk k j
Rn
Z j,k=1
X
for all f, g W1,2(Rn). See, for example, Chapter IV of [14]. The weak Huygens’
∈
principle is well known for these operators. See, for example, [18]. Degenerate
elliptic operators are also treated in [10].
Remark 5.5. Our methods work also for systems where the functions are CN-valued
for some N N, and A L∞(CnN) satisfies G˚arding’s inequality:
∈ ∈
J (f,f) λ f 2 f W1,2(Rn,CN) .
A ≥ k∇ kL2(Rn,CnN) ∀ ∈
The proof that iBD generates a C group needs to be modified as follows.
0
The positivity condition on B L∞(Rn, (CN+nN)) is weakened to (BDu,Du)
∈ L ≥
λ u 2 for all u Dom(D). Then, following [3], L2(Rn,CN+nN) = N(D) R(BD)
wkithkcorrespond∈ing projections PN and PR. (Here N(D) denotes the nu⊕llspace of
D and R(BD) denotes the range of BD) The projections, which are typically non-
orthogonal, commute with BD. Moreover B : R(D) R(BD) has a bounded
→
inverse B−1 : R(BD) R(D). The operator BD is self-adjoint in L2(Rn,CN+nN)
under the inner produ→ct (u,v)B := (PNu,PNv)+(B−1PRu,PRv), whose associated
norm u is equivalent to u . Thus iBD generates a C group. Proceed as in
B 0
k k k k
Case II, though note that the bound on κ is not the same as before.
BD
Finally we consider inhomogeneous hyperbolic equations on a domain Ω Rn.
⊂
As in Case III, suppose that V is a closed subspace of W1,2(Ω) which contains
C∞(Ω), and which has the property that ηf V for all η Lip(Ω) and f V .
c ∈ ∈ ∈
FINITE PROPAGATION SPEED FOR FIRST ORDER SYSTEMS 10
Suppose also that a,A L∞(Rn) (j,k = 0,...,n) with a(x) λ > 0 and
jk
∈ ≥
n A ζ ζ λ ζ 2 for a.e. x Ω and all ζ Cn.
j,k=1 jk k j ≥ | | ∈ ∈
Define : V L2(Ω) L2(Ω,Cn) by f = (∂ f) , and set div = ∗.
V V j j V V
PThat is, d∇iv u =⊂divu for→all u Dom(div∇ ) = u L2(Ω,Cn); divu −L∇2(Ω)
V V
∈ { ∈ ∈
and ( f ,u) = (f ,divu) f V .
−∇ ∀ ∈ }
Define the operator L in L2(Ω) by
n n n
Lu = a ∂ A ∂ u a ∂ A u+a A ∂ u+aA u
j jk k j j0 0k k 00
− −
j,k=1 j=1 k=1
X X X
A (A ) I
= a I div 00 0k u
− V (A ) (A )
j0 jk V
(cid:20) (cid:21)(cid:20) ∇ (cid:21)
with Dom(L) = u(cid:2) V ; ( n (cid:3)A ∂ u+A u) Dom(div ) .
{ ∈ k=1 jk k j0 j ∈ V }
Theorem 5.6. If f V ,Pg L2(Ω) with sppt(f) sppt(g) K Ω, then the
∈ ∈ ∪ ⊂ ⊂
solution
t
(5.1) F(t) = cos(t√L)f + cos(s√L)gds
Z0
of the Cauchy problem
∂2 F(t)+LF(t) = 0 (t > 0)
∂t2
limF(0) = f
t→0
lim ∂ F(0) = g
t→0 ∂t
has support sppt(F(t)) K := x Ω; dist(x,K) α˜t for some finite α˜.
α˜t
⊂ { ∈ ≤ }
Proof. With B and D specified as in Case III, we have shown that iBD generates
a C0 group (eitBD)t∈R with finite propagation speed that we now denote by α˜. The
cosine family cos(tBD) = 1(eitBD +e−itBD) = cos(t (BD)2) (t 0) has the same
2 ≥
propagation speed. Noting that
p
L 0 0
(BD)2 = 0 L˜ (L˜ )
00 0k
 
0 (L˜ ) (L˜ )
j0 jk
 
for suitable operators L˜ , it follows that the first component (cos(t√L)) , acting
jk t≥0
on L2(Rn), has finite propagation speed bounded by α˜. The result follows. (cid:3)
Remark 5.7. Theuseofthesquarerootsymbol isagainpurelysymbolic, as cos(t√L)
is really just the leading component of cos(tBD). It is not the case in general that
an operator √L is defined. See Remark 5.3. In the language of cosine families, it
is said that (BD)2, and hence L, are generators of their respective cosine families.
In the usual treatment of cosine families associated with L, one adds a positive
constant to A if necessary to ensure that ReJ (f,f) f 2+ u 2, and notes
00 A ≥ k∇ k2 k k2
that the numerical range of A is contained in a parabola. This ensures that L
generates a cosine family [12], and that √Lf f 2+ f 2 [15]. See also [1].
k k2 ≈ k∇ k2 k k2
Our treatment is consistent with this approach, but we do not need to apply it,
as we use the more straightforward fact that, since iBD generates a C group,
0
then (BD)2 generates a cosine family, and so then does the restriction to its first
component L in L2(Rn).