Inverse Problem for a Curved Quantum Guide

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Volume 2012, Article ID 651390,12pages doi:10.1155/2012/651390

Research Article

Inverse Problem for a Curved Quantum Guide

Laure Cardoulis

1, 2

_{and Michel Cristofol}

3

1_{UT1 Ceremath, Université de Toulouse, 21 Allées de Brienne, 31042 Toulouse Cedex 9, France} 2_{Institut de Mathématiques de Toulouse, Université Paul Sabatier, UMR 5219,}

31062 Toulouse Cedex 9, France

3_{LATP, Universit´e d’Aix-Marseille, UMR 7353, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France}

Correspondence should be addressed to Laure Cardoulis,[email protected]

Received 28 March 2012; Accepted 22 June 2012

Academic Editor: Wolfgang Castell

Copyrightq2012 L. Cardoulis and M. Cristofol. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the Dirichlet Laplacian operator−Δon a curved quantum guide inRn _n_2,₃_with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method.

1. Introduction and Main Results in Dimension

n

2

The spectral properties of curved quantum guides have been studied intensively for several years, because of their applications in quantum mechanics electron motion. We can cite among several papers1–7.

However, inverse problems associated with curved quantum guides have not been

studied to our knowledge, except in8. Our aim is to establish uniqueness results for the

inverse problem of the reconstruction of the curvature of the quantum guide: the data of one eigenpair determines uniquely the curvature up to its sign and similar results are obtained by considering the knowledge of a solution of Poisson’s equation in the guide.

We consider the Laplacian operator on a nontrivially curved quantum guideΩ⊂ R2

which is not self-intersecting, with Dirichlet’s boundary conditions, denoted by −ΔΩ_D. We

proceed as in1. We denote byΓ Γ1,Γ2the functionC3-smoothsee7, Remark 5which

characterizes the reference curve and byN N1, N2the outgoing normal to the boundary

ofΩ. We denote bydthe fixed width ofΩand byΩ0 :R×−d/2, d/2. Each pointx, yof

Ωis described by the curvilinear coordinatess, uas follows:

f:Ω0−→Ω with

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We assumeΓ₁s2 Γ₂s21 and we recall that the signed curvatureγofΓis defined by

γs −Γ₁sΓ₂s Γ₂sΓ₁s, 1.2

named so because|γs|represents the curvature of the reference curve ats. We recall that a

guide is called simply bent ifγdoes not change sign inR. We assume throughout this paper

the following.

Assumption 1.1. One has the following.

ifis injective.

iiγ∈C2_R_∩_L∞_R_{, γ /}_≡₀_, _i.e.,_Ω_{is nontrivially curved}_.

iiid/2<1/ γ ∞, where γ ∞:sups∈R|γs| γ L∞R.

ivγs → 0 as|s| → ∞i.e.,Ωis asymptotically straight.

Note that by the inverse function theorem, the mapfdefined by 1.1 is a local

diﬀeomorphism provided 1−uγs/0, for allu, s, which is guaranteed by Assumption1.1

and sincefis assumed to be injective, the mapfis a global diﬀeomorphism. Note also that

1−uγs>0 for alluands. More precisely, 0<1−d/2 γ _∞≤1−uγs≤1 d/2 γ _∞for

allu, s. The curvilinear coordinatess, uare locally orthogonal, so by virtue of the

Frenet-Serret formulae, the metric inΩis expressed with respect to them through a diagonal metric

tensor,e.g.,4

gij

1−uγs2 0

0 1

. 1.3

The transition to the curvilinear coordinates represents an isometric map of L2_Ω _to

L2_Ω

0, g1/2ds duwhere

gs, u1/2:1−uγs 1.4

is the Jacobian∂x, y/∂s, u. So we can replace the Laplacian operator−ΔΩ_Dacting onL2Ω

by the Laplace-Beltrami operatorHgacting onL2Ω0, g1/2dsdurelative to the given metric

tensorgij see1.3and1.4where

Hg:−g−1/2∂s

g−1/2_∂

s

−g−1/2_∂

u

g1/2_∂

u

. 1.5

We rewriteHg defined by1.5into a Schr ¨odinger-type operator acting onL2Ω0, ds du.

Indeed, using the unitary transformation

Ug :L2Ω0, g1/2ds du−→L2Ω0, ds du

ψ−→g1/4ψ,

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setting

Hγ :UgHgU−g1, 1.7

we get

Hγ−∂s

cγs, u∂s

−∂2_uVγs, u, 1.8

with

cγs, u

1

1−uγs2, 1.9

Vγs, u − γ

2_s

41−uγs2 −

uγs

21−uγs3 −

5u2_γ2_s

41−uγs4. 1.10

We will assume throughout all this paper that the following assumption is satisfied.

Assumption 1.2. γ ∈ C2Randγk ∈ L∞Rfor eachk 0,1,2 where γk denotes thekth derivative ofγ.

Remarks 1. SinceΩis nontrivially curved and asymptotically straight, the operator−ΔΩ_Dhas at least one eigenvalue of finite multiplicity below its essential spectrumsee4,7; see also

1under the additional assumptions that the widthdis suﬃciently small and the curvature

γ is rapidly decaying at infinity; see3under the assumption that the curvature γ has a

compact support.

Furthermore, note that such operatorHγadmits bound states and that the minimum

eigenvalueλ1is simple and associated with a positive eigenfunctionφ1see9, Section 8.17.

Then, note that by 10, Theorem 7.1 any eigenfunction of Hγ is continuous and by 11,

Remark 25 page 182any eigenfunction ofHγbelongs toH2Ω0.

Finally, note also thatλ, φis an eigenpairi.e., an eigenfunction associated with its

eigenvalueof the operatorHγacting onL2_Ω_{0, ds du}_{means that}_{λ, U}−1

g φis an eigenpair

of−ΔΩ_Dacting onL2_Ω_{. So the data of one eigenfunction of the operator}_Hγ_{is equivalent to}

the data of one eigenfunction of−ΔΩ_D.

We first prove that the data of one eigenpair determines uniquely the curvature.

Theorem 1.3. LetΩbe the curved guide inR2_{defined as above. Let}_γ_{be the signed curvature defined}

by1.2and satisfying Assumptions1.1and1.2. LetHγ be the operator defined by1.8andλ, φ

be an eigenpair ofHγ.

Then

γ2s −4Δφs,0

φs,0 −4λ, 1.11

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Note that the conditionφs,0/0 in Theorem 1.3is satisfied for the positive

eigen-functionφ1and for alls∈R. Then, we prove later in the paper under the following

assump-tion.

Assumption 1.4. γ∈C5_R_and_γk_∈_L∞_R_{for each}_k₀_{, . . . ,}_{5, that one weak solution}_φ_of

the problem

Hγφf inΩ0

φ0 on∂Ω0, 1.12

wherefis a known given functionis in fact a classical solution and the data ofφdetermines

uniquely the curvatureγ.

by1.2and satisfying Assumptions1.1and1.4. LetHγ be the operator defined by1.8. Letf ∈

H3_Ω

0∩CΩ0and letφ∈H01Ω0be a weak solution of1.12.

Then we haveγ2_s ₋₄_Δ_φ_s,₀_/φ_s,₀₋₄_f_s,₀_/φ_s,₀_{for all}_s_when_φ_s,₀_/_0.

In the case of a simply bent guidei.e., whenγ does not change sign inR, we can

restrain the hypotheses upon the regularity ofγ. We obtain the following result.

by1.2and satisfying Assumptions1.1and1.2. We assume also thatγis a nonnegative function. Let

Hγbe the operator defined by1.8. Letf∈L2Ω0be a non null function and letφbe a weak solution

inH₀1Ω0of1.12. Assume that there exists a positive constantMsuch that|fs, u| ≤M|φs, u|

a.e. inΩ0. Thenf, φdetermines uniquely the curvatureγ.

Note that the above result is still valid for a nonpositive functionγ.

This paper is organized as follows. InSection 2, we prove Theorems1.3,1.5, and1.6.

In Sections3and4, we extend our results to the case of a curved quantum guide defined in

R3_.

2. Proofs of Theorems

1.3 ,

1.5 , and

1.6 2.1. Proof of

Theorem 1.3

Recall thatφis an eigenfunction ofHγ, belonging toH2_Ω

0. Sinceφis continuous andHγφ

λφ, thenHγφis continuous too. Thus, noticing thatcγs,0 1, we deduce the continuity of

the functions,0→Δφs,0and from1.8to1.10, we get

−Δφs,0−γ

2_s

4 φs,0 λφs,0, 2.1

and equivalently,

γ2s −4Δφs,0

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2.2. Proof of

Theorem 1.5

First, we recall from11, Remark 25 page 182the following lemma.

Lemma 2.1. For a second-order elliptic operator defined in a domainω⊂Rn_{, if}_φ_∈_H1

0ωsatisfies

ω

i,j

aij

∂φ ∂xi

∂ψ

∂xj _ωfψ ∀ψ∈H

1

0ω 2.3

then ifωis of classC2

f ∈L2_ω_{, a}

ij ∈C1ω, Dαaij ∈L∞ω∀i, j, ∀α,|α| ≤1

implyφ∈H2ω

2.4

and form≥1, ifωis of classCm2

f∈Hm_ω_{, a}

ij∈Cm1ω, Dαaij ∈L∞ω∀i, j, ∀α,|α| ≤m1

implyφ∈Hm2ω.

2.5

Now we can proveTheorem 1.5.

We haveHγφf, so

Ω0

cγ∂sφ∂sψ∂uφ∂uψ

Ω0

f−Vγφψ, ∀ψ∈H₀1Ω0 2.6

withcγdefined by1.9andVγ defined by1.10.

UsingAssumption 1.4, sinceγk ∈L∞Ω0fork 0,1,2 thenVγ ∈ L∞Ω0andf −

Vγφ ∈ L2Ω0. From the hypothesesγ ∈ C1Randγ ∈ L∞R, we get that cγ ∈ C1Ω0,

Dα_cγ _∈ _L∞_Ω₀_{for any} _α,_{|α| ≤} _{1, and so, using} _{Lemma 2.1}_for _2.6_{, we obtain that} _φ _∈

H2Ω0.

By the same way, we get thatf−Vγφ ∈H1Ω0, cγ ∈C2Ω0andDαcγ ∈L∞Ω0for

anyα,|α| ≤2fromγ∈C3_R_{, γ}k_∈_L∞_R_{for any}_k₀_{, . . . ,}₃_{. Using}_{Lemma 2.1}_{, we obtain}

thatφ∈H3Ω0.

We apply againLemma 2.1 to get thatφ ∈ H4_Ω

0 sincef −Vγφ ∈ H2Ω0, cγ ∈

C3_Ω

0, Dαcγ∈L∞Ω0for allα,|α| ≤3, from the hypothesesγ ∈C4Randγk∈L∞Rfor

k0, . . . ,4.

Finally, usingAssumption 1.4andLemma 2.1, we obtain thatφ∈H5_Ω

0.

Due to the regularity ofΩ0, we haveφ ∈ H5R2andΔφ ∈ H3R2. Since∇Δφ ∈

H2_R22 _and_H2_R2 _⊂ _L∞_R2_{, we can deduce that}_Δ_φ _{is continuous}_see₁₁_{, Remark 8}

page 154.

Therefore, we can conclude by using the continuity of the function

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Therefore, we get−Δφs,0−γ2_s_/₄_φ_s,₀ _f_s,₀_{and equivalently,}

γ2s −4Δφs,0

φs,0 −4

fs,0

φs,0 ifφs,0/0. 2.8

2.3. Proof of

Theorem 1.6

We prove here thatf, φdetermines uniquelyγwhenγis a nonnegative function.

For that, assume thatΩ1andΩ2are two quantum guides inR2with same widthd. We

denote byγ1andγ2the curvatures, respectively, associated withΩ1andΩ2, and we suppose

that eachγisatisfiesAssumption 1.2and is a nonnegative function. Assume thatHγ1φf

Hγ2φ.

Thenφsatisfies

−∂s

cγ1s, u−cγ2s, u

∂sφs, u

Vγ1s, u−Vγ2s, u

φs, u 0. 2.9

Assume thatγ1/≡γ2.

Step 1. First, we consider the case wherefor exampleγ1s< γ2sfor alls∈R.

Let >0, ω :R×I withI − ,0. Multiplying2.9byφand integrating over

ω , we get

ω

cγ1−cγ2

∂sφ2−

∂ω

cγ1−cγ2

∂sφφνs

ω

Vγ1−Vγ2

φ20. 2.10

Since 1,Vγis, u −γi2s/4 fori1,2, and soVγ1s, u−Vγ2s, u>0 inω .

Moreover, since

cγ1s, u−cγ2s, u

uγ1s−γ2s2−uγ1s γ2s

1−uγ1s21−uγ2s2 , 2.11

we havecγ1s, u> cγ2s, uinω .

Since

∂ω

cγ1−cγ2

∂sφ

φνs0 2.12

Thus, from2.10−2.12, we get

ω

cγ1−cγ2

∂sφ

2

ω

Vγ1−Vγ2

φ20, 2.13

withcγ1−cγ2>0 inω andVγ1−Vγ2>0 inω . We can deduce thatφ0 inω .

Using a unique continuation theorem see 12, Theorem XIII.63 page 240, from

Hγφ f, noting that−ΔU−1

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so by|f| ≤ M|φ|we have|ΔU−1

g φ| ≤ M|g−1/4φ|withg > 0 a.e., and we can deduce that

φ 0 inΩ0. So we get a contradictionsinceHγφ f andf is assumed to be a non null

function.

Step 2. FromStep 1, we obtain that there exists at least one points0 ∈ Rsuch thatγ1s0 γ2s0. Sinceγ1/≡γ2, we can choosea∈Randb∈R∪ {∞}such thatfor exampleγ1a

γ2a, γ1s< γ2sfor alls∈a, bandγ1b γ2bifb∈R.

We proceed as in Step 1, considering, in this case, ω : a, b×I . We study again

2.10and as inStep 1, we have

∂ω

cγ1−cγ2

∂sφφνs0. 2.14

Indeed from2.11andγ1a γ2awe havecγ1a, u cγ2a, uand so

0

−

cγ1a, u−cγ2a, u

∂sφa, uφa, udu0. 2.15

By the same way, ifb∈R, we also havecγ1b, u cγ2b, u. Thus,2.10becomes2.13with

cγ1−cγ2 > 0 inω andVγ1−Vγ2 > 0 inω . Soφ 0 inω and as inStep 1, by a unique

con-tinuation theorem, we obtain thatφ0 inΩ0. Therefore, we get a contradiction.

Note that the previous theorem is true if we replace the hypothesis “γis nonnegative”

by the hypothesis “γis nonpositive.” Indeed, in this last case, we just have to takeI 0,

and the proof rests valid.

3. Uniqueness Result for a

R

3

_{-Quantum Guide}

Now, we apply the same ideas for a tubeΩinR3_{. We proceed here as in}₇_{. Let}_s_→_Γ_s_,_Γ

Γ1,Γ2,Γ3, be a curve inR3. We assume thatΓ:R → R3is aC4-smooth curve satisfying the

following hypotheses

Assumption 3.1. Γpossesses a positively oriented Frenet frame{e1, e2, e3}with the following

properties

ie1 Γ,

iifor alli∈ {1,2,3}, ei∈C1R,R3,

iiifor alli∈ {1,2}, for alls∈R, e_islies in the span ofe1s, . . . , ei1s.

Recall that a suﬃcient condition to ensure the existence of the Frenet frame of

Assumption 3.1is to require that for alls∈Rthe vectorsΓs,Γsare linearly independent.

Then we define the moving frame{e1,e2,e3}alongΓ by following7. This moving

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Given aC5_{bounded open connected neighborhood}_ω_of₀_,₀_∈_R2_{, let}_Ω

0denote the

straight tubeR×ω. We define the curved tubeΩof cross-sectionωaboutΓby

Ω:fR×ω fΩ0,

fs, u2, u3: Γs

3

i2

ui

3

j2

Rijsejs Γs

3

i2

uieis,

3.1

withu u2, u3∈ωand

Rs:Rijs_{i, j}_∈{_2,3_}

cosθs −sinθs

sinθs cosθs

, 3.2

θbeing a real-valued diﬀerentiable function such thatθs τsthe torsion ofΓ. This

dif-ferential equation is a consequence of the definition of the moving Tang frame see 7,

Remark 3.

Note thatRis a rotation matrix inR2_{chosen in such a way that}_{s, u}

2, u3are

ortho-gonal “coordinates” inΩ. Letkbe the first curvature function ofΩ. Recall that sinceΩ⊂R3_,

kis a nonnegative function. We assume throughout all this section that the following

hypo-thesis holds:

k ∈ C2_R_∩_L∞_R_{, a} _: _sup

u∈ω u R2 < 1/ k ∞, ks → 0 as |s| → ∞Ωdoes not

overlap.

Assumption 3.2assures that the mapfdefined by3.1is a diﬀeomorphismsee7

in order to identifyΩwith the Riemannian manifoldΩ0,gijwheregijis the metric tensor

induced byf, that is,gij:tJf·Jf,Jfdenoting the Jacobian matrix off. Recall that

gij diagh2_,₁_,₁ _see₇_with

hs, u2, u3:1−kscosθsu2sinθsu3. 3.3

Note thatAssumption 3.2implies that 0<1−a k _∞≤1−hs, u2, u3≤1a k ∞for alls∈R

andu u2, u3∈ω. Moreover, setting

g:h2, 3.4

we can replace the Dirichlet Laplacian operator−ΔΩ_Dacting onL2_Ω_{by the Laplace-Beltrami}

operatorKg acting onL2Ω0, hdsdurelative to the metric tensor gij. We can rewriteKg

into a Schr ¨odinger-type operator acting on L2_Ω_{0, ds du}_{. Indeed, using the unitary}

trans-formation

Wg :L2Ω0, hds du−→L2Ω0, ds du

ψ −→g1/4ψ,

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setting

Hk:WgKgW_g−1, 3.6

we get

Hk−∂sh−2∂s−∂_u2₂−∂2_u₃Vk, 3.7

where∂sdenotes the derivative relative tosand∂uidenotes the derivative relative touiand

with

Vk:−k

2

4h2

∂2

sh

2h3 −

5∂sh2

4h4 . 3.8

We assume also throughout all this section that the following hypotheses hold:

ik∈L∞R, k ∈L∞R

iiθ∈C2_R_{, θ}_τ _∈_L∞_R_{, θ} _∈_L∞_R_.

Remarks 2. Note that as for the 2-dimensional case, such operatorHkdefined by3.3−3.8

admits bound states and that the minimum eigenvalue λ1 is simple and associated with a

positive eigenfunctionφ1 see 7, 9. Still note that λ, φ is an eigenpair of the operator

Hk acting on L2Ω0, ds du means thatλ, Wg−1φ is an eigenpair of−ΔΩD acting onL2Ω

withWgdefined by3.5. Finally, note that by10, Theorem 7.1any eigenfunction ofHkis

continuous and by11, Remark 25 page 182any eigenfunction ofHkbelongs toH2_Ω

0.

As for the 2-dimensional case, first we prove that the data of one eigenpair determines uniquely the curvature.

Theorem 3.4. LetΩbe the curved guide inR3_{defined as above. Let}_k_{be the first curvature function}

ofΩ. Assume that Assumptions3.1to3.3are satisfied. LetHkbe the operator defined by3.3−3.8

andλ, φbe an eigenpair ofHk.

Thenk2_s ₋₄_Δ_φ_s,₀_,₀_/φ_s,₀_,₀₋₄_λ_{for all}_s_when_φ_s,₀_,₀_/_0.

Then, One has the following.

ik∈C5_R_{, k}i_∈_L∞_R_{for all}_i₀_{, . . . ,}_5,

iiθ∈C5_R_{, θ}i_∈_L∞_R_{for all}_i₁_{, . . . ,}₅_,

whereki resp.,θi denotes the ith derivative ofk resp. ofθ, we obtain the following

result.

ofΩ. Assume that Assumptions3.1to3.5are satisfied. LetHkbe the operator defined by3.3−3.8.

Letf∈H3_Ω

0∩CΩ0and letφ∈H01Ω0be a weak solution ofHkφfinΩ0.

Thenφis a classical solution andk2_s ₋₄_Δ_φ_s,₀_,₀_/φ_s,₀_,₀₋₄_f_s,₀_,₀_/φ_s,₀_,₀

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Remarks 3. Recall that inR3_,_k_{is a nonnegative function and that the condition imposed on}_φ

φs,0,0/0in Theorems3.4and3.6is satisfied by the positive eigenfunctionφ1.

As for the two-dimensional case, we can restrain the hypotheses upon the regularity of the functionskandθ.

For a guide with a known torsion, we obtain the following result.

ofΩand letτ be the second curvature function (i.e., the torsion) ofΩ. Denote byθa primitive ofτ

and suppose that 0≤θs≤π/2 for alls∈R. Assume that Assumptions3.1to3.3are satisfied. Let

Hkbe the operator defined by3.3−3.8. Letf ∈L2Ω0be a non null function and letφ∈H01Ω0

be a weak solution of Hkφ f in Ω0. Assume that there exists a positive constant Msuch that |fs, u| ≤M|φs, u|a.e. inΩ0.

Then the dataf, φdetermines uniquely the first curvature functionkif the torsionτis given.

4. Proofs of Theorems

3.4 ,

3.6 , and

3.7 4.1. Proof of

Theorem 3.4

Recall thatφis an eigenfunction ofHk. Sinceφ is continuous,Hkφ λφandφ ∈ H2Ω0

thenHkφ is continuous. Therefore, foru u2, u3 0,0, we get:−Δφs,0,0−k2s/4 φs,0,0 λφs,0,0and equivalently,k2_s ₋₄_Δ_φ_s,₀_,₀_/φ_s,₀_,₀₋₄_λ_if_φ_s,₀_,₀_/_0.

4.2. Proof of

Theorem 3.6

We follow the proof ofTheorem 1.5. We haveHkφfwithφ∈H₀1Ω0. So

Ω0

h−2∂sφ

∂sψ

∂u2φ

∂u2ψ

∂u3φ

∂u3ψ

Ω0

f−Vkφ

ψ, ∀ψ∈H₀1Ω0,

4.1

withhdefined by3.3andVkdefined by3.8.

From Assumptions 3.2and3.3, sincek, k, k, θ, θ are bounded, we deduce that

Vk ∈ L∞Ω0. Therefore, f −Vkφ ∈ L2Ω0. Moreover, we have also h−2 ∈ C1Ω0 and

Dα_h−2 _∈ _L∞_Ω

0 for any α,|α| ≤ 1. Thus, using Lemma 2.1 for 4.1, we obtain that

φ∈H2_Ω

0.

By the same way, we get thatf−Vkφ∈H1Ω0, h−2 ∈C2Ω0andDαh−2∈L∞Ω0

for anyα,|α| ≤2sincek∈C3_R_{, θ}_∈_C3_R_{and all of their derivatives are bounded}_{. Using}

Lemma 2.1, we obtain thatφ∈H3_Ω

0.

We apply againLemma 2.1 to get thatφ ∈ H4_Ω

0 sincef −Vγφ ∈ H2Ω0, cγ ∈

C3_Ω

0, Dαcγ ∈L∞Ω0for allα,|α| ≤3, from the hypothesesγ ∈C4Randγk∈L∞Rfor

k0, . . . ,4.

Finally, usingAssumption 3.5andLemma 2.1, we obtain thatφ∈H5_Ω

0. Due to the

regularity ofΩ0 see11, Note page 169, we haveφ ∈ H5R3and Δφ ∈ H3R3. Since

∇Δφ ∈ H2_R33 _and_H2_R3 _⊂ _L∞_R3_{, we can deduce that}_Δ_φ_{is continuous} _see₁₁_,

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Thus, we conclude as inTheorem 1.5and foru u2, u3 0,0, we get−Δφs,0,0−

k2_s_/₄_φ_s,₀_,₀ _f_s,₀_,₀_{and equivalently,}_k2_s ₋₄_Δ_φ_s,₀_,₀_/φ_s,₀_,₀₋₄_f_s,₀_,₀_/

φs,0,0ifφs,0,0/0.

4.3. Proof of

Theorem 3.7

We prove here thatf, φ, θdetermines uniquelyk.

Assume thatΩ1 andΩ2 are two guides inR3. We denote by k1 andk2 the first

cur-vatures functions associated withΩ1andΩ2and we denote byθa primitive ofτthe common

torsion ofΩ1andΩ2. We suppose thatk1, k2andθsatisfy Assumptions3.2and3.3and that

0≤θs≤π/2 for alls∈R. Assume thatHk1φfHk2φ.

Thenφsatisfies

−∂sh−₁2s, u2, u3−h₂−2s, u2, u3∂sφs, u2, u3

Vk1s, u2, u3−Vk2s, u2, u3φs, u2, u3 0,

4.2

whereh1associated withk1is defined by3.3,Vk1is defined by3.8,h2associated with

k2is defined by3.3, andVk2is defined by3.8.

Assume thatk1/≡k2.

Step 1. First, we consider the case wherefor examplek1s< k2sfor alls∈R. Recall that

eachkiis a nonnegative function.

Let >0 and denote byJ : − , 0×− , 0, O :R×J with small enough to

haveJ ⊂ωrecall thatΩ0R×ω.

Multiplying4.2byφand integrating overO , we get

O

h−₁2−h−₂2∂sφ2

∂O

h−₁2−h−₂2∂sφφνs

O

Vk1−Vk2φ

2₀_. _4.3

Since 1, Vki −ki2s/4 fori1,2, and soVk1s, u2, u3−Vk2s, u2, u3>0 inO .

Moreover, note that

h−₁2s, u2, u3−h−₂2s, u2, u3

αs, u2, u3k1s−k2sh1s, u2, u3 h2s, u2, u3

h2₁s, u2, u3h2₂s, u2, u3 ,

4.4

withαs, u2, u3:cosθsu2sinθsu3.

Sinceu2, u3∈J and 0≤θs≤π/2 for alls∈R, we haveαs, u2, u3<0. Therefore, by4.4, we deduce thath−2

1 −h−22>0 inO .

Thus,_O h−₁2−h−₂2∂sφ2_OVk1−Vk2φ2≥0.

Note also that

∂O

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Therefore, from4.3and4.5we get

O

h−₁2−h−₂2∂sφ2

O

Vk1−Vk2φ

2₀_, _4.6

withh−₁2−h−₂2>0 inO andVk1−Vk2 >0 inO .

From4.6we can deduce thatφ0 inO . Using a unique continuation theoremsee

12, Theorem XIII.63 page 240, fromHk1φf, noting that−ΔW−

1

g φ Wg−1f g−1/4f, by

|f| ≤ M|φ|a.e. inΩ0, we can deduce thatφ 0 inΩ0. So we get a contradiction sincef is

assumed to be a non null function.

Step 2. FromStep 1, we obtain that there exists at least one points0 ∈ Rsuch thatk1s0 k2s0. Sincek1/≡k2, we can choosea∈Randb∈R∪ {∞}such thatfor examplek1a

k2a, k1s< k2sfor alls∈a, bandk1b k2bifb∈R. We proceed as inStep 1,

con-sidering in this case O : a, b×J . From k1a k2a, we get that h−₁2a, u2, u3

h−2

2 a, u2, u3. Therefore, we obtain

∂O h−12−h−22∂sφφνs 0. So4.3becomes4.6with h−₁2−h−₂2>0 inO andVk1−Vk2 >0 inO . Soφ0 inO and as inStep 1, by a unique

contin-uation theorem, we obtain thatφ0 inΩ0. Therefore, we get a contradiction.

References

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