Concentration and lack of observability of waves in highly heterogeneous media C. Castro, E. Zuazua Abstract We construct rapidly oscillating H¨older continuous coefficients for which the corresponding 1 − d wave equation lacks of the classical observability property guaranteeing that the total energy of solutions may be bounded above by the energy localized in an open subset of the domain where the equation holds, if the observation time is large enough. The coefficients we build oscillate arbitrarily fast around two accumulation points. This allows us to build quasi-eigenfunctions for the corresponding eigenvalue problem that concentrate the energy away from the observation region as much as we wish. This example may be extended to several space dimensions by separation of variables and illustrates why the well-known controllability and dispersive properties for wave equations with smooth coefficients fail in the class of H¨older continuous coefficients. In particular we show that for such coefficients no Strichartz type estimate holds.

1. Introduction and main result Let us consider the following variable coefficient 1 − d wave equation  0 < x < 1, 0 < t < T  ρ(x)utt − uxx = 0, u(0, t) = u(1, t) = 0, 0
(1.1)

We assume that ρ is measurable and that it is bounded above and below by finite, positive constants, i.e. 0 < ρ0 ≤ ρ(x) ≤ ρ1 < ∞ a.e. x ∈ (0, 1).

(1.2)

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C. Castro, E. Zuazua

Under these conditions system (1.1) is well-posed in the sense that for any pair of initial data (u0 , u1 ) ∈ H01 (0, 1) × L2 (0, 1) there exists a unique solution ¡ ¢ ¡ ¢ u ∈ C [0, T ]; H01 (0, 1) ∩ C 1 [0, T ]; L2 (0, 1) . (1.3) Moreover, the energy of solutions Z ¤ 1 1£ E(t) = ρ(x) | ut (x, t) |2 + | ux (x, t) |2 dx 2 0

(1.4)

is constant in time. When ρ ∈ BV (0, 1), the following observability properties are well known to hold: √ 1. Boundary observability: If T > ρ1 there exists C(T ) > 0 such that Z

T

E(0) ≤ C

h

2

2

|ux (0, t)| + |ux (1, t)|

i dt,

(1.5)

0

for every solution of (1.1). 2. Internal observability: For any subinterval (α, β) ⊂ (0, 1), if T > √ 2 ρ1 max(α, 1 − β), there exists C > 0 such that Z

T

Z

β

E(0) ≤ C 0

α

£ ¤ ρ(x)u2t + u2x dxdt,

(1.6)

for every solution of (1.1). These results may be proved easily using sidewise energy estimates for the wave equation in which the role of space and time are interchanged. We refer to [6] for the details of the proof. These observability estimates are relevant in the context of controllability. In fact they are equivalent to the controllability of the system with controls acting on the boundary or in the interior of the domain respectively (see [10]). For a long time the problem of whether these estimates do hold for less regular coefficients (say ρ ∈ L∞ (0, 1) or ρ ∈ C([0, 1])) has been open. In [1] the problem of homogenization was considered. It was shown that, for a suitable ρ, due to concentration effects of high frequency solutions, the constant C on the observability inequalities (1.5)-(1.6) blows up when ρ is replaced by ρ² (x) = ρ(x/²) and ² → 0. This result shows that the constant C in (1.5)-(1.6) does not only depend on the lower and upper bounds ρ0 and ρ1 of ρ. The results in [1] were an evidence of the possible lack of observability for highly oscillatory density functions ρ. But, up to now, there was not proof of this negative result in the literature. In this paper we definitely answer to the problem by the negative. More precisely, we prove that the following holds: Theorem 1. There exist H¨ older continuous density functions ρ ∈ C 0,s ([0, 1]) for all 0 < s < 1, for which (1.5) and (1.6) fail for all T > 0 and for all subinterval (α, β) ⊂ (0, 1) ((α, β) 6= (0, 1)).

Title Suppressed Due to Excessive Length

3

Remark 1. 1.- The density functions we build are in fact of class C ∞ everywhere in (0, 1) except at the extremes x = 0 and x = 1 of the interval. 2.- Obviously, the density functions we obtain are not of finite total variation. The total variation blows up on the two extremes x = 0, 1. 3.- A similar construction may be done by means of piecewise constant density functions (see Remark 3). 4.- In fact, given any smooth density ρ in [0, 1], one can perturb it in a subinterval of arbitrarily small length so that inequalities (1.5) and (1.6) fail for the new density ρ˜. More precisely, given a smooth ρ and a subinterval [x0 , x1 ] of [0, 1] one can find a H¨older continuous function ² with support in (x0 , x1 ) and L∞ −norm of arbitrarily small size, and such that the inequalities (1.5) and (1.6) fail for the new density ρ˜ = ρ + ². The proof of Theorem 1 is based in an argument introduced in [5] in a different context that allows to construct a density ρ for which there exists a sequence of pairs (ϕk (x), λk ) satisfying ϕ00k + λ2k ρ(x)ϕk = 0,

(1.7)

and such that ϕk is exponentially concentrated on any given point of the closed interval [0, 1]. In fact we construct a double sequence so that part of it is concentrated on x = 0, while the other one is concentrated on x = 1. Let us explain the main idea behind this argument. We consider density functions ρ(x) which oscillate more and more as x approaches the extremes of the interval [0, 1]. The sequence of pairs (ϕk (x), λk ) is constituted by functions ϕk (x) which oscillate at the same order as ρ in a small region inside [0, 1] close to x = 0 or x = 1. In this region a resonance-type phenomenum occurs and ϕk (x) becomes exponentially larger than in the rest of the interval [0, 1]. Note that (1.7) together with ϕk (0) = ϕk (1) = 0,

(1.8)

constitute the eigenvalue problem associated to (1.1). There is no reason for the functions ϕk above, exponentially concentrated near the boundary, to satisfy (1.8). However by choosing in an appropriate way the point where the energy concentrates, close to one of the extremes x = 0, 1, the values of ϕk and ϕ0k at x = 0, 1 may be guaranteed to be exponentially small. This motivates referring to these functions ϕk as quasi-eigenfunctions. These quasi-eigenfunctions allow us to construct a sequence of solutions uk of (1.1) of the form uk (x, t) = eiλk t ϕk (x) + v˜k (x, t) where v˜k (x, t) is a correction introduced to make uk satisfy the boundary conditions. The solutions uk concentrate in the interior of (0, 1) along the time and therefore constitute an obstacle to the boundary observability. Recall that we are dealing with a double sequence of quasi-eigenfunctions and therefore with a double family uk of solutions of (1.1), one of them being concentrated near x = 0 while the other one is concentrated near

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C. Castro, E. Zuazua

x = 1. In this way one can guarantee that neither (1.5) nor (1.6) hold and the latter, whatever the interval (α, β) ⊂ (0, 1) ((α, β) 6= (0, 1)) is. The rest of the paper is organized as follows: in section 2 we state two O.D.E. lemmas introduced in [5] that we use to construct the density ρ. In section 3 we build the pathological density ρ and the sequence of quasieigenfunctions associated to ρ. In sections 4 and 5 we prove the lack of boundary and interior observability respectively for this choice of ρ. In section 6 we extend these results to the multi-dimensional case. In section 7 we state the results on the lack of controllability that one may derive from Theorem 1. Finally, in section 8 we comment some related results. In particular we generalize our result to the case where the variable coefficient is in the principal part of the operator in the wave equation, and to the corresponding Schr¨odinger model.

2. Preliminary lemmas In this section we recall the following two lemmas proved in [5]. Lemma 1. There exists ²¯ > 0 such that for all ² ∈ (0, ²¯), it is possible to find two even real functions, α² (x) and w² (x), of class C ∞ on IR, satisfying ½

w²00 + α² (x)w² = 0, w² (0) = 1, w²0 (0) = 0,

(2.1)

in such a way that α² (x) is 1−periodic on x < 0 and on x > 0 α² (x) ≡ 4π 2 in a neighborhood of x = 0 |α² (x) − 4π 2 | ≤ M ², |α²0 (x)| ≤ M ² ½ w² (x) = p² (x)e−²|x| for some p² (x) 1−periodic on x < 0 and on x > 0

(2.2) (2.3) (2.4)

|w² | + |w²0 | + |w²00 | ≤ C Z 1 w² (x)dx ≥ γ², (γ > 0)

(2.6)

Z

(2.5)

(2.7)

0 1

|w² |2 dx ≥ γ,

(2.8)

0

where M, C and γ are constants independent of ². Remark 2. As a consequence of (2.5), (2.1) and (2.7), we have in particular for all integers n ≥ 0, w² (x) = e−²|x| ,

w²0 (x) = 0,

w²00 (x) = −4π 2 e−²|x| ,

for x = ±n.(2.9)

Title Suppressed Due to Excessive Length

↓ α(x)

5

← w(x)

Fig. 1. Construction of α and w satisfying (2.11).

Remark 3. The parameter ² in (2.1) allows us to introduce a family of coefficients α² approaching to a constant (see (2.4)), and for which we know explicitely the decay rate of the solution of (2.1) w² as |x| → ∞ (see (2.5)). As we will see this is important to guarantee the H¨older continuity of the density ρ that we construct in the next section. Remark 4. For a fixed ² > 0, Lemma 1 establishes the existence of a coefficient α(x) and a solution w(x) of ½ 00 w + α(x)w = 0, (2.10) w(0) = 1, w0 (0) = 0, satisfying α(x) is 1−periodic on x < 0 and on x > 0 ½ w(x) = p(x)e−k|x| (2.11) for some p(x) 1−periodic on x < 0 and on x > 0 and k > 0. Explicit examples of piecewise constant functions α and solutions w with the above properties may be built easily (see [1]). The main idea is to consider a periodic coefficient α1 such that the solution w1 of (2.10) with α = α1 satisfies w1 (x) = p(x)e−kx for some 1−periodic function p and k > 0. Let xs ∈ IR be a point where the solution satisfies w10 (xs ) = 0. Then α2 , w2 , the even reflection of α1 and w1 with respect to xs respectively, satisfies (2.10) and w2 (x) = p(x)e−k|x−xs | with p(x) periodic for x > xs and x < xs . Finally we can translate α2 and w2 in order to have (2.11). This construction is illustrated in Figure 1. In [5] Lemma 1 is stated with α² being 2π−periodic. A straightforward change of variables shows that α² may be taken to be 1−periodic as well, as stated in Lemma 1. Let us explain briefly the result in Lemma 1. If we restrict the first equation in (2.1) to IR+ we obtain the Hill equation w00 + λq(x)w = 0,

x ∈ IR+ , λ ∈ IR and q being 1-periodic.

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C. Castro, E. Zuazua

It is well known that for any periodic function q there exist some positive values of λ for which this equation has a solution of the form w = p(x)e−ax where p is 1-periodic and a > 0 (see [7]). The proof of Lemma 1 in [5] relies on a suitable choice of q, λ and w satisfying these properties. The values of α and w for x < 0 are obtained by even extension. Note that the condition (2.3) assures the regularity of this extension. The explicit choice of q, λ and w is as follows: λ = 4π 2 q(x) = 1 − 4²r(2πx) sin(4πx) + 2²r0 (2πx) cos2 (2πx) − 4²2 r2 (2πx) cos4 (2πx) µ ¶ Z 2πx w(x) = cos(2πx) exp −2² r(s) cos2 (s)ds , 0

where r(s) ≥ 0 is a fixed 2π-periodic function, of class C ∞ , vanishing in a neighborhood of s = 0 and satisfying the conditions Z 2π Z 2π 1 2 r(s) cos2 s sin s ds > 0. r(s) cos s ds = , 2 0 0 Lemma 2. Let φ(x) be a solution of the equation φ00 + h2 a(x)φ = 0,

x ∈ IR

where h ∈ Z and a(x) is a strictly positive function of class C 1 , and let us consider the energy functions Eφ (x) = 4π 2 h2 |φ(x)|2 + |φ0 (x)|2 ˜φ (x) = h2 a(x)|φ(x)|2 + |φ0 (x)|2 . E Then, for all t1 and t2 , the following estimates hold ¯ Z x2 ¯ ¯ ¯ 2 ¯ Eφ (x2 ) ≤ Eφ (x1 ) exp ¯h |4π − a(x)|dx¯¯ x ¯Z x21 0 ¯ ¯ |a (x)| ¯¯ ˜φ (x2 ) ≤ E ˜φ (x1 ) exp ¯ E dx ¯ ¯. a(x) x1

(2.12) (2.13)

To prove this result it is sufficient to differentiate the energy functions and to apply Gronwall’s Lemma.

3. Construction of the density and the quasi-eigenfunctions In this section we make the main construction of the paper. We build simultaneously the density ρ and the sequence of quasi-eigenfunctions that exhibit the concentration effect we are looking for. Our construction is inspired on that in [5]. Let us consider the sequences rj = 2−j ,

hj = 2 2

Nj

,

2 ²j = h−1 j (log hj )

(3.1)

Title Suppressed Due to Excessive Length

7

-

I4

0

+

I4

1/8

1/2

1/4

7/8

3/4

I 3-

I

I

-

I

2

1

+ 3

+ 2

Fig. 2. Partition of the interval [0, 1] in the subintervals Ij±

where N > 1 is a fixed, large enough integer (with respect to the constant M in Lemma 1) so that the following inequalities hold: 1 2M k−1 X 4M ²j hj rj ≤ ²k hk rk

²k ≤

2M

j=1 ∞ X

²j rj ≤ ²k rk .

(3.2) (3.3) (3.4)

j=k+1

Note that such inequalities are true for large N due to the following: {²k }k≥2 is a decreasing sequence for N ≥ 2 and ²1 → 0 as N → ∞; the sequence ²j hj rj (²k hk rk )−1 = (AN )j−k with AN converging to infinity as N → ∞ and finally ²j rj (²k rk )−1 = (δN )j−k for some δN → 0 for N → ∞. With this choice of the sequence rk we define a partition of the interval (0, 1): (0, 1/2] =

[

Ij− ,

j≥2

[1/2, 1) =

[

Ij+

rj rj rj rj , m− ], Ij+ = [m+ , m+ ), j + j + j − 2 2 2 2 ∞ X rj − m− = + rk , m+ j ≥ 2. j j = 1 − mj , 2 Ij− = (m− j −

(3.5)

j≥2

j ≥ 2 (3.6) (3.7)

k=j+1

± We observe that m± j is the center of the interval Ij with length rj (see Figure 1). The super-index + (respectively −) indicates that the interval is to the right (respectively left) of x = 1/2. This notation is convenient to distinguish the two singularities, at x = 0 and x = 1, of the density that we are going to construct.

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C. Castro, E. Zuazua

Now, we define the density ρ as follows  −   α²2j (h2j (x − m2j )) + ρ(x) = α²2j+1 (h2j+1 (x − m2j+1 ))   4π 2

− for x ∈ I2j + for x ∈ I2j+1 ³S ´ − + for x ∈ [0, 1]\ (I ∪ I ) 2j+1 j≥1 2j (3.8) where α²j are the functions introduced in Lemma 1. The density ρ oscillates near x = 0 and x = 1 but with different frequencies so that, in some sense, as we will see, these two oscillations do not interact. Note that ρ(x) ∈ C ∞ (0, 1) because α² ∈ C ∞ (IR) and

ρ(x) ≡ 4π 2 in a neighborhood of the extremes of Ij± ,

(3.9)

h r

since j2 j is an integer and properties (2.2)-(2.3) hold. On the other hand ρ(x) ∈ C 0,s ([0, 1]) for all 0 < s < 1. Indeed, by (2.2) and the bound (2.4) for |α²0 | we have |α²j |C 0,s (IR) = max x,y∈IR

|α²j (x) − α²j (y)| |α²j (x) − α²j (y)| ≤ max s |x − y| |x − y|s x,y∈[0,1]

≤ M ²j max |x − y|1−s ≤ M ²j x,y∈[0,1]

(3.10)

for 0 < s < 1. Note that here | · |C 0,s (IR) represents the H¨older semi-norm. Therefore |ρ|C 0,s (I ± ) ≤ M ²j hsj j

for 0 < s < 1.

(3.11)

The term on the right is uniformly bounded in j in view of (3.1), i.e. sup ²j hsj < ∞

for 0 < s < 1.

(3.12)

j

From (3.9)-(3.12) we conclude that ρ(x) ∈ C 0,s ([0, 1]) for all s < 1. In fact, if we define Ix as the interval of the family {Ij± }j≥2 such that x ∈ Ix and lx , rx are the left and right extremes of Ix respectively, then |ρ(x) − ρ(y)| |ρ(x) − ρ(rx )| + |ρ(ly ) − ρ(y)| ≤ max 0≤x≤y≤1 |x − y|s |x − y|s |ρ(x) − ρ(rx )| |ρ(ly ) − ρ(y)| ≤ max + max ≤ 2 sup |ρ|C 0,s (I ± ) . j |x − rx |s |ly − y|s x∈[0,1] y∈[0,1] j≥2

max

x,y∈[0,1]

Here we have used the fact that, in view of (3.9), ρ(rx ) = ρ(ly ). Finally, we observe that ρ is bounded above and below with positive constants in view of (2.4) and (3.2). In fact we have 2π 2 ≤ ρ(x) ≤ 8π 2 .

(3.13)

Title Suppressed Due to Excessive Length

9

+ Let us define two sequences of quasi-eigenfunctions {ϕ− 2j }j≥1 and {ϕ2j+1 }j≥1 as the solutions of the following initial value problems ½ − 00 (ϕ2j ) + h22j ρ(x)ϕ− 0 < x < 1, 2j = 0, (3.14) − − 0 − ϕ− (m ) = 1, (ϕ ) (m ) 2j 2j 2j 2j = 0 ½ + (ϕ2j+1 )00 + h22j+1 ρ(x)ϕ+ 0 < x < 1, 2j+1 = 0, (3.15) + + + 0 ϕ+ (m ) = 1, (ϕ ) (m 2j+1 2j+1 2j+1 2j+1 ) = 0.

These are simply quasi-eigenfunctions since the boundary conditions at x = 0, 1 are not necessarily fulfilled. For example, there is no reason for ϕ− 2j to − vanish neither at x = 0 nor at x = 1. However, we will see that ϕ2j is mainly − concentrated in the interior of I2j so that the values of ϕ− 2j at x = 0, 1 are − exponentially small. This justifies referring to ϕ2j as quasi-eigenfunctions in the sense that the missing boundary conditions are almost satisfied. The − same argument applies to ϕ+ 2j+1 . To see that ϕ2j is concentrated in the − interior of I2j we observe that it satisfies ½ − 00 − − x ∈ I2j (ϕ2j ) + h22j α²2j (h2j (x − m− 2j ))ϕ2j = 0, (3.16) − − − 0 − ϕ2j (m2j ) = 1, (ϕ2j ) (m2j ) = 0. Therefore,

− ϕ− 2j (x) = w²2j (h2j (x − m2j ))

(3.17)

where w²2j is the function in Lemma 1 associated to α²2j . Combining Remark 2 and the fact that h2j r2j /2 are integers we deduce that Z Z r2j h2j Z 1 1 1 − 2 2 |ϕ2j (x)| dx = |w²2j (s)| ds ≥ |w²2j (s)|2 ds − h h 2j −r2j h2j 2j 0 I2j γ 2 ²22j C ≥ 3 h2j h2j ¯ r2j ¯¯2 ¯¯ − 0 − r2j ¯¯2 ¯ − − )¯ + ¯(ϕ2j ) (m2j − )¯ = e−²2j h2j r2j , ¯ϕ2j (m2j − 2 2 ¯ r2j ¯¯2 ¯¯ − 0 − r2j ¯¯2 ¯ − − )¯ + ¯(ϕ2j ) (m2j + )¯ = e−²2j h2j r2j , ¯ϕ2j (m2j + 2 2 ≥

(3.18) (3.19) (3.20)

− −3 i.e. the L2 -norm of ϕ− 2j in I2j is of the order h2j but exponentially larger − − than the values of ϕ2j at the extremes of I2j in the sense that h32j e−²2j h2j r2j ≤ Cp h−p 2j for all p > 0. In fact, for any p > 0 −rj log(hj )

hpj e−²j hj rj = hpj hj

p−2−j+N j log(2)

= hj

→ 0,

as j → ∞.

(3.21)

Roughly speaking, we have checked that ϕ− 2j is concentrated in the interior − of I2j . Now, one may expect that the energy of ϕ− 2j will still be small outside − I2j . We claim that this is the case, i.e. ϕ− is small at x = 0, 1. The analysis 2j we shall make at each extreme (x = 0 and x = 1) will be of different nature. Therefore, at this point we divide our analysis in these two cases.

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Analysis at x = 0: We estimate, for x ≤ m− 2j − r2j /2, the energy function ¯ ¯2 ¯ − 0 ¯2 ¯ ¯ ¯ Eϕ− (x) = 4π 2 h22j ¯ϕ− (3.22) 2j (x) + (ϕ2j ) (x) . 2j

In view of (3.19) we have Eϕ− (m− 2j − 2j

r2j ) ≤ 4π 2 h22j e−²2j h2j r2j . 2

(3.23)

On the other hand, taking (2.12) into account, the definition of ρ in (3.8) and the estimate for α² in (2.4), we have à ! Z m2j −r2j /2 r2j |4π 2 − ρ| Eϕ− (x) ≤ − ) exp 4π 2 h22j 2j 2 x   ∞ X r 2j ≤ Eϕ− (m− ) exp 4π 2 M h2j ²2k r2k  . 2j − 2j 2 Eϕ− (m− 2j 2j

k=j+1

This last term can be estimated by the aid of (3.4) and (3.23) and therefore 

∞ X



r2j ) exp 4π 2 M h2j ²2k r2k  2 k=j+1 ¶ µ 1 2 2 ≤ 4π h2j exp −h2j ²2j r2j + h2j ²2j r2j 2 µ ¶ ¡ ¢ 1 ≤ 4π 2 h22j exp − h2j ²2j r2j ≤ 4π 2 h22j exp −2−2j−1 (log h2j )2 2 Eϕ− (x) ≤ Eϕ− (m− 2j − 2j

2j

for all x ≤ m− 2j − r2j . Hence ¯ − ¯ ¯ ¯ ¯ϕ (x)¯2 + ¯(ϕ− )0 (x)¯2 ≤ Cp h−p , 2j 2j 2j

∀p > 0,

∀x ≤ m− 2j − r2j /2.

(3.24)

In particular (3.24) holds for x = 0. Analysis at x = 1: Here we first estimate, for m− 2j + r2j /2 ≤ x ≤ + m2j+1 − r2j+1 /2, the energy function ¯ ¯ ¯ ¯ e − (x) = h22j ρ(x) ¯ϕ− (x)¯2 + ¯(ϕ− )0 (x)¯2 . E 2j 2j ϕ 2j

(3.25)

In view of (3.19) we have ˜ − (m− + r2j ) ≤ 4π 2 h2 e−²2j h2j r2j . E 2j 2j ϕ2j 2

(3.26)

Title Suppressed Due to Excessive Length

11

On the other hand, we get, by the aid of Lemma 2 (estimate (2.13)), the definition of ρ in (3.8), the estimate for α²0 in (2.4) and the fact that ρ(x) ≥ 2π 2 : ÃZ ! x |ρ0 (s)| − e e Eϕ− (x) ≤ Eϕ− (m2j + r2j /2) exp ds 2j 2j m− +r2j /2 |ρ(s)| 2j à 2j−1 ! X − e ≤ E − (m + r2j /2) exp M ²k rk hk . ϕ2j

2j

k=1

This last term can be estimated with (3.26) and (3.19) and therefore µ ¶ µ ¶ 1 3 2 2 e Eϕ− (x) ≤ h2j exp −h2j ²2j r2j + h2j ²2j r2j = h2j exp − h2j ²2j r2j 2j 4 4 + for all x ∈ [m− 2j − r2j /2, m2j+1 − r2j+1 /2]. In particular, taking (3.13) into account, we have

˜ (m+ − r2j+1 /2) Eϕ− (m+ 2j+1 − r2j+1 /2) ≤ 2Eϕ− 2j+1 2j 2j ¶ µ 3 ≤ 2h22j exp − h2j ²2j r2j . 4 For x ≥ m+ 2j+1 − r2j+1 /2 we use the energy function (3.22),   ∞ X  2 Eϕ− (x) ≤ Eϕ− (m+ ²2k+1 r2k+1  2j+1 − r2j+1 /2) exp 4π M h2j 2j

2j

k=j

µ ¶ 3 1 2 ≤ 2h2j exp − h2j ²2j r2j + h2j ²2j r2j 4 2 µ ¶ ¡ ¢ 1 ≤ 2h22j exp − h2j ²2j r2j ≤ 2h22j exp −2−2j−2 (log h2j )2 4 for all x ≥ m+ 2j+1 − r2j+1 /2. Therefore ¯ − ¯ ¯ ¯ ¯ϕ (x)¯2 + ¯(ϕ− )0 (x)¯2 ≤ Cp h−p , ∀p > 0, 2j 2j 2j

∀x ≥ m+ 2j+1 − r2j+1 /2, (3.27)

which holds in particular for x = 1. We have proved the existence of a sequence of quasi-eigenfunctions ϕ− 2j − concentrated in the interior of I2j , i.e. a sequence of solutions of (3.16) which satisfy Z C 2 (3.28) |ϕ− 2j (x)| dx ≥ 3 , − h I2j 2j ¯ − ¯2 ¯ − 0 ¯2 ¯ϕ (0)¯ + ¯(ϕ ) (0)¯ ≤ Cp h−p , ∀p > 0, (3.29) 2j 2j 2j ¯ − ¯2 ¯ − 0 ¯2 −p ¯ϕ (1)¯ + ¯(ϕ ) (1)¯ ≤ Cp h , ∀p > 0, (3.30) 2j 2j 2j Z β £ − 0 ¤ −p 2 |(ϕ2j ) (x)|2 + h22j |ϕ− ∀p > 0, (3.31) 2j (x)| dx ≤ Cp h2j , α

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C. Castro, E. Zuazua

− for all (α, β) ⊂ (0, 1) with α 6= 0 and j ≥ J large enough to have I2j ∩ + (α, β) = ∅ for all j ≥ J. A similar result can be obtained for ϕ2j+1 . More precisely, Z C 2 , (3.32) |ϕ+ 2j+1 (x)| dx ≥ 3 + h I2j+1 2j+1 ¯ + ¯2 ¯ + ¯2 0 ¯ϕ ¯ ¯ ¯ ≤ Cp h−p , ∀p > 0, (3.33) 2j+1 (0) + (ϕ2j+1 ) (0) 2j+1 ¯2 ¯ + ¯ ¯ + 2 0 ¯ ≤ Cp h−p , ∀p > 0, ¯ϕ ¯ ¯ (3.34) 2j+1 2j+1 (1) + (ϕ2j+1 ) (1) Z β £ + ¤ −p 2 |(ϕ2j+1 )0 (x)|2 + h22j |ϕ+ ∀p > 0, (3.35) 2j+1 (x)| dx ≤ Cp h2j , α

+ ∩ for all (α, β) ⊂ (0, 1) with β 6= 1 and j ≥ J large enough to have I2j+1 (α, β) = ∅ for all j ≥ J.

Remark 5. Let us describe how the claim of point 4 in Remark 1.1 may be proved. First of all, note that the construction above (in this section) may be done with a H¨older continuous density ² of arbitrarily small support [l1 , l2 ], with an arbitrary value of ²(l1 ) = ²(l2 ) and with k²(x)−²(l2 )kL∞ (l1 ,l2 ) arbitrarily small. On the other hand, for a given smooth density function ρ and any subinterval [x0 , x1 ] of [0, 1] we can find a smooth function ρˆ such that ½ ρc constant, in a compact set [l1 , l2 ] ⊂⊂ (x0 , x1 ) ρˆ = ρ(x) in [0, x0 ] ∪ [x1 , 1]. We now choose ² as above with ²(l2 ) = ρc and define ½ ²(x) in [l1 , l2 ] ⊂⊂ (x0 , x1 ) ρ˜ = ρˆ(x) in [0, l0 ) ∪ (l1 , 1]. In this way we obtain a H¨older continuous density function ρ˜ with localized quasi-eigenfunctions within the interval [l1 , l2 ] and such that ρ = ρ˜ outside [x0 , x1 ], and kρ − ρ˜kL∞ (0,1) is arbitrarily small. 4. Lack of boundary observability In this section we prove that (1.5) fails for all T > 0 for the density function ρ we have built in section 3. Consider the sequence of quasieigenfunctions {ϕ− 2j }j≥1 . We can construct the following sequence of solutions of the first equation in (1.1) vj (x, t) = eih2j t ϕ− 2j (x).

(4.1)

Note that vj does not satisfy the boundary conditions in (1.1) due to the fact that ϕ− 2j are not true eigenfunctions, i.e. they do not vanish at x = 0, 1. However, we can correct vj with a function vej in such a way that uj = vj + vej

(4.2)

Title Suppressed Due to Excessive Length

13

satisfies all the equations in system (1.1). To this end we define vej as the unique solution of  ρ(x)e vtt − vexx = 0, 0 < x < 1, 0 < t < T    ve(0, t) = −v (0, t) = −eih2j t ϕ− (0), 0 < t < T j 2j (4.3) ve(1, t) = −vj (1, t) = −eih2j t ϕ−  2j (1), 0 < t < T   ve(x, 0) = vet (x, 0) = 0, 0 < x < 1. i.e.

In the rest of the section we prove that (1.5) fails for the sequence uj , ¤ R1£ |uj,x (x, 0)|2 + ρ(x)|uj,t (x, 0)|2 dx 0 lim = ∞. (4.4) RT j→∞ [|uj,x (0, t)|2 + |uj,x (1, t)|2 ] dt 0 We first estimate the numerator in (4.4): Z 1 £ ¤ |uj,x (x, 0)|2 + ρ(x)|uj,t (x, 0)|2 dx 0

Z

1

£ ¤ |vj,x (x, 0)|2 + ρ(x)|vj,t (x, 0)|2 dx

1

£ − 02 ¤ 2 |(ϕ2j ) | + h22j ρ(x)|ϕ− 2j | dx

= 0

Z =

0



h22j ρm

Z 0

1

Z 2 |ϕ− 2j | dx



h22j ρm

− I2j

2 |ϕ− 2j | dx.

(4.5)

Concerning the denominator in (4.4) we have Z T Z T £ ¤ £ ¤ 2 2 |uj,x (0, t)| + |uj,x (1, t)| dt ≤ 2 |vj,x (0, t)|2 + |vj,x (1, t)|2 dt 0

Z

0 T

£ ¤ +2 |e vj,x (0, t)|2 + |e vj,x (1, t)|2 dt £ 0 0 ¤ − 0 2 2 = 2T |(ϕ− 2j ) (0)| + |(ϕ2j ) (1)| Z T £ ¤ +2 |e vj,x (0, t)|2 + |e vj,x (1, t)|2 dt

(4.6)

0

The last term can be estimated applying the following result: Proposition 1. Consider the following system:  0 < x < 1, 0 < t < T  ρ(x)vtt − vxx = 0, v(0, t) = f1 (t), v(1, t) = f2 (t), 0 < t < T  v(x, 0) = vt (x, 0) = 0, 0
(4.7)

where ρ ∈ L∞ (0, 1), 0 < ρ0 ≤ ρ(x) ≤ ρ1 < ∞. Given T > 0, there exists C(T ) > 0 such that Z TZ 1 £ ¤ |vx |2 + ρ(x)|vt |2 dxdt 0

0

14

C. Castro, E. Zuazua

³ ´ ≤ C(T )||ρ||∞ ||f1 ||2W 2,∞ (0,T ) + ||f2 ||2W 2,∞ (0,T ) Z

(4.8)

T

£ ¤ |vx (0, t)|2 + |vx (1, t)|2 0 ³ ´ ≤ C(T )||ρ||∞ ||f1 ||2W 3,∞ (0,T ) + ||f2 ||2W 3,∞ (0,T )

(4.9)

We prove this proposition at the end of the section. When applying Proposition 1 to v˜ in (4.6) we obtain Z

T

£ ¤ |uj,x (0, t)|2 + |uj,x (1, t)|2 dt 0 £ ¤ − − 0 − 0 2 2 2 2 ≤ 2C(T )h62j |(ϕ− 2j ) (0)| + |ϕ2j (0)| + |(ϕ2j ) (1)| + |ϕ2j (1)| . (4.10)

Finally, combining (4.5), (4.10) and the estimates for ϕ− 2j of the previous section we easily obtain ¤ R1£ h−1 |uj,x (x, 0)|2 + ρ(x)|uj,t (x, 0)|2 dx 2j ρm C 0 ≥ RT 2T (1 + h62j )Cp h−p [|uj,x (0, t)|2 + |uj,x (1, t)|2 ] dt 2j 0 which converges to infinity for j → ∞ and p > 7. Remark 6. The lack of boundary observability that we have proved©above ª relies on the existence of a unique sequence of quasi-eigenfunctions ϕ− . © −ª © + 2j ªj≥1 Therefore, the double sequence of eigenfunctions ϕ2j and ϕ2j−1 j≥1 constructed in section 3 is not necessary here. It will be used later to prove the lack of interior observability. Proof of Proposition 1 We introduce h(x, t) = (1 − x)f1 (t) + xf2 (t).

(4.11)

w(x, t) = v(x, t) − h(x, t)

(4.12)

Then, satisfies  ρ(x)wtt − wxx = −ρ(x) [(1 − x)f100 (t) + xf200 (t)] ,    w(0, t) = w(1, t) = 0, w(x, 0) = − [(1 − x)f1 (0) + xf2 (0)] ,    wt (x, 0) = − [(1 − x)f10 (0) + xf20 (0)] ,

0 < x < 1, 0 < t < T 0
T

Z

1 0

£ ¤ ¡ ¢ |wx |2 + ρ(x)|wt |2 dxdt ≤ C(T )||ρ||∞ ||f100 ||2∞ + ||f200 ||2∞ . (4.14)

Title Suppressed Due to Excessive Length

15

On the other hand, Z

Z

T

1

£ ¤ |hx |2 + ρ(x)|ht |2 dxdt 0 0 ¡ ¢ ≤ 2T ||ρ||∞ ||f10 ||2∞ + ||f1 ||2∞ + ||f20 ||2∞ + ||f2 ||2∞ .

(4.15)

This last estimate and (4.14) allow us to obtain easily the first inequality (4.8). For the boundary inequality (4.9) we first obtain a corresponding boundary estimate for system (4.13). The classical procedure for such estimates is to multiply the first equation in (4.13) by the multiplier xwx and integrate: Z

Z

T

Z

1

T

Z

1

ρ(x)wtt xwx dxdt −

0= 0

0 T Z 1

Z

+ 0

0

wxx xwx dxdt 0

0

ρ(x) [(1 − x)f100 (t) + xf200 (t)] xwx .

(4.16)

Now we integrate by parts in the second term of the right hand side, Z

T

Z

Z

1

T

Z

1

wxx xwx dxdt = 0

0

Z

T

Z

=− 0

0

0 1

2

|wx | dxdt + 2

0

Z

T 0

x d 2 |wx | dxdt 2 dx 2

|wx (1, t)| dt. 2

(4.17)

Combining (4.16) and (4.17) we easily find the following estimate Z

T

2

|wx (1, t)| dt 0 ³ ´ ≤ CT kρk∞ kwtt k2L2 (0,1) + kwk2H 1 (0,1) + ||f100 ||2∞ + ||f200 ||2∞ . (4.18) 0

Here we have to remove the L2 -norm of wtt from the right hand side. To do this we observe that w ˜ = wt satisfies the system  0 < x < 1, ρ(x)w ˜tt − w ˜xx = −ρ(x) [(1 − x)f1000 (t) + xf2000 (t)] ,     w(0, ˜ t) = w(1, ˜ t) = 0, 0
T

Z

1

£ ¤ |w ˜x |2 + ρ(x)|w ˜t |2 dxdt 0 0 ³ ´ ≤ C(T )||ρ||∞ ||f1 ||2W 3,∞ (0,T ) + ||f2 ||2W 3,∞ (0,T ) .

(4.20)

16

C. Castro, E. Zuazua

This inequality allows us to estimate the term with wtt = w ˜t in (4.18). Then we have Z T 2 |wx (1, t)| dt 0 ´ ³ ≤ CT kρk∞ kwk2H 1 (0,1) + ||f1 ||2W 3,∞ (0,T ) + ||f2 ||2W 3,∞ (0,T ) , (4.21) 0

for some constant C > 0. A similar estimate can be obtained for the L2 -norm of wx (0, t). Therefore Z 0

T

£ ¤ |wj,x (0, t)|2 + |wj,x (1, t)|2 dt ³ ´ ≤ CT kρk∞ kwk2H 1 (0,1) + ||f1 ||2W 3,∞ (0,T ) + ||f2 ||2W 3,∞ (0,T ) . (4.22) 0

On the other hand, Z

T

0

£ ¤ ¡ ¢ |hx (0, t)|2 + |hx (1, t)|2 ≤ T ||f1 ||2∞ + ||f2 ||2∞ .

Combining this last estimate with (4.22) we easily obtain the inequality (4.9) for v = h + w. u t Remark 7. We recall that when ρ ∈ W 1,∞ one may proceed differently in the proof of Proposition 1. Indeed, the term Z

T

Z

1

I=

ρ(x)wtt xwx dxdt 0

0

can be bounded as follows. Integrating by parts with respect to time we obtain Z

T

Z

I=−

Z

1

1

ρ(x)wt xwxt dxdt + 0

0

¸T ρ(x)wt xwx dx .

0

0

The first integral in this identity may be rewritten as Z

T

Z

1

ρ(x)wt xwxt dxdt = 0

0

1 2

Z

T

Z

0

1 0

¡ ¢ ρ(x) |wt |2 x dxdt

and, after integrating by parts, Z

T

Z

1

ρ(x)wt xwxt dxdt = − 0

0

1 2

Z

T 0

Z

1

ρx |wt |2 dxdt.

0

Obviously this argument can not be applied in our case since ρx is not bounded.

Title Suppressed Due to Excessive Length

17

5. Lack of internal observability This section is devoted to prove the lack of observability from any subinterval (α, β) of (0, 1), provided (α, β) 6= (0, 1). We first assume that α > 0 and consider the sequence of quasi-eigenfunctions {ϕ− 2j }j≥1 from which we can construct the following sequence of solutions of the first equation in (1.1) vj (x) = eih2j t ϕ− 2j (x).

(5.1)

Note that vj does not satisfy the boundary conditions in (1.1) due to the fact that ϕ− 2j are not true eigenfunctions. As in the previous case, we correct vj with a function vej in such a way that uj = vj + vej

(5.2)

satisfies all the equations in system (1.1). To this end we define vej as the unique solution of (4.3) Now we prove that for the sequence uj , (1.6) fails, i.e. ¤ R1£ |uj,x (x, 0)|2 + ρ(x)|uj,t (x, 0)|2 dx 0 = ∞. lim R R j→∞ T β [|u 2 2 j,x (x, t)| + |uj,t (x, t)| ] dxdt 0 α The numerator in (5.3) can be estimated by (4.5). Concerning the denominator in (5.3) we have Z

T

Z

0

β

α

£ ¤ |uj,x (x, t)|2 + |uj,t (x, t)|2 dxdt

Z

T

Z

Z

T

β

≤2 0

£ ¤ |vj,x (x, t)|2 + |vj,t (x, t)|2 dxdt

α

Z

β

+2 0

Z

β

α

£ − 0 ¤ 2 |(ϕ2j ) (x)|2 + h22j |ϕ− 2j (x)| dx

≤ 2T α

£ ¤ |e vj,x (x, t)|2 + |e vj,t (x, t)|2 dxdt

Z

T

Z

+2 0

1

£ ¤ |e vj,x (x, t)|2 + |e vj,t (x, t)|2 dxdt.

0

Then, by Proposition 1 applied to v˜ Z 0

T

Z

β α

£ ¤ |uj,x (x, t)|2 + |uj,t (x, t)|2 dxdt Z

β

¤ £ − 0 2 |(ϕ2j ) (x)|2 + h22j |ϕ− 2j (x)| dx α £ ¤ − 2 2 +C(T )h42j |ϕ− 2j (0)| + |ϕ2j (1)| .

≤ 2T

(5.3)

18

C. Castro, E. Zuazua

Finally we obtain ¤ R1£ |uj,x (x, 0)|2 + ρ(x)|uj,t (x, 0)|2 dx 0 RT Rβ [|uj,x (x, t)|2 + |uj,t (x, t)|2 ] dxdt 0 α R 2 h22j ρm I − |ϕ− 2j | dx 2j ≥ ¤ £ − ¤ Rβ£ − − 2 4 2 2 2 0 2 2T α |(ϕ− 2j ) (x)| + h2j |ϕ2j (x)| dx + C(T )h2j |ϕ2j (0)| + |ϕ2j (1)| ≥

−1 h2j ρm C

2(T + C(T )h42j )Cp h−p 2j

,

which converges to infinity for j → ∞ when p > 5. When α = 0 we have β 6= 1 and we can argue in a similar way with the sequence of quasi-eigenfunctions ϕ+ 2j+1 which concentrates near x = 1 instead of x = 0.

6. The multi-dimensional case In this section we show that the result of Theorem 1 can be easily extended to higher dimensional wave equations. The main idea is that, based in the 1 − d construction above, we can construct densities ρ in separated variables, which oscillate in a neighborhood of any point of the domain or of the boundary. For these densities we also construct a sequence of quasieigenfunctions concentrated inside the domain. Let Ω be an open set of Rd , d ≥ 2, with boundary ∂Ω of class C 3 and consider the wave equation  x ∈ Ω, 0 < t < T,  ρ(x)utt − ∆u = 0, u(x, t) = 0, x ∈ ∂Ω, 0 < t < T, (6.1)  u(x, 0) = u0 (x), ut (x, 0) = u1 (x) x ∈ Ω. The energy of the system is given by Z £ ¤ 1 E(t) = ρ(x)|ut (x, t)|2 + |∇u(x, t)|2 dx, 2 Ω and the boundary and internal observability properties read: ¯2 Z TZ ¯ ¯ ∂u ¯ ¯ ¯ E(0) ≤ C ¯ ∂n (x, t)¯ dσdt, 0 ∂Ω Z TZ £ ¤ E(0) ≤ C ρ(x)|ut (x, t)|2 + |∇u(x, t)|2 dxdt, 0

(6.2)

(6.3) (6.4)

ω

respectively. Here ω ⊂ Ω is an open subset and derivative.

∂ ∂n

represents the normal

Title Suppressed Due to Excessive Length

19

¯ and Ω is of class C 3 inequalities (6.3)-(6.4) hold proWhen ρ ∈ C 2 (Ω) vided a Geometric Control Condition is fulfilled (see [2] and [3]). This condition requires that every ray of Geometric Optics enters the set where the observation is being made (the boundary ∂Ω in (6.3) and the open subset ω ⊂ Ω in (6.4)) in time less than T . When ρ ∈ C 1 (Ω) the existence of rays is guaranteed but uniqueness fails in general and the analysis of inequalities (6.3)-(6.4) remains to be done in this more general setting, except for the pace dimension n = 1 in which we know that these inequalities hold even when the coefficient is in BV . In this section we show how to construct H¨older continuous density functions ρ such that the above observability inequalities fail for a large class of subsets ω. In fact, the density functions we build are such that there exists a sequence of solutions for which the energy is concentrated around a given point as much as we wish and for time intervals of arbitrary length. However, this result can not be easily interpreted in terms of Geometric Optics, since one needs C 1 coefficients in order to build solutions of the Hamiltonian system that yields the bicharacteristic rays. In any case, since for the density functions we build there exists a sequence of solutions that concentrates its energy around a point in space as much as we wish, the only possibility of getting inequalities of the form (6.3)-(6.4) is that this point belongs to the observed region. Thus, this is in contrast with the microlocal results that apply for density functions that are in C 2 . In the latter case the total energy of solutions along the ray is captured by measuring the energy at any point of the ray. To simplify things we construct here a density ρ with only one singular point at the boundary, although we could also construct densities with a finite number of singular points. This particular choice provides the following negative result: Theorem 2. Given any point xs ∈ ∂Ω, there exist H¨ older continuous density functions ρ ∈ C 0,s (Ω) for all 0 < s < 1, for which (6.3) and (6.4) fail for all T > 0 and for all subset ω ⊂ Ω such that xs does not belong to the closure of ω. Remark 8. As we mentioned in the introduction, in one space dimension, the lack of observability may also be shown for piecewise constant coefficients oscillating arbitrarily fast between two given values at some point (or several points). By separation of variables, as in the proof of Theorem 2 we present below, this can be extended to several space dimensions. However, in dimensions d ≥ 2, the lack of observability for piecewise constant densities is not new. Indeed, following Snel’s Law, one can show that, when the interface between two media with different speeds of propagation has a suitable geometry, there exist rays of Geometric Optics that are trapped in one of these media. This shows that observability fails when making measurements in the other medium. We refer to [12] for the technical details.

20

C. Castro, E. Zuazua

Nevertheless, the counterexamples we give here are more drammatic since solutions are not concentrated along rays but, in some sense, at a standing point in space. Proof: We assume, without loss of generality, the following two conditions for the singular point xs : ½

xs = 0 ∈ ∂Ω, (0, a)d ⊂ Ω, for a small enough a > 0.

(6.5)

For the first condition to hold it is sufficient to translate Ω so that the singular point is at the origin. The second one holds after a suitable rotation. With the notation introduced in section 3 we define ρ(x) = ρb(x1 ) + ρb(x2 ) + ... + ρb(xd ), where

½ ρb(s) =

− αεj (hj (s − m− j )), for s ∈ Ij , j ≥ 2 2 4π , for s ∈ K\(∪j≥2 Ij− ),

(6.6)

(6.7)

where K is a large compact set with Ω ⊂ K d . Observe that ρ is defined in separated variables from a one-dimensional function ρb as the one introduced in section 3 for the one dimensional case. The only difference is that ρb oscillates around a unique point s = 0 instead of two points which is the case for the density function in section 3. Note that ρ(x) ∈ C 0,s (K d ) and it is bounded above and below by positive constants that we will refer as ρ0 and ρ1 respectively. We now construct the sequence of quasi-eigenfunctions ϕj (x). Consider ϕj (x) = ϕ bj (x1 )ϕ bj (x2 )...ϕ bj (xd )

(6.8)

where ϕ bj is the solution of ½

(ϕ bj )00 (s) + h2j ρb(s)ϕ bj (s) = 0, s ∈ K, ϕ bj (m− (ϕ) b 0j (m− j ) = 1, j ) = 0.

(6.9)

Therefore, ϕj satisfies  2 x ∈ K d,  ∆ϕj + hj ρ(x)ϕj = 0, − − − ϕj (mj , mj , ..., mj ) = 1,  − − ∇ϕj (m− j , mj , ..., mj ) = 0.

(6.10)

We refer to ϕj as quasi-eigenfunctions of (6.1). They are not true eigenfunctions because their restrictions to Ω do not satisfy the boundary condition ϕj (x) = 0,

on ∂Ω.

(6.11)

Title Suppressed Due to Excessive Length

21

However, we show that ϕj is mainly concentrated in (Ij− )d . This is a consequence of the fact that ϕ bj is mainly concentrated in Ij− . In fact, we can argue as in section 3 for the one dimensional case to obtain Z |ϕ bj (s)|2 ds ≥ Ch−3 j , − I2j

¯ rj ¯¯2 ¯¯ ¯ bj (m− − )¯ + ¯(ϕ bj )0 (m− ¯ϕ j − j 2 ¯ ¯ ¯ r j ¯2 ¯ ¯ bj (m− )¯ + ¯(ϕ bj )0 (m− ¯ϕ j + j + 2

rj ¯¯2 )¯ = e−εj hj rj , 2 rj ¯¯2 )¯ = e−εj hj rj . 2

Following the ideas in section 3 we now estimate ϕ bj in any compact interval to the left and to right of Ij− from the estimates in the extremes of Ij− . In particular we have 2

2

|ϕ bj (s)| + |(ϕ bj )0 (s)| ≤ Cp h−p j ,

∀p > 0,

s ∈ K\Ij− .

(6.12)

Using these estimates for ϕ bj we prove that ϕj (x) = ϕ bj (x1 )ϕ bj (x2 )...ϕ bj (xd ) is concentrated in (Ij− )d , i.e. Z |ϕj (x)|2 dx ≥ Ch−3d , j − d (I2j )

2

2

|ϕj (x)| + |∇ϕj (x)| ≤ Cp h−p j ,

∀p > 0,

x ∈ K d \(Ij− )d . (6.13)

Once we have constructed the density and a sequence of quasi-eigenfunctions we introduce the following sequence of solutions of the first equation in (6.1) vj (x, t) = eihj t ϕj (x).

(6.14)

Note that vj does not satisfy the boundary conditions in (6.1). However we correct vj with a new function v˜j in such a way that uj = vj + v˜j

(6.15)

satisfies all the equations in (6.1). Therefore, we define v˜j as the unique solution of  vtt − ∆˜ v = 0, x ∈ Ω, 0 < t < T,  ρ(x)˜ v˜(x, t) = −vj (x, t) = −eihj t ϕj (x), x ∈ ∂Ω, 0 < t < T, (6.16)  v˜(x, 0) = v˜t (x, 0) = 0 x ∈ Ω. We claim that for the sequence uj observability inequality (6.3) fails. Moreover, (6.4) fails as well for those observability zones ω for which xs does not belong to ω ¯ . The proof is a straightforward generalization of the one dimensional case discussed in sections 4 and 5. However, for the sake of completeness we make the proof for the boundary observability case. We claim that ¤ R £ |∇uj (x, 0)|2 + ρ(x)|uj,t (x, 0)|2 dx Ω lim = ∞. (6.17) RT R ∂u j→∞ | j (x, t)|2 dσdt 0 ∂Ω ∂n

22

C. Castro, E. Zuazua

Indeed the numerator in (6.17) can be bounded below as in (4.5) Z Z £ ¤ 2 2 2 |∇uj (x, 0)| + ρ(x)|uj,t (x, 0)| dx ≥ hj ρm |ϕj |2 dx. (Ij− )d



(6.18)

Concerning the denominator we proceed as in (4.6) to obtain ¯2 ¯ ¯2 ¯ Z ¯ Z TZ ¯ ¯ ∂uj ¯ ¯ ∂ϕj ¯2 ¯ ∂˜ ¯ vj ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∂n (x, t)¯ dσdt ≤ 2T ¯ ∂n ¯ dσ + 2 ¯ ∂n (x, t)¯ dσdt. 0 ∂Ω ∂Ω 0 ∂Ω (6.19) To estimate the last term we introduce the generalization of Proposition 1 to the d-dimensional case. Z

T

Z

Proposition 2. Consider the following system  x ∈ Ω, 0 < t < T,  ρ(x)vtt − ∆v = 0, v(x, t) = f (x)eith , x ∈ ∂Ω, 0 < t < T,  v(x, 0) = vt (x, 0) = 0 x ∈ Ω,

(6.20)

where ρ ∈ L∞ (Ω), 0 < ρm ≤ ρ(x) ≤ ρM < ∞ a.e. and f is the restriction to ∂Ω of a function F (x) ∈ H 2 (Ω). Given T > 0, there exists C(T ) > 0 such that Z TZ £ ¤ |∇v|2 + ρ(x)|vt |2 dxdt 0 Ω ³ ´ ≤ C(T ) kh2 ρF + ∆F k2L2 (Ω) + kF k2H 1 (Ω) + h2 kF k2L2 (Ω) , Z TZ 2 |∇v| dxdt 0 ∂Ω ³ ´ ≤ C(T )h kh2 ρF + ∆F k2L2 (Ω) + kF k2H 1 (Ω) + h2 kF k2L2 (Ω) . (6.21) We prove this result at the end of the section. We apply Proposition 2 to the solution v˜ of (6.16). Note that ϕj |∂Ω satisfies the hypothesis of the proposition because it is the restriction to ∂Ω of a function ϕj ∈ H 2 (Ω). However, the estimates in (6.21) depend on the extension we choose of the boundary values to the interior of Ω. It would be natural to choose ϕj itself as extension but this is not convenient since this function exhibits a concentration of energy inside Ω. To avoid this problem we remove from ϕj its energy concentrated in (Ij− )d with a suitable cutoff function. Let us introduce the cutoff functions ψj ∈ H 2 (Ω) with the following conditions: ½ − − 1 if x ∈ Ω\(Ij−1 ∪ Ij− ∪ Ij+1 )d ψj (x) = − d 0 if x ∈ (Ij ) |ψj (x)| ≤ 1,

|∇ψj (x)| ≤ Chj ,

|∆ψj (x)| ≤ Ch2j for all x ∈ Ω,

Title Suppressed Due to Excessive Length

23

for some constant C > 0. The sequence ψj with the above properties can be − − for example ψj (x) = ψ(xrj + m− j ) where xrj + mj = (x1 rj + mj , ..., xd rj + − 2 d mj ) and ψ ∈ H (R ) is a fixed function satisfying ½ 1 if x ∈ Rd \(−3/4, 5/2)d ψ(x) = ; |ψ(x)| ≤ 1, for all x ∈ Rd . 0 if x ∈ (−1/2, 1/2)d We apply Proposition 2 to the solution v˜ of (6.16) with ψj ϕj as the extension of ϕj |∂Ω to Ω. Note that kh2j ρψj ϕj + ∆ (ψj ϕj ) k2L2 (Ω) = k2∇ψj · ∇ϕj + ϕj ∆ψj k2L2 (Ω\(I − )d ) j ³ ´ 2 2 2 ≤ C k∇ψj kL∞ (Ω) + k∆ψj kL∞ (Ω) kϕj kH 1 (Ω\(I − )d ) j



Ch4j kϕj k2H 1 (Ω\(I − )d ) , j

kψj ϕj k2H 1 (Ω) h2j

≤ kψj k2H 1 (Ω) kϕj k2H 1 (Ω\(I − )d ) h2j ≤ Ckϕj k2H 1 (Ω\(I − )d ) h4j ,

kψj ϕj k2L2 (Ω)

kψj k2L2 (Ω) kϕj k2L2 (Ω\(I − )d )



j

j

j



Ckϕj k2H 1 (Ω\(I − )d ) j

,

for some constant C > 0. Therefore we deduce that Z 0

T

¯ ¯2 ¯ ∂˜ ¯ ¯ vj (x, t)¯ dσdt ≤ C(T )h4j kϕj k2 1 . ¯ ∂n ¯ H (Ω\(Ij− )d ) ∂Ω

Z

Finally, combining (6.18) and (6.22) we have ¤ R £ |∇uj (x, 0)|2 + ρ(x)|uj,t (x, 0)|2 dx Ω RT R ∂u | j (x, t)|2 dσdt 0 ∂Ω ∂n R |ϕj |2 dx (Ij− )d ≥ C(ρ0 , T ) R ¯ . ¯ ¯ ∂ϕj ¯2 2 4 2 hj ∂Ω ¯ ∂n ¯ dσ + hj kϕj kH 1 (Ω\(I − )d )

(6.22)

(6.23)

j

Now, taking into account the estimates (6.13) for ϕj we obtain that the above quantity converges to infinity as j → ∞. Proof of Proposition 2: We generalize the proof of Proposition 1. Let us introduce g(x, t) = F (x)eiht . (6.24) Then, w(x, t) = v(x, t) − g(x, t)

(6.25)

satisfies  ρ(x)wtt − ∆w = ρ(x)h2 eiht F (x) + eiht ∆F (x), x ∈ Ω, 0 < t < T    w(x, t) = 0, x ∈ ∂Ω, 0 < t < T w(x, 0) = −F (x), x∈Ω    wt (x, 0) = −ihF (x), x ∈ Ω. (6.26)

24

C. Castro, E. Zuazua

Classical estimates on non-homogeneous wave equations show that Z TZ £ ¤ |∇w|2 + ρ(x)|wt |2 dxdt 0 Ω ´ ³ ≤ C(T ) kh2 ρF + ∆F k2L2 (Ω) + kF k2H 1 (Ω) + h2 kF k2L2 (Ω) . (6.27) On the other hand, Z TZ ³ ´ £ ¤ |∇g|2 + ρ(x)|gt |2 dxdt ≤ 2T ||ρ||∞ kF k2H 1 (Ω) + h2 kF k2L2 (Ω) . 0

0



(6.28) This last estimate and (6.27) allow us to obtain easily the first inequality in (6.21). For the boundary inequality in (6.21) we first obtain a boundary estimate for system (6.26). The classical procedure to get such estimates is to multiply ¯ d is a the first equation in (6.26) by the multiplier ν · ∇w, where ν ∈ C 1 (Ω) vector field which coincides with the outward normal of Ω at the boundary, i.e. ν = n on ∂Ω. The existence of such a vector field is proved in [10] (Lemma 3.1, Ch. 1). Then we have Z TZ Z TZ 0= ρ(x)wtt ν · ∇wdxdt − ∆wν · ∇wdxdt 0

Z

Ω T Z

+

0



2

ρ(x)h F ν · ∇wdxdt. 0

(6.29)



Now we integrate by parts in the second term of the right hand side, Z TZ 1 Z Z 1 T 2 ∆wν · ∇wdxdt = − div ν |∇w| dxdt 2 0 Ω 0 0 Z Z 1 T 2 + |∇w| dσdt. (6.30) 2 0 ∂Ω Combining (6.29) and (6) we easily find the following estimate Z TZ ³ ´ 2 |∇w| dσdt ≤ CT kρk∞ kwtt k2L2 (0,1) + kwk2H 1 (0,1) + h2 ||F ||2L2 (Ω) . 0

∂Ω

0

(6.31) Here the constant C > 0 only depends on kνkW 1,∞ , i.e. the geometry of the domain. To remove the L2 -norm of wtt from the right hand side we observe that w ˜ = wt satisfies the system  ρ(x)w ˜tt − ∆w ˜ = ρ(x)ih3 eiht F (x) + iheiht ∆F (x), x ∈ Ω, 0 < t < T    w(x,  x ∈ ∂Ω, 0 < t < T  ˜ t) = 0, w(x, ˜ 0) = wt (x, 0) = −ihF (x), x∈Ω   w ˜t (x, 0) = wtt (x, 0)    = 1 £∆w(x, 0) + ρ(x)h2 F (x) + ∆F ¤ = h2 F (x), x ∈ Ω. ρ(x) (6.32)

Title Suppressed Due to Excessive Length

25

Once again the energy estimate for the non-homogeneous problem provides Z TZ £ ¤ |w ˜x |2 + ρ(x)|w ˜t |2 dxdt 0 Ω ³ ´ ≤ C(T )h kh2 ρF + ∆F k2L2 (Ω) + kF k2H 1 (Ω) + h2 kF k2L2 (Ω) . (6.33) This inequality allows us to estimate the term wtt = w ˜t in (6.31). Then we have Z TZ 2 |∇w| dσdt 0 ∂Ω ³ ´ ≤ C(T )h kh2 ρF + ∆F k2L2 (Ω) + kF k2H 1 (Ω) + h2 kF k2L2 (Ω) , (6.34) for some constant C(T ) > 0. On the other hand, Z TZ 2 |∇g| dσdt ≤ T ||F ||2H 2 (Ω) . 0

∂Ω

Combining this last estimate with (6) we easily obtain the inequality (6.21) for v = g + w. t u 7. On the lack of controllability of the wave equation Consider the controlled system:   ρ(x)utt (x, t) − uxx (x, t) = f (x, t)χ(α,β) , 0 < x < 1, u(0, t) = g1 (t), u(1, t) = g2 (t), 0 < t < T,  u(x, 0) = u0 (x), ut (x, 0) = u1 (x).

0 < t < T,

(7.1) Here χ(α,β) represents the characteristic function of the interval (α, β) ⊂ (0, 1), f is an internal control acting on (α, β) and g1 , g2 are boundary controls acting on the extremes x = 0, 1 respectively. When ρ ∈ BV (0, 1) and 0 < ρ0 ≤ ρ(x) ≤ ρ1 < ∞ a.e. the following controllability result holds: Given √ T > Tm = ρ1 max{1 − β, α} and the initial data (u0 , u1 ) ∈ L2 (0, 1) × H −1 (0, 1) there exists f ∈ L2 (0, T ; H −1 (α, β)) and g1 , g2 ∈ L2 (0, T ) such that the solution of (7.1) satisfies u(x, t) = ut (x, t) = 0,

∀t ≥ T.

(7.2)

In fact, only one among the controls f, g1 , g2 is sufficient to guarantee the controllability if T is sufficiently large and ρ is BV. The proof of this result is a consequence of a corresponding observability inequality for the uncontrolled equation and the well-known HUM method (see [10]). The non-observability result stated in Theorem 1 provides H¨older continuous density functions ρ ∈ C 0,s ([0, 1]) for all 0 < s < 1, for which the above controllability property does not hold. To simplify things we restrict ourselves to the boundary controllability case which is the most delicate one. We prove the following

26

C. Castro, E. Zuazua

Theorem 3. There exist H¨ older continuous functions ρ ∈ C 0,s ([0, 1]) for all 0 < s < 1 with 0 < ρ0 ≤ ρ(x) ≤ ρ1 < ∞ such that, for any T > 0, there exists initial data (u0 , u1 ), in the class (u0 , ρu1 ) ∈ L2 (0, 1) × H −1 (0, 1),

(7.3)

such that, for any g2 ∈ L2 (0, T ) the solution u of (7.1) (with f = 0 and g1 = 0) does not satisfy (7.2). Remark 9. The proof of Theorem 3 that we present here can be adapted to the higher dimensional case, taking into account the lack of internal observability stated in Theorem 2. Proof: We argue by contradiction. Assume that for T > 0 and any initial data (u0 , u1 ) in the class (7.3) there exists a control g2 ∈ L2 (0, T ) such that u reaches the equilibrium at time t = T . We are going to show that, when this controllability property holds, the corresponding observability inequality for the adjoint system will hold as well which is in contradiction with the negative result of Theorem 1. We proceed in two steps: Step 1. We prove that the following operator is linear and continuous: S : L2 (0, 1) × H −1 (0, 1) → L2 (0, T ) (u0 , ρ(x)u1 ) → g2 , where g2 is the control with minimal L2 -norm which makes u to reach the equilibrium in time t = T . Note that S is well-defined because we are assuming that the controllability property holds and the control of minimal L2 −norm is unique due to the convexity of the norm. Next we prove that S is linear. Let us introduce N ⊂ L2 (0, T ), the subset of controls of the trivial initial data (0, 0). The subset N can be characterized as follows: g ∈ N if and only if Z T g(t)vx (1, t) = 0, ∀(v0 , v1 ) ∈ H 2 ∩ H01 (0, 1) × H01 (0, 1), (7.4) 0

where v is the solution of the adjoint system with initial data (v0 , v1 ):  0 < x < 1, 0 < t < T  ρ(x)vtt − vxx = 0, v(0, t) = v(1, t) = 0, 0H01 − Z − < ρu1 , v0 >H01 +

0

1

Z

ρ(x)u0 (x)v1 (x)dx + 0

T

g2 (t)vx (1, t) = 0.(7.6) 0

Title Suppressed Due to Excessive Length

27

Taking into account that the initial data are (0, 0) we see that (7.4) is equivalent to u(T ) ≡ ut (T ) ≡ 0. When ρ is regular we can consider less regular initial data (v0 , v1 ) ∈ H01 (0, 1) × L2 (0, 1) in (7.4), because solutions with these initial data satisfy the extra regularity property vx (1, t) ∈ L2 (0, T ). It is easy to see that N is a non-empty closed linear subset of L2 (0, T ) and therefore we can decompose L2 (0, T ) in a direct sum as follows L2 (0, T ) = N + N ⊥ . Then, any L2 −control gˆ2 of the initial data (u0 , u1 ) can be uniquely decomposed as gˆ2 = g2 + n, where g2 ∈ N ⊥ is the minimal L2 −norm control and n ∈ N . We deduce that S(u0 , ρu1 ) is the unique control g2 of (u0 , u1 ) satisfying g2 ∈ N ⊥ . Now we assume that S(u0 , ρu1 ) = g2 and S(v0 , ρv1 ) = h2 . Then g2 +h2 ∈ N ⊥ and it is a control of (u0 +v0 , u1 +v1 ). Therefore S(u0 +v0 , ρu1 +ρv1 ) = g2 + h2 and S is linear. Finally, to prove the continuity of S we use the closed graph theorem. Let us consider a sequence of initial data (u0,j , u1,j ) and a sequence of associated minimal L2 −controls g2,j such that (u0,j , ρ(x)u1,j ) → (u0 , ρ(x)u1 ), in L2 (0, 1) × H −1 (0, 1), g2,j → g2 , in L2 (0, T ).

(7.7) (7.8)

Observe that g2 ∈ N ⊥ because {g2,j }j∈N ⊂ N ⊥ and N ⊥ is closed. On the other hand, as g2,j is a control of (u0,j , u1,j ), we have Z − < ρu1,j , v(x, 0) >H01 +

Z

1

ρ(x)u0,j (x)vt (x, 0)dx + 0

T

gj (t)vx (1, t) = 0, 0

(7.9) for all (v0 , v1 ) ∈ H 2 ∩ H01 (0, 1) × H01 (0, 1). Indeed, in (7.9) we are simply writing that u(T ) ≡ ut (T ) ≡ 0 in a weak form. Passing to the limit in (7.9) we find that g2 is a control for (u0 , u1 ) and therefore S(u0 , ρu1 ) = g2 . We have proved that S is a linear operator with closed graph. By the closed graph theorem it is continuous, i.e. there exists a constant C > 0 such that kg2 kL2 (0,T ) ≤ C k(u0 , ρ(x)u1 )kL2 (0,1)×H −1 (0,1) . (7.10) Step 2. We prove that (7.10) is equivalent to the corresponding observability inequality for the adjoint system, i.e. Z E(0) ≤ C

T

|vx (1, t)|2 dt

0

where v is a solution of (7.5) with initial data (v0 , v1 ).

28

C. Castro, E. Zuazua

From (7.6) and taking into account that u(T ) ≡ ut (T ) ≡ 0, we have ¯ ¯ Z 1 ¯ ¯ ¯− < ρu1 , v0 >H 1 + ρ(x)u0 (x)v1 (x)dx¯¯ ¯ 0 0

≤ C k(u0 , ρ(x)u1 )kL2 (0,1)×H −1 (0,1) kvx (1, t)kL2 (0,T ) for all (u0 , ρ(x)u1 ) ∈ L2 (0, T ) × H −1 (0, T ), and therefore k(v0 , v1 )kH 1 (0,1)×L2 (0,1) ≤ Ckvx (1, t)kL2 (0,T ) . 0

8. On the lack of dispersive properties and Strichartz inequalities for the wave equation In this section we consider the wave equation in the whole space ½ ρ(x)utt − ∆u = 0, x ∈ Rd , t > 0, (8.1) u(x, 0) = u0 , ut (x, 0) = u1 (x), x ∈ Rd . For ρ constant and d ≥ 2, the Strichartz estimates establish space-time integrability properties of the solutions of this system due to dispersive effects. One version of these estimates is kukL2 (R;Lqx (Rd )) ≤ c ku0 kH r (Rd ) + ku1 kH r−1 (Rd ) t

(8.2)

provided that

2d 2(d + 1) , ≤ q < ∞. (8.3) d − 2r − 1 d−1 When ρ is smooth the above estimates (8.2) hold locally in time and they are sharp (see [13]). For low regularity coefficients, ρ ∈ C 1,s , there exist weakened versions of estimates (8.2) (see [14] and [15]). Note that the above estimates cannot be obtained by classical Sobolev embeddings. In fact, due to the conservation of energy we easily deduce that the solutions of (8.1) belong to the class u ∈ C(0, ∞; H r (Rd )). Therefore, the Sobolev embeddings in space allow us to obtain: q=

2d

u ∈ C(0, ∞; L d−2r (Rd ).

(8.4)

For ρ smooth, estimates (8.2) establish, in particular, that the solution 2d 2d u(·, t) ∈ Lq (Rd ) with d−2r < q ≤ d−2r−1 almost everywhere in t ∈ R. According to the constructions of the previos sections, for the C 0,s density function we have built, and due to the existence of a sequence of solutions that concentrates its energy around a point as much as one wishes, no (8.2) nor any weakened version of it may hold except of course for the integrability properties that Sobolev’s embedings provide. Therefore, one may say that in the class C 0,s density functions there is no Strichartz type estimates even locally in space-time and for weaker integrability requeriments. More precisely, we have the following result:

Title Suppressed Due to Excessive Length

29

Theorem 4. Given any point xs ∈ Rd , there exist H¨ older continuous density functions ρ ∈ C 0,s (Rd ) for all 0 < s < 1, and a sequence of solutions uj of (8.1) for which ¡R

¢1/p |uj (·, t)|p dx lim =∞ j→∞ kuj (·, 0)k r d + k∂t uj (·, 0)k r−1 d H (R ) H (R ) Id

for any p > with δ > 0.

2d d−2r ,

(8.5)

t ∈ R and for all d-dimensional cube I d = [ xs , xs + δ]d

Proof: We assume without loss of generality that xs = 0. Consider the density ρ introduced in (6.6) and defined by (6.6)-(6.7), that we extend to Rd \K d by the constant 4π 2 . As we have seen, there exists a sequence (h2j , ϕj ) of eigenpairs of (8.1) concentrated around x = 0 associated to ρ, i.e. solutions of ∆ϕj + h2j ρ(x)ϕj = 0, x ∈ K d ,

(8.6)

satisfying (6.13). Note that uj (x, t) = eihj t ϕj (x)

(8.7)

constitute a sequence of solutions of (8.1) over K d that we can extend to Rd in such a way that (uj , ∂t uj ) is uniformly bounded in H r × H r−1 as j → ∞. We prove that, due to the concentration of energy of ϕj near x = 0, the sequence uj satisfies (8.5). Observe that |uj (·, t)| = |ϕj (·)| for all t and x ∈ K d . Therefore it is sufficient to prove the following ¡R

¢1/p |ϕj (x)|p dx lim = ∞, j→∞ kϕj k r H (K d ) + hj kϕj kH r−1 (K d ) Id

∀p >

2d . d − 2r

(8.8)

Recall that ϕj (x) is defined in separated variables over each Ij− as follows − ϕj (x) = ϕ bj (x1 )...ϕ bj (xd ) = wεj (hj (x1 − m− j ))...wεj (hj (xd − mj )).

Therefore, the change of variables yα = hj (xα − m− j ) in (8.8) provides ¡R

¢1/p |ϕj (x)|p dx kϕj kH r (K d ) + hj kϕj kH r−1 (K d ) ³R ´d/p p |w (y)| dy d d ε j Ij − + −r , ≥ hj q 2 ° ° ° ° −p °wεj °d r−1 °wεj °d r + h + O(h ) j j H (Ij ) H (Ij ) Id

where the interval I j = hj (Ij− − m− j ), and for all p ≥ 0.

(8.9)

30

C. Castro, E. Zuazua

For j sufficiently large, [0, 1] ⊂ I j and the numerator in (8.9) can be bounded below by a constant C(d, p) which does not depend on εj , i.e. !d/p µZ ÃZ ¶d/p Ij

1

|wεj (y)|p dy

≥ 0

|wεj (y)|p dy

≥ C(d, p) > 0

in view of (2.8). On the other hand, the denominator in (8.9) is bounded above, by a constant which does not depend on ²j , due to the properties of wε in Lemma 1. It follows that the left hand side of (8.9) cannot be uniformly bounded 2d in j → ∞ for p > d−2r . 9. Comments In this section we mention a number of applications and remarks related with the result stated in Theorem 1. 1. The density function that we have constructed is singular at both extremes in the sense that it oscillates more and more as we approach x = 0 and x = 1. Similar constructions can be done to obtain densities with singularities at one interior point or a finite number of interior and boundary points. We have chosen to present the construction above since it is the simplest one that provides at the same time a non-observability result for the boundary case and for the interior one when we restrict ourselves to connected intervals (α, β) for the observation. Note that a more general result, i.e. a density function for which any observability inequality fails without any restriction on the observability zone, would imply a construction of the density with infinitely many singular points localized in a dense set of [0, 1]. Whether such density functions exist is still an open problem. 2. Variable coefficients in the principal part of the operator. Theorem 1 can be easily adapted to systems of the form  0 < x < 1, 0 < t < T,  utt − (a(x)ux )x = 0, u(0, t) = u(1, t) = 0, 0 < t < T, (9.10)  u(x, 0) = u0 (x), ut (x, 0) = u1 (x) 0 < x < 1. In this case the energy of the system is given by Z ¤ 1 1£ E(t) = |ut (x, t)|2 + a(x)|ux (x, t)|2 dx, 2 0 and the boundary and internal observability properties read: Z T £ ¤ E(0) ≤ C |ux (0, t)|2 + |ux (1, t)|2 dx, 0

Z

T

Z

β

E(0) ≤ C 0

α

£ ¤ |ut (x, t)|2 + a(x)|ux (x, t)|2 dx,

(9.11)

(9.12) (9.13)

Title Suppressed Due to Excessive Length

31

respectively. Once again, when a ∈ BV (0, 1) and 0 < a0 ≤ a(x) ≤ a1 < ∞ a.e. x ∈ (0, 1), both observability inequalities hold when T is large enough. Theorem 5. There exist continuous functions a ∈ C 0,s ([0, 1]) for all 0 < s < 1 with 0 < a0 ≤ a(x) ≤ a1 < ∞ for which (9.12) and (9.13) fail for all T > 0 and for all subinterval (α, β) ⊂ (0, 1) ((α, β) 6= (0, 1)). Proof: We center our attention in the interior observability inequality (9.13) since the other one is similar. Consider the following change of variables R x dr µZ 1 ¶2 dr 0 a(r) , ρ(y) = y(x) = R 1 a(x(y)) dr 0 a(r) 0 a(r)

v(y, t) = u(x(y), t), v(y, 0) = u0 (x(y)),

vt (y, 0) = u1 (x(y))

(9.14)

where x(y) represents the inverse function of y(x). Note that y(x) ∈ C 1 (0, 1) and y(x) is invertible because y 0 (x) 6= 0 for all x ∈ [0, 1]. Hence, x(y) ∈ C 1 (0, 1) and ρ(y) has the same regularity as a(x). With the above change of variables (9.10) is transformed into  0 < y < 1, 0 < t < T,  ρ(y)vtt − vyy = 0, v(0, t) = v(1, t) = 0, 0 < t < T, (9.15)  v(y, 0) = v0 (y), vt (y, 0) = v1 (y) 0 < y < 1. For this system we can apply Theorem 1 and find ρ for which the internal observability fails. More precisely, there exists a sequence vj such that ¤ R1£ |vj,y (y, 0)|2 + ρ(y)|vj,t (y, 0)|2 dy 0 lim R R =∞ (9.16) j→∞ T y(β) [|v (y, t)|2 + |v (y, t)|2 ] dydt j,y j,t 0 y(α) for any 0 < α < β < 1. Coming back to the original variables we easily obtain the following ¤ R1£ 2 a (x)|uj,x (x, 0)|2 + a(x)|uj,t (x, 0)|2 dx 0 lim R R = ∞, (9.17) j→∞ T β [a2 (x)|u 2 a2 |uj,t (x, t)|2 ] dxdt j,x (x, t)| + b 0 α ³R ´−1 1 1 where b a = 0 a(x) dx . Note that a(x) is bounded above and below with positive constants because ρ(y) satisfies this property. This fact combined with (9.17) provides the result in Theorem 5. 3. Schr¨ odinger equation. The singular densities introduced for the wave equation produce lack of observability for the linear Schr¨odinger equation   iρ(x)ut + ∆u = 0, x ∈ Ω, 0 < t < T, u(x, t) = 0, x ∈ ∂Ω, 0 < t < T, (9.18)  u(x, 0) = u0 (x), x ∈ Ω,

32

C. Castro, E. Zuazua

where Ω is a bounded domain of Rd with boundary ∂Ω of class C 3 and T > 0. The energy of the system is given by Z 1 E(t) = |∇u(x, t)|2 dx, (9.19) 2 Ω and the boundary and internal observability properties read: Z

T

Z |

E(0) ≤ C 0

Z

T

Z

∂Ω

|∇u(x, t)|2 dx,

E(0) ≤ C 0

∂u (x, t)|2 dx, ∂n

(9.20) (9.21)

ω

∂ respectively. Here ∂n represents the normal derivative. 2 When ρ ∈ C (Ω) and ω satisfies the geometrical control condition, the above observability inequalities hold for any T > 0. We refer to [9] for the proof of this result. Theorem 2 can be also adapted for the Schr¨odinger equation:

Theorem 6. Given any point xs ∈ ∂Ω, there exist H¨ older continuous density functions ρ ∈ C 0,s (Ω) for all 0 < s < 1, for which (9.20) and (9.21) fail for all T > 0 and for all subset ω ⊂ Ω such that xs does not belong to the closure of ω. The proof of this theorem is a straightforward generalization of the proof of Theorem 2. In this result the density ρ has only one singular point xs in the sense that it oscillates very much near this point. We observe that, as in the wave equation, more general results can be proved with densities oscillating around several different points. Note that the Strichartz type estimates fail also in (9.18). 4. Heat equation. In this section we observe that the singular densities introduced for the wave equation do not produce lack of observability for the linear heat equation   ρ(x)ut − ∆u = 0, x ∈ Ω, 0 < t < T, u(x, t) = 0, x ∈ ∂Ω, 0 < t < T, (9.22)  u(x, 0) = u0 (x), x ∈ Ω, where Ω is a bounded domain of Rd with boundary ∂Ω of class C 3 and T > 0. The energy of the system is given by Z 1 E(t) = |u(x, t)|2 dx, (9.23) 2 Ω

Title Suppressed Due to Excessive Length

and the boundary and internal observability properties read: Z TZ ∂u E(T ) ≤ C | (x, t)|2 dx, ∂n 0 ∂Ω Z TZ E(T ) ≤ C |∇u(x, t)|2 dx, 0

33

(9.24) (9.25)

ω

respectively. When ρ ∈ C 2 (Ω) the above observability inequalities hold for any T > 0 (see [8]). We briefly sketch why our construction, that can be perfectly adapted to the heat equation, does not produce lack of observability. Observe that the quasi-eigenfunctions ϕj that we constructed for the wave equation in (6.10) are also quasi-eigenfunctions of the heat equation in the sense that 2 vj (x, t) = e−hj t ϕj (x) satisfy the differential equation in (9.22) and they are exponentially concentrated in the interior of Ω. However, due to the non reversibility of the heat equation, in (9.24) and (9.25) we must estimate the energy of the solution in time T > 0. Therefore we have to estimate a much smaller function than for the wave equation since the solution of the heat equation at time T is multiplied by the factor 2 e−hj T . To our knowledge there is no example in the literature of L∞ coefficients for linear parabolic equations for which the observability inequalities (9.24) and (9.25) fail. This is an interesting open problem. Acknowledgements. This work has been supported by grant PB96-0663 of the DGES (Spain) and the TMR Project “Homogenization and Multiple Scales” of the EU. The second author is grateful to F. Colombini and S. Spagnolo for enlightenling discussions on their previous work [5].

References 1. M. Avellaneda, C. Bardos and J. Rauch, Contrˆ olabilit´e exacte, homog´en´eisation et localisation d’ondes dans un milieu non-homog`ene, Asymptotic Analysis, 5 (1992), 481-494. 2. C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control and Optim. 30 (5) (1992), 1024-1075. 3. N. Burq, Contrˆ olabilit´e exacte des ondes dans des ouverts peu r´eguliers, Asymptot. Anal. 14 (2) (1997), 157-191. 4. C. Castro and E. Zuazua, Low frequency asymptotic analysis of a string with rapidly oscillating density, SIAM J. Appl. Math., 60 (2000), 1205-1233. 5. F. Colombini and S. Spagnolo, Some examples of hyperbolic equations without ´ Norm. Sup., 22 (4), (1989), 109-125. local solvability, Ann. Scient. Ec. 6. S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (2), (1995) 545-573.

34

C. Castro, E. Zuazua

7. M.S.P. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press, Edinbourgh and London, 1973. 8. A.V. Fursikov and O.Y. Imanunilov, Controllability of evolution equations, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, Korea, 1996. 9. G. Lebeau, Contrˆ ole de l’´equation de Schr¨ odinger, Journal Math. Pures et Appl. 71 (1992), 267-291. 10. J.L. Lions, Contrˆ olabilit´e exacte, stabilisation et perturbations de syst`emes distribu´es. Tomes 1 & 2. Masson, RMA 8 & 9, Paris, 1988. 11. J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag, 1972. 12. F. Macia and E. Zuazua, Some applications of gaussian beams to the controllability of waves, Fifth international conference on mathematical and numerical aspects of wave propagation, SIAM, 2000, 1011-1015. 13. H.F. Smith and C.D. Sogge, On the critical semilinear wave equation outside convex obstacles, Journal of the American Math. Society 8 (4), (1995) 879-916. 14. H.F. Smith and D. Tataru, Sharp counterexamples for Strichartz estimates for low frequency metrics, preprint. 15. D. Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients II, preprint. Departamento de Matem´ atica e Inform´ atica ETSI Caminos, Canales y Puertos Univ. Polit´ecnica de Madrid 28040 Madrid, Spain. [email protected] and Departamento de Matem´ aticaAplicada Universidad Complutense 28040 Madrid, Spain. [email protected]

Concentration and lack of observability of waves in ...

interior of the domain respectively (see [10]). For a long time the problem of whether these estimates do hold for less regular coefficients (say ρ ∈ L∞(0, 1) or ρ ...

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