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Some uniqueness results for the determination of forces acting over Germain-Lagrange plates

Three different techniques

• A. Kawano

Escola Politecnica da Universidade de Sao Paulo

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 1 / 49

Introduction Presentation of the Germain-Lagrange Operator

Plan

1

Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems

2

Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 2 / 49

Introduction Presentation of the Germain-Lagrange Operator

Basic Equation

The Germain-Lagrange equation models the vibration of a plate. Is is given by ∂2u ∂t2 + △2u = h, where u is the displacement and h is the loading.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 3 / 49

Introduction Presentation of the Germain-Lagrange Operator

Fundamental solution

We can work out an explicit expression for a fundamental solution φ of the Germain-Lagrange operator. It is given by φ(t, x) = H(t) 2π +∞ sen[r 2t] r J0(r | x |) dr. (1)

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 4 / 49

Introduction Presentation of the Germain-Lagrange Operator

The Germain-Lagrange operator is not hypo-elliptic

We have two properties: The operator P = △2 + ∂2

∂t2 is not hypo-elliptic.

Proof

The speed of propagation of signals in a Germain-Lagrange plate is infinite.

Proof

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 5 / 49

Introduction Presentation of the inverse problems

Plan

1

Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems

2

Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 6 / 49

Introduction Presentation of the inverse problems

Inverse problems

Statement of the problems presented here

We consider the problem of proving that loads h of the form h(t, x) = g(t)f(x) in the equation ∂2u ∂t2 + △2u = h = g(t)f(x), can be determined uniquely from data about the displacement of a set

• f points in the plate.

In general, f is a distribution and g a function of time that is class C1.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 7 / 49

Introduction Presentation of the inverse problems

Inverse problems

Statement of the problems presented here

Given the fact that the speed of propagation is infinite, the data for the inverse problem is taken in arbitrary small intervals of time and space. The problems we are going to present are:

1

Unbounded plate that extends over all R2.

2

Rectangular plate.

3

Bounded plate with arbitrary shape.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 8 / 49

Inverse Problems Unbounded plate

Plan

1

Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems

2

Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 9 / 49

Inverse Problems Unbounded plate

Unbounded Plate

    

∂2u ∂t2 + △2u = g ⊗ f,

in ]0, +∞) × R2, u(0) = 0,

∂u ∂t (0) = 0,

(2) where f ∈ E′(R2) (To be determined); ∃T0 > 0 such that g ∈ C([0, T0]) and g(0) = 0.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 10 / 49

Inverse Problems Unbounded plate

Data for the identification of f ∈ E′(R2)

Theorem 1 (For the unbounded case) If ∃T0 > 0 such that g ∈ C([0, T0]) and g(0) = 0, and f ∈ E′(R2), then for arbitrary 0 < T < T0, the knowledge of the set Γ]0,T[×Ω = {u , ψ ⊗ ϕ : ψ ∈ C∞

c (]0, T[), ϕ ∈ C∞ c (Ω)} ,

where Ω ⊂ R2 is arbitrary, is enough to determine uniquely f ∈ E′(R2).

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 11 / 49

Inverse Problems Unbounded plate

Idea of the proof

Technique: spherical means

The steps for the proof are:

1 Factor into two Schrodinger operators 2

Write explicitly the solution in terms of a convolution with a fundamental solution of the equation. It is well known that the only solution w ∈ C∞([0, +∞), S′(R2)), supported in [0, +∞), that satisfies

(9) is given by

w(t, x) = Et ∗ f, where the convolution is performed only in the spatial variable, and for t > 0, Et(x) = 1 4πı t eı x 2

4t . 3 Aplication of a spherical means result

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 12 / 49

Inverse Problems Rectangular Plate

Plan

1

Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems

2

Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 13 / 49

Inverse Problems Rectangular Plate

Rectangular Plate

       ρh ∂2˜

u ∂˜ t2 + D△2˜

u = g ⊗ ˜ Q, in ]0, +∞) × (]0, L1[×]0, L2[), ˜ u = ∂˜

u ∂˜ t = 0,

at t = 0, γ∂ ˜

R

˜ u(˜ t)

• = γ∂ ˜

R

• ∂2˜

u ∂˜ ν2 (˜

t)

• = 0,

∀˜ t ≥ 0. (3) where g : [0, +∞) → R is a C1 function with g(0) = 0, ˜ u stands for the vertical displacement, ∂ ˜ R is the boundary of the rectangle ˜ R =]0, L1[×]0, L2[, ν is the normal to ∂ ˜ R, where it is defined, and γ∂ ˜

R : H1(˜

R) → H

1 2 (∂ ˜

R) is the trace operator u → u|∂ ˜

• R. The constants

ρ, h, D > 0 that appear in

(3) stand respectively for mass density, plate

thickness and flexural rigidity.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 14 / 49

Inverse Problems Rectangular Plate

Data for the rectangular plate problem

The set Γa

˜ Ta,˜ Ω =

˜ u(˜ t) , ψ

• : ˜

t ∈ [0, ˜ Ta], ψ ∈ C∞

c (˜

Ω)

• ,

(4) where ˜ Ω ⊂ ˜ R is any arbitrary non-empty line segment parallel to

• ne of the rectangle sides, which we take as being the Ox axis,

containing any point (x, y0) ∈ ˜ R such that sen(y0n) = 0, ∀n ∈ N, is enough for the identification of ˜ Q ∈ L2(˜ R), provided that ˜ Ta > L1L2

4π

• ρh

D .

The set Γc

˜ Tc,[0,L2] =

∂˜ u ∂˜ x (˜ t, 0, ·) , ψ

• : ˜

t ∈ [0, ˜ Tc], ψ ∈ C∞

c ([0, L2])

• , (5)

where ˜ Tc > 0 can be arbitrarily small, is enough for the determination of ˜ Q ∈ L2(˜ R).

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 15 / 49

Inverse Problems Rectangular Plate

Steps in the proof

Technique: almost periodic distributions

1.

Representation of the loading

2.

Representation of the solution

3.

Application of the data

• 4. The main ingredient:

Almost Periodic Distributions

5.

Application of almost periodic distributions property

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 16 / 49

Inverse Problems Plates with arbitrary shapes

Plan

1

Introduction Presentation of the Germain-Lagrange Operator Presentation of the inverse problems

2

Inverse Problems Unbounded plate Rectangular Plate Plates with arbitrary shapes

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 17 / 49

Inverse Problems Plates with arbitrary shapes

Plates with arbitrary shapes

Let Ω ⊂ R2 be any bounded subset with smooth and regular boundary. Let T0 > 0 and N ∈ N. Consider the plate equation        ∂2u ∂t2 + △2u = N

n=1 gn ⊗ fn,

in ]0, T0[×Ω, u = ∂u

∂t = 0,

at t = 0, γ∂Ω (u(t)) = γ∂Ω (△u(t)) = 0, ∀t ∈ [0, T0[, (6) where the set {gn : n ∈ {1, . . . , N}} ⊂ C1([0, T0]) is linearly independent, and fn ∈ H−1(Ω), ∀n ∈ {1, . . . , N}. The vertical displacement is represented by u, and γ∂Ω : H1(˜ R) → H

1 2 (∂ ˜

R) is the trace operator u → u|∂Ω.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 18 / 49

Inverse Problems Plates with arbitrary shapes

Data for plates with arbitrary shapes

Let O ⊂ R2 be any open neighborhood of G+(x) ⊂ R2. Now define VG+(x) = O ∩ Ω. Also, define for any x0 ∈ R2 \ Ω and T < T0 the set ˜ ΓT,G+(x0)(u) = {(t, φ, u(t, ·) , φ) : t ∈ [0, T], φ ∈ C∞

c (VG+(x0))} . (7)

Observe that from ˜ ΓT,G+(x0)(u) it is not possible to extract information about derivatives of u with respect to the space variable at the boundary, since all φ ∈ C∞

c (VG+(x0)) are already null near ∂Ω.

˜ ΓT,G+(x0)(u) is in fact interior data. Define for any x0 ∈ R2 \ Ω and T < T0 the set ˆ ΓT,G+(x0)(u) =

• t, φ,

∂u(t, ·) ∂ν , φ

• : t ∈ [0, T], φ ∈ C∞

c (G+(x0))

• ,

(8)

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 19 / 49

Inverse Problems Plates with arbitrary shapes

Theorem

Theorem 1 For Ω ⊂ R2 open and bounded, given x0 ∈ R2 \ Ω, and any 0 < T < T0, then if u is a solution of

(6) , any one of the sets

˜ ΓT,G+(x0)(u) or ˆ ΓT,G+(x0)(u) is enough to uniquely determine the set {f1, . . . , fN} ⊂ H−1(Ω) in

(6) .

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 20 / 49

Inverse Problems Plates with arbitrary shapes

Steps in the proof

Technique: Comparison with the wave equation

1.

Representation of the loading

2.

Representation of the solution

3.

Application of the data

• 4. The main ingredient:

Comparison with the wave equation

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 21 / 49

Thanks

Thank you for your attention!

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 22 / 49

Thanks

P is not hypo-elliptic

By Hörmander’s theorem, P is hypo-elliptic if and only if its symbol P satisfies There are δ, C > 0 such that |P(α)(ξ)|/|P(ξ)| ≤ C|ξ|−|α|δ for ξ ∈ Rm and |ξ| sufficiently big. The symbol is P(ξ) = (ξ2

1 + ξ2 2)2 − ξ2 3.

Take α = (0, 1, 0) to see that

• ξ2

1 + ξ2 2

• | ξ2 | ≤ C | ξ |−δ
• (ξ2

1 + ξ2 2)2 − ξ2 3

• is not possible for any δ, C > 0.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 23 / 49

Thanks

Speed of signal propagation is infinite

The proof is just an application of the Paley-Wiener Theorem. If the speed of propagation of signals were finite, then freezing the fundamental solution φ

(1) at any fixed t > 0, the distribution

x → φ(t, x) would have compact support. And by the Paley-Wiener Theorem, its Fourier Transform would have controlled growth: There would exist C > 0, N0 ∈ N and M > 0 such that lim sup

|z|→+∞

| φ(z, t)| (1 + | z |)N0eM| Imz | ≤ C. However, it is possible that it is false. And therefore, the speed of propagation of signals is in fact infinite.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 24 / 49

Unbounded plate Factorization

Factor into two Schrodinger operators

P1 . = ∂ ∂t − ı △

• ,

P2 . = ∂ ∂t + ı △

• ,

∂2 ∂t2 + △2 = P1 P2. The original problem implies

• P1w = 0,

in ]0, +∞) × R2, w(0) = f, (9) where w = P2u is supported in [0, +∞). Now the set for the identification of f is ˘ Γ]0,T[×Ω . = {w(t) , ϕ : ϕ ∈ C∞

c (Ω), t ∈]0, T[} ,

where T > 0 is arbitrarily small.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 25 / 49

Unbounded plate Spherical Means

Spherical Means Result

If φ ∈ C∞

c (Rn), φx denotes the function y → φ(y − x).

Theorem 2 Suppose that Ω ⊂ Rn is an open connected subset, Q1 ⊂ Q2 ⊂ R and that for any t ∈ Q2, Tt ∈ D′(Ω). Suppose also that there is U ⊂ Ω such that Tt|U ≡ 0, ∀t ∈ Q1. Suppose that there exists a neighborhood of the origin V ⊂ Rn such that for all x ∈ Ω and all φ ∈ C∞

c (V) such that

supp(φx) ⊂ Ω, it is true that Tt , φx = 0, ∀t ∈ Q1 ⇒

• S1(0)
• Tt , φx+ ˙

ξR

• dS ˙

ξ = 0, ∀t ∈ Q2,

(10) for all R > 0 such that supp(φx+ ˙

ξR) ⊂ Ω, ∀ ˙

ξ ∈ S1(0). Then Tt ≡ 0, ∀t ∈ Q2.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 26 / 49

Rectangular plate Representation of the loading

Representation of the loading

We represent Q ∈ L2(R) by Q =

+∞

• m=1

+∞

• n=1

Qm,nsen(mx)sen(ny). (11) Assuming that Qm,n ∈ R, ∀m, n ∈ N, because we deal only with real valued loading.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 27 / 49

Rectangular plate Representation of the solution

Representation of the solution

We prove that the solution of the direct problem is u(t, x, y) =

+∞

• m=1

+∞

• n=1

Qm,nsen(mx)sen(ny) km,n t g(t −τ)sen(km,nτ) dτ. (12)

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 28 / 49

Rectangular plate Application of the data

Application of the data

After some manipulation, and invoking the data available, we have Proposition 5.1 Consider u given by

(12) . Let Ω ⊂ R be any line segment parallel to

the Ox axis that contains a point (x0, y0) ∈ R and T0 > 0. If for any φ ∈ C∞

c (Ω), u(t) , φ = 0, ∀t ∈ [0, T0], then +∞

• m=1

+∞

• n=1

Qm,n km,n sen(km,nτ)sen(ny0) sen(mx) , φ = 0, ∀τ ∈ [0, T0]. (13)

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 29 / 49

Rectangular plate Almost periodic distributions

Almost Periodic Distributions

To analyse the uniqueness conditions, we shall use some results concerning almost periodic distributions [1]. Consider a series of the form w(t) =

• n∈N

aneı λnt, (14) where the coefficients (an)n∈N and exponents Λ = (λn)n∈N satisfy the following conditions:

1

There is a q ∈ Z+ such that (n−qan)n∈N ∈ ℓ1, that is, (an)n∈N ∈ s′, the space of slowing growing sequences.

2

For Λ, we suppose that there are n0 ∈ N, C > 0, and α > 0 such that n ≥ n0 ⇒ | λn | ≥ Cnα.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 30 / 49

Rectangular plate Almost periodic distributions

Definition 5.1 (Uniform discrete set) The set {dn : n ∈ N} is a uniformly discrete set if there exists δ > 0 such that i = j ⇒

• di − dj
• ≥ δ.

(15) Definition 5.2 (upper uniform density) Given Λ = (λn)n∈N, uniformly discrete, the upper uniform density of Λ, denoted by u. u. d.(Λ), is defined as

• u. u. d.(Λ) =

lim

r→+∞ max x∈R

♯(Λ ∩ [x, x + r]) r , where ♯(Λ ∩ [x, x + r]) denotes the cardinality of the set Λ ∩ [x, x + r].

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 31 / 49

Rectangular plate Almost periodic distributions

The following result is from [1]. Theorem 3 For w of the form

(14) , given Λ uniformly discrete, if α > 1, then for any

τ > 0, w|[−τ,τ] = 0 ⇒ w ≡ 0. If α = 1, Λ = O(n), then if τ > π u. u. d.(Λ), then w|[−τ,τ] = 0 ⇒ w ≡ 0. Remember: w(t) =

• n∈N

aneı λnt.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 32 / 49

Rectangular plate Application of APD to the problem

Application of the uniqueness property of almost periodic distributions

We order the exponents (km,n) non-decreasingly and call the ordered sequence ˜ Λ = (˜ λν)ν∈N, we will have ˜ Λ = O(ν). Let Ω ⊂ R be an arbitrary line segment parallel to the Ox axis. Defining for φ ∈ C∞

c (Ω),

Bν(φ) =

• m,n∈N

km,n=|λν|

(−1)ν Qm,n 2ı km,n sen(ny) sen(mx) , φ (16) and Λ = (λν)ν∈N, λν = (−1)ν˜ λ⌈ ν

2 ⌉,

where ⌈x⌉ is the least natural number a such that x ≤ a, equation

(13)

becomes

• ν∈N

Bν(φ)eı λντ = 0, ∀τ ∈ [0, T0], ∀φ ∈ C∞

c (Ω).

(17)

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 33 / 49

Rectangular plate Application of APD to the problem

Proposition 5.2 The set Λ = (λn)n∈N is uniformly discrete if and only if (L1/L2)2 ∈ Q. If (L1/L2)2 ∈ Q, then

• u. u. d.((λν)ν∈N) ≤ L1L2

4πh2 . Remember: L1 and L2 are the sides of the rectangle and h is the thickness of the plate.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 34 / 49

Rectangular plate Application of APD to the problem

Theorem 4 For a given rectangular plate with sides (L1, L2) ∈ R2

+, the set Γa Ta,Ω,

where Ω ⊂ R is an arbitrary line segment parallel to the Ox axis that contains any point (x, y0) ∈ R such that sen(ny0) = 0, ∀n ∈ N, and Ta > L1L2

4πh2 , is enough for the identification of Q ∈ L2(R).

Proof (When L2

1/L2 2 ∈ Q).

The proof is divided in two parts. Here we deal only with the situation when L2

1/L2 2 ∈ Q.

From

Proposition 5.2 , we know that the set (λν)ν∈N is uniformly discrete.

Then by an application of

Theorem 3 to equation (17) , we obtain that for

each φ ∈ C∞

c (Ω), the coefficients Bν(φ) = 0, ∀ν ∈ N. For each ν ∈ N,

we have an linear equation. Now, it suffices to use a enumerable quantity of φ’s in C∞

c (Ω), or to take enough derivatives in the direction

• f the line segment Ω, to determine Qm,n, ∀m, n ∈ N.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 35 / 49

Plates with arbitrary shapes Representation of the loading

Representation of the loading

Engenvectors

Consider the eigenproblem      △2S = µ2S, in Ω, S(x) = 0, ∀x ∈ ∂Ω, △u(x) = 0, ∀x ∈ ∂Ω, (18) for S ∈ H1

0(Ω) ∩ H3(Ω). From the standard theory, we know that if S

satisfies

(18) , then S ∈ C∞(Ω).

The eigenvectors of

(18) form an orthogonal Hilbert basis

(Sn) ⊂ H1

0(Ω). We normalize Sn so that Sn H1

0(Ω) = 1, ∀n ∈ N. The

eigenvalues are all positive and µn = O(n).

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 36 / 49

Plates with arbitrary shapes Representation of the loading

Representation of the loading

The loading is of the form h(t, x) =

N

• j=1

gn(t) fn(x), fn(x) =

+∞

• m=1

bn,m Sm(x), where for each n ∈ {1, . . . , N}, bn,m µm

• m∈N

∈ ℓ2.

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 37 / 49

Plates with arbitrary shapes Representation of the solution

Representation of the solution

After some manipulation, the solution of the direct plate problem can be expressed as u(t, x) = t

N

• n=1

gn(t − τ) +∞

• m=1

−bn,m µm sen(µmτ)Sm(x)

• dτ.

(19)

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 38 / 49

Plates with arbitrary shapes Application of the data

Application of the data

Applying data ˜ ΓT,G+(x0)(u) = {(t, φ, u(t, ·) , φ) : t ∈ [0, T], φ ∈ C∞

c (VG+(x0))}

to the problem, we conclude that ∀˜ T > 0,

+∞

• m=1

bn,m µm sen(tµm) Sm , φ = 0, ∀t ∈ [0, ˜ T]. (20) Taking the time derivative of the above expression, we obtain that for any n ∈ {1, . . . , N},

+∞

• m=1

bn,m cos(tµm) Sm , φ = 0, ∀t ∈ [˜ T], ∀φ ∈ C∞

c (VG+(x0)).

(21)

Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 39 / 49

Plates with arbitrary shapes Comparison with the wave equation

Results for the wave equation

For plates with arbitrary shapes, we do not have an explicit formula for the eigenvalues. To circumvent this difficulty, we look for, surprisingly, a uniqueness result for the wave equation. Consider now the wave equation problem              ∂2˜ v ∂t2 − △˜ v = 0, in ]0, +∞) × Ω, ˜ v = 0, at t = 0,

∂˜ v ∂t = Q,

at t = 0, γ∂Ω (˜ v(t)) = 0, ∀t ≥ 0. (22)

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 40 / 49

Plates with arbitrary shapes Comparison with the wave equation

For Q ∈ H−1(Ω) expressed as Q =

+∞

• m=1

amSm, with (am/µm)m∈N ∈ ℓ2, the solution of

(22) is given by

˜ v(t, x) =

+∞

• m=1

am õm sen(õmt)Sm(x). (23) Formally, the only difference with the one that appears in

(20) is that in

the place of exponents µm,

(23) has õm instead.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 41 / 49

Plates with arbitrary shapes Comparison with the wave equation

The proof of the following result for the wave equation is found in [2]. Theorem 2 For Ω ⊂ R2 convex or with boundary of class C1,1, given x0 ∈ R2 \ Ω, and any T1 > supx∈Ω\G+(x0) |x − x0|, there exists C > 0 such that Q 2

H−1(Ω) < C

T1

• G+(x0)

| ˜ v(t, x) |2 dx dt, (24) where ˜ v and Q ∈ H−1(Ω) are related by equation

(22) . Theorem 2 means that the set ˜

ΓT1,G+(x0)(˜ v) is enough for the unique determination of Q ∈ H−1(Ω) in the WAVE problem

(22) .

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 42 / 49

Plates with arbitrary shapes Comparison with the wave equation

In other words,

Theorem 2 states that

+∞

• m=1

am √µm sen(√µmt) Sm , φ = 0, ∀t ∈ [0, T1], ∀φ ∈ C∞

c (VG+(x0))

implies (am)m∈N = {0}. Equivalently, noting that the above series converges absolutely, if we take the time derivative of the above distribution, we conclude that

+∞

• m=1

am cos(√µm t) Sm , φ = 0, ∀t ∈ [0, T1], ∀φ ∈ C∞

c (VG+(x0))

⇒ (am)m∈N = {0}. (25)

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 43 / 49

Plates with arbitrary shapes Comparison with the wave equation

Translated to the case of the Germain-Lagrange equation, our problem is to prove that

(25) leads to the implication

+∞

• m=1

bn,m cos(µm t) Sm , φ = 0, ∀t ∈ [0, T − tN], ∀φ ∈ C∞

c (VG+(x0))

⇒ (bn,m)m∈N = {0}, ∀n ∈ {1, . . . , N} . (26) Observe that the tricky part is the absence of the square root of µm in the expression appearing in

(26) .

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 44 / 49

Plates with arbitrary shapes Comparison with the wave equation

In what follows, we call Zτ the space of functions in S that have an extension to an entire function and whose Fourier Transforms are in C∞

c (] − τ, τ[). By the Paley–Wiener Theorem, Zτ is the space of entire

functions ϕ such that for all k ∈ Z+, there is a Ck > 0 such that | ϕ(z) | ≤ Ck (1 + | z |)k eτ| Imz |, ∀z ∈ Z.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 45 / 49

Plates with arbitrary shapes Comparison with the wave equation

Now we obtain from

(25) the implication:

+∞

• m=1

amϕ(√µm t) Sm , φ = 0, ∀ϕ ∈ ZT1, ∀φ ∈ C∞

c (VG+(x0))

⇒ (am)m∈N = {0}. (27) we must prove that

+∞

• m=1

bn,mϕ(µm t) Sm , φ = 0, ∀ϕ ∈ Z(T−tN), ∀φ ∈ C∞

c (VG+(x0)).

⇒ (bn,m)m∈N = {0}. (28) The implication in

(27) means that the set

A . =

• (µmϕ(√µm) Sm , φ)m∈N : ϕ ∈ ZT1, φ ∈ C∞

c (VG+(x0))

• (29)

is dense in ℓ2.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 46 / 49

Plates with arbitrary shapes Comparison with the wave equation

Important lemma

Lemma 3 Let T > 0 and (λi)i∈N ⊂ C. If ϕ ∈ ZT is an entire function such that ϕ|R is even, then z → ϕ(√z) is also an entire function and

• (ϕ(
• λi))i∈N : ϕ ∈ ZT
• ⊂
• ( ˜

ϕ(λi))i∈N : ˜ ϕ ∈ Z˜

T

• , ∀˜

T > 0. (30)

Lemma 3 and the fact that the set in (29) is dense in ℓ2 imply that the set

• (µmϕ(µm) Sm , φ)m∈N : ϕ ∈ ZT2, φ ∈ C∞

c (VG+(x0))

• is also dense in ℓ2, ∀T2 > 0.

This proves the implication in

(28) , (26) and Theorem 1 . Back

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 47 / 49

Plates with arbitrary shapes Comparison with the wave equation

Algumas referências I

Kawano, A. and Zine, A., Uniqueness and nonuniqueness results for a certain class of almost periodic distributions. SIAM Journal of Mathematical Analysis, n.1, V. 43, pp. 135–152, 2011. Yamamoto, Masahiro and Zhang, Xu. Global Uniqueness and Stability for an Inverse Wave Source Problem for Less Regular Data. Journal of Mathematical Analysis and Applications, n.2, V.263, pp. 479–500, 2001.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 48 / 49

Plates with arbitrary shapes Comparison with the wave equation

Algumas referências II

Stephen P . Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells. McGraw-Hill, “2nd” edition, 1970. Kawano, Alexandre. Uniqueness in the determination of vibration sources in rectangular Germain–Lagrange plates using displacement measurements over line segments with arbitrary small length. Inverse Problems, n. 8, V.29, pages 085002, 2013. Kawano, Alexandre. Uniqueness results in the identification of distributed sources over Germain-Lagrange plates by boundary measurements. Submitted, 2014.

• A. Kawano (Poli-USP)

Germain-Lagrange plates May 2014 49 / 49