Quantitative unique continuation for wave operators with a jump discontinuity across an interface and applications to approximate control

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Quantitative unique continuation for wave operators with a jump

discontinuity across an interface and applications to approximate

control

Spyridon Filippas

October 2022

Abstract

In this article we prove quantitative unique continuation results for wave operators of the form

∂2

t−div(c(x)∇·) where the scalar coeﬃcient cis discontinuous across an interface of codimension one

in a bounded domain or on a compact Riemannian manifold. We do not make any assumptions on

the geometry of the interface or on the sign of the jumps of the coeﬃcient c. The key ingredient is a

local Carleman estimate for a wave operator with discontinuous coeﬃcients. We then combine this

estimate with the recent techniques of Laurent-L´eautaud [LL19] to propagate local unique continua-

tion estimates and obtain a global stability inequality. As a consequence, we deduce the cost of the

approximate controllability for waves propagating in this geometry.

Keywords

Unique continuation, Carleman estimate, wave equation, jumps across an interface, approximate

control, stability estimates

2010 Mathematics Subject Classiﬁcation: 35B60, 47F05, 35L05, 93B07, 93B05, 35Q93

Contents

1 Introduction 2

1.1 Setting and statement of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Strategy of the proof and organization of the paper . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Some notations and deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 The Carleman estimate 8

2.1 General transmission conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Local setting in a neighborhood of the interface . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Proof of Theorem 2.2 from Proposition 2.6 ........................... 11

3 Proof of Proposition 2.6 for a toy model 13

3.1 Factorization and ﬁrst estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 End of the proof for the toy model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Proof of Proposition 2.6 for the general case 21

4.1 Notation, microlocal regions and ﬁrst estimates . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Microlocal estimates in the non-elliptic regions . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Microlocal estimate in the elliptic region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Patching together microlocal estimates: End of the proof of Proposition 2.6 ........ 37

4.5 Convexiﬁcation: A perturbation argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

arXiv:2210.04634v1 [math.AP] 10 Oct 2022

5 The quantitative estimates 44

5.1 Some deﬁnitions and statement of the local estimate . . . . . . . . . . . . . . . . . . . . . 44

5.2 Proof of Theorem 5.1 ....................................... 45

5.3 Propagation of information and applications . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A A few facts on pseudodiﬀerential calculus 61

A.1 Diﬀerentialoperators....................................... 61

A.2 Standardtangentialclasses ................................... 61

A.3 Tangential classes with a large parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B Some lemmata used in the quantitative estimates 63

1 Introduction

For a wave operator Pthe question of unique continuation consists in asking whether a partial observation

of a wave on a small set ωis suﬃcient to determine the whole wave. If this property holds, then the next

natural question is if we can quantify it. This is expressed via a stability estimate of the form

kukΩ.φ(kukω,kP ukΩ,kukΩ),(1.1)

with φsatisfying

φ(a, b, c)a,b→0

−→ 0,with cbounded.

Such estimates have numerous applications in control theory, spectral geometry and inverse problems.

Concerning the wave operator a seminal unique continuation result was obtained by Robbiano in [Rob91]

and reﬁned by H¨ormander in [H¨or92]. The optimal version of this qualitative result was ﬁnally attained in

the so called Tataru, H¨ormander, Robbiano-Zuily Theorem [Tat95,Tat99,H¨or97,RZ98]. This theorem

deals in fact with the more general case of operators with partially analytic coeﬃcients and, in the

particular case of a wave operator with coeﬃcients independent of time, gives uniqueness across any

non characteristic hypersurface. Recently, in [LL19] the authors proved a quantitative version of the

latter theorem which, for the wave equation, is optimal with respect to the observation time and the

stability estimate obtained. Note that, a qualitative uniqueness result is equivalent to an approximate

controllability result, and a quantiﬁed version of it gives an estimate of the control cost. The quantitative

unique continuation result of [LL19] applies to (variants of) the operator ∂2

t−∆gwhere ∆gis an elliptic

operator with C∞coeﬃcients. See also [BKL16] for a related set of estimates concerning the wave

operator.

However, in many contexts, waves propagate through singular media and therefore in the presence of

non smooth coeﬃcients. E.g. in the case of seismic waves [Sym83] or acoustic waves [YDdH+17,AdHG17,

CdHKU19] propagating through the Earth’s crust. Models proposing to describe such phenomena use

discontinuous metrics and more precisely metrics which are piece-wise regular but presenting jumps along

some hypersurfaces. See for instance the Mohoroviˇci´c discontinuity between the Earth’s crust and the

mantle. Another example arises in medical imaging. The human brain [FdLKBH01,MCCP17] has two

main components: white and grey matter. These two have very diﬀerent electric conductivities and

models describing the situation are very similar to the preceding example.

The question of quantitative unique continuation across a jump discontinuity seems to be well under-

stood in the elliptic/parabolic context. One of the ﬁrst results (in the parabolic case) is [DOP02] where

the operator ∂2

t−div(c∇·) is studied with a monotonicity assumption imposed on the scalar coeﬃcient

c=c(x): the observation should take place in the region where the coeﬃcient cis smaller. In this article

a global Carleman estimate was proved. Later, in the elliptic case in [LR10] a similar result was obtained

but without any restriction on the sign of the jump of the coeﬃcient. These techniques were extended

to the parabolic context in [LR11]. The most recent (and general) result to the best of our knowledge is

proved by Le Rousseau and Lerner [LRL13] where the anisotropic case (−div(A(x)∇·), with Aa matrix

jumping across an interface) is treated.

The question of exact control for waves with jumps at an interface has already been addressed in

the book [Lio88]. A controllability result is proved for the operator ∂2

t−div(c∇·) with ca piece-wise

constant coeﬃcient under a geometric assumption on the jump hypersurface and a sign condition on

the jump. One of the ﬁrst Carleman estimates was proved in the discontinuous setting in [BMO07].

With the same assumption on the coeﬃcient and assuming that the interface is convex the authors prove

linear quantitative stability estimates. Recently, in [BGM21] quantitative results were proved as well

for interfaces that interpolate between star-shaped and convex. Other related works are [Gag18] and

[BCDCG20].

However, to our knowledge the question of stability estimates without any particular geometric as-

sumption on the interface has not been studied yet. This is the main object of this article.

1.1 Setting and statement of main results

Let (M, g) be a smooth connected compact n-dimensional Riemannian manifold with or without bound-

ary. We consider San (n−1)-dimensional submanifold of Mwithout boundary. We assume that

M\S= Ω−∪Ω+with Ω−∩Ω+=∅.

We consider a scalar coeﬃcient c(x) = 1Ω−c−(x) + 1Ω+c+(x) with c±∈C∞(Ω±) satisfying 0 <

cmin < c(x)< cmax uniformly on Ω−∪Ω+to ensure ellipticity. We shall work with the wave operator P

deﬁned as

P=∂2

t−divg(c(x)∇g),on Rt×Ω−∪Ω+.(1.2)

We consider for (u0, u1)∈H1

0(M)×L2(M) the following evolution problem:











P u = 0 in (0, T )×Ω−∪Ω+

u|S−=u|S+in (0, T )×S

(c∂νu)|S−= (c∂νu)|S+in (0, T )×S

u= 0 in (0, T )×∂M

(u, ∂tu)|t=0 = (u0, u1) in M,

(1.3)

where we denote by ∂νa nonvanishing vector ﬁeld deﬁned in a neighborhood of S, normal to S(for the

metric g), pointing into Ω+and normalized for g. We denote as well by u|S±the traces of u|Ω±on the

hypersurface S.

Notice that there are two extra equations in our system. These are some natural transmission con-

ditions that we impose in the interface. These conditions imply that the underlying elliptic operator is

self-adjoint on its domain and one can show using classical methods (for instance with the Hill-Yosida

Theorem) that the system (1.3) is well posed. For more details on this we refer to Section 2.

Our ﬁrst result provides a quantitative unique continuation result from an observation region ωfor

the discontinuous wave operator P.

In section 1.3 we introduce L(M, ω) = supx∈M dist(x, ω),the “largest distance” of the subset ωto a

point of M, where dist is a distance function adapted to (M, g, c).

Theorem 1.1. Consider (M, g), S, Ω±and Pas deﬁned in (1.2). Then for any nonempty open subset

ωof Mand any T > 2L(M, ω), there exist C, κ, µ0such that for any (u0, u1)∈H1

0(M)×L2(M)and

usolving (1.3)one has, for any µ≥µ0,

k(u0, u1)kL2×H−1≤Ceκµ kukL2((0,T )×ω)+C

µk(u0, u1)kH1×L2.

If moreover ∂M 6=∅and Γis a non empty open subset of ∂M, for any T > 2L(M,Γ), there exist

C, κ, µ0>0such that for any (u0, u1)∈H1

0(M)×L2(M)and usolving (1.3), we have

k(u0, u1)kL2×H−1≤Ceκµ k∂νΓukL2((0,T )×Γ) +C

µk(u0, u1)kH1×L2.

Remark 1.2. In fact one can take µ > 0 in the statement of the above theorem. However we preferred

to state it in this way in order to stress out the fact that this estimate is interesting only when µis large.

With the above one can recover the following qualitative result: “If we do not see anything from ω

during a time Tstrictly larger than 2L(M, ω), then there is no wave at all.” Indeed, if kukL2((0,T )×ω)= 0,

then letting µ→+∞in the above inequality implies that (u0, u1) = 0.

(a) The observation takes place inside Ω−.

(b) The observation takes place inside Ω+. If c−< c+

then a part of the wave may be trapped inside Ω−. Nev-

ertheless, the quantitative unique continuation and its

consequences still hold.

An important aspect of this theorem is that there is no assumption on the sign of the jump of the

coeﬃcient cand consequently the observation region ωcan be chosen indiﬀerently on Ω−or Ω+. Let us

explain why this is quite surprising. Suppose, to ﬁx ideas, that c−< c+are two constants. We can then

interpret c−and c+as the the speed of propagation of a wave travelling through two isotropic media Ω−

and Ω+with diﬀerent refractive indices, n−and n+respectively (recall that n±= 1/c±). Imagine that

a wave starts travelling from a region that is inside Ω−. One has c−

c+=n+

n−and therefore the assumption

c−< c+translates to n−> n+. Then Snell-Descartes law states that when a wave travels from a medium

with a higher refractive index to one with a lower refractive index there is a critical angle from which

there is total internal reﬂection, that is no refraction at all. At the level of geometric optics, that is to

say, in the high frequency regime such a wave stays trapped inside Ω−. Therefore one expects that, at

least at high frequency, no information propagates from Ω−to Ω+, following the laws of geometric optics.

Our result (see Theorem 1.3) states that the intensity of waves in Ω+is at least exponentially small in

terms of the typical frequency Λ of the wave.

We can reformulate Theorem 1.1 in a way closer to quantitative estimates such as (1.1). Indeed,

optimizing the inequalities of Theorem 1.1 with respect to µyields the following result (which we state

only in the interior observation case):

Theorem 1.3. Under the assumptions of Theorem 1.1 there exists C > 0such that for all (u0, u1)∈

0(M)×L2(M)with (u0, u1)6= (0,0) one has:

k(u0, u1)kH1×L2≤CeCΛkukL2((0,T )×ω),

k(u0, u1)kL2×H−1≤Ck(u0, u1)kH1×L2

log 1 + k(u0,u1)kH1×L2

kukL2((0,T )×ω).(1.4)

where Λ = k(u0,u1)kH1×L2

k(u0,u1)kL2×H−1.

Note that Λ represents the typical frequency of the initial data. Theorem 1.3 is a direct consequence

of Theorem 1.1 and Lemma A.3 in [LL19]. Notice that the function

x7→ 1

log(1 + 1/x),

appearing in the right hand side of (1.4) has been tacitly extended by continuity by 0 when x= 0.

In [Rob95, proof of Theorem 2, Section 3] it is shown that such a quantitative information can lead

to an estimate for the cost of the approximate controllability. We state the case of internal control, a

similar result holds for approximate boundary controllability as well.

Theorem 1.4 (Cost of approximate interior control).Consider M,Sand ω⊂ M as before. Then for

any T > 2L(M, ω)there exist C, c > 0such that for any  > 0and any (u0, u1)∈H1

0(M)×L2(M),

there exists f∈L2((0, T )×ω)with

kfkL2((0,T )×ω)≤Cec/ k(u0, u1)kH1

0(M)×L2(M)

such that the solution of











P u =1ωfin (0, T )×Ω−∪Ω+

u|S−=u|S+in (0, T )×S

(c∂νu)|S−= (c∂νu)|S+in (0, T )×S

u= 0 in (0, T )×∂M

(u, ∂tu)|t=0 = (u0, u1) in M,

satisﬁes 

(u, ∂tu)|t=T

L2×H−1≤k(u0, u1)kH1

0×L2.

In other words, if we act on the region ωduring a time T > 2L(M, ω) we can drive our solution

from energy 1 (in H1×L2) to close to 0 (in L2×H−1). Additionally, this comes with an estimate of

the energy of the control which is of the order of ec/. In the analytic context and without the presence

of an interface it was shown in [Leb92] that this form of exponential cost is optimal in the absence of

Geometric Control Condition [BLR92].

In the more general hypoelliptic context of [LL21] the result of Theorem 1.4 is stated as approximate

observability for the wave equation. It is shown by the authors of this article that such a property

implies some resolvent estimates (Proposition 1.11 in [LL21]) which in turn give a logarithmic energy

decay estimate for the damped wave equation (see Theorem 1.5 in [LL21]). Consequently, Theorem 1.4

combined with the results of [LL21] provides a diﬀerent proof for theorems that were already obtained

using Carleman estimates for elliptic/parabolic operators (see [LR10] or [LRL13]).

Remark 1.5. We have assumed that the interface Sdecomposes Min two disjoint parts Ω+and Ω−.

However the same results can be obtained for other more general geometric situations as well. This comes

from the fact that the key ingredient is a local quantitative estimate (see Theorem 5.1). See also the

ﬁgure in [LRLR13, Section 1.3.2]

1.2 Strategy of the proof and organization of the paper

1.2.1 The Carleman estimate

One of the main tools for dealing with problems of local unique continuation across a hypersurface {φ= 0}

is Carleman estimates. The idea, introduced by Carleman in [Car39], is to prove an inequality involving

a weight function ψand a large parameter τ, of the form



eτψP u

L2&

eτψu

L2, τ ≥τ0,

uniform in τ. The weight function ψis closely related to the level sets of the function φwhich deﬁnes

implicitly the hypersurface. In an heuristic way, the chosen weight re-enforces the sets where uis zero

and propagates smallness from sets where ψis big to sets where ψis small. Since Carleman estimates

are already quantitative in nature they provide a good starting point for results of the form (1.1). We

point out the fact that this is a local problem. In order to obtain a global result one needs in general to

propagate the local one by passing through an appropriate family of hypersurfaces.

The core of this article is to prove a local Carleman estimate in a neighborhood of the interface,

containing a microlocal weight in the spirit of [Tat95,H¨or97,RZ98]. The presence of discontinuous

coeﬃcients complicates signiﬁcantly this task. In general, for a Carleman estimate to hold a condition

involving the principal symbol of the operator and the hypersurface needs to hold, the so-called pseudo-

convexity condition (see for instance [H¨or67]). These results are based on microlocal analysis arguments

and some regularity is necessary for the estimate to hold. In our case we explicitly construct an appro-

priate weight function and show our estimate for this particular weight. Our proof is inspired by that of

Lerner-Le Rousseau in the elliptic case [LRL13] and relies in a factorization argument. Even though the

behavior of our (hyperbolic) operator may be very diﬀerent we consider in our context the wave operator

as a “perturbation” of a Laplacian, in the spirit of [Tat95,H¨or97,RZ98,LL19]. Let us explain why. For

the sake of exposition, consider that M=Rn,gis the Euclidean metric and cis piecewise constant with

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QuantitativeuniquecontinuationforwaveoperatorswithajumpdiscontinuityacrossaninterfaceandapplicationstoapproximatecontrolSpyridonFilippasOctober2022AbstractInthisarticleweprovequantitativeuniquecontinuationresultsforwaveoperatorsoftheform@2tdiv(c(x)r)wherethescalarcoecientcisdiscontinuousacrossanin...

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Quantitative unique continuation for wave operators with a jump discontinuity across an interface and applications to approximate control

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