# Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

Authors: Fagueye Ndiaye; Idrissa Ly
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

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## Abstract

In this paper, we are interested in the inverse problem of the determination of the unknown part∂Ω,Γ0 of the boundary of a uniformly Lipschitzian domain Ω included in ℝN from the measurement of the normal derivative ∂nv on suitable part Γ0 of its boundary, where v is the solution of the wave equation ∂ttvx,t−Δvx,t+pxvx=0 in Ω×0,T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂Ω. From necessary conditions, we estimate a Lagrange multiplier kΩ which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.


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## Body

## 1. Introduction and Main Result

The inverse problem in this paper means the problem of reconstructing object from observation data. We restrict ourselves to the case when the observation data are given as a boundary of the Cauchy data of a solution of a wave equation and the unknown object is a boundary. LetN∈ℕ,T>0 and let Ω⊂ℝN be a bounded domain with smooth boundary ∂Ω. Moreover, let us consider a partition of this boundary ∂Ω=Γ0∪Γ,Γ0∩Γ=∅, where Γ0 is the accessible regular part, for example, C2, and it satisfies the interior sphere condition (see [1]) and Γ=∂Ω/Γ0 is the unknown part of boundary. Throughout this paper, let us take the functional v=vx,t with x∈Ω,t∈0,T. We use the following notations:(1)∂nv=∂v∂n,∂ttvx,t=∂2v∂t2.We consider the following wave equation:(2)∂2v∂t2+pxv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ.First of all, assume thatp∈L∞Ω,v0∈H1Ω , and v1∈L2Ω are given and verified the compatibility condition v0x=0 for all x∈∂Ω. The Cauchy problem (2) is known to be well posed and one can also prove the solution v∈C0,T;H01Ω∩C10,T;L2Ω; this result can be found in [2].Our inverse problem consists of determiningΓ=∂Ω,Γ0, the unknown part of boundary from Cauchy data v0|∂Ω,h of a weak solution v of the following problem (3) with a given potential px.(3)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ,∂v∂nx,t=hxonΓ0×0,T,where h is a given function, and the corresponding Neumann data measured on Γ0 and n is outer normal vector unit. In this case, Ω and v are unknowns and we assume that the normal derivatives of function v can be measured by h.In Section1, we present the inverse problem which consists of finding a formula reconstructing the part of boundary Γ from the Cauchy data. The remainder of the paper is organized as follows.In Section2, we establish the shape optimization problem and prove the existence results. In Section 3, we study the derivation with respect to the domain and we prove the necessary conditions of optimality, that is, the existence of a Lagrange multiplier. Section 4 is devoted to auxiliary lemmas based on maximum principle theory for hyperbolic equations; see [3]. In Section 5, we give by a monotonicity result and under geometrical assumptions a uniqueness result of our inverse problem.The questions for the wave equation have all already received positive answers since the uniqueness result for the linear inverse problem has been proved by Klibanov (see [4]) and Lipschitz stability results (for both linear and nonlinear inverse problems) of Yamamoto (see [5]). Many results, to which we can refer concerning the wave equation, are related to the same type of inverse problem of determining a potential px. Some of them can be found in [6], for example. These references are all based upon local Carleman estimates for the wave operator (see [5]) or global Carleman estimates for Schrödinger equation (see [7]) to prove uniqueness and stability estimate solution. Nevertheless, in our approach, for a given potential, the reconstruction of Γ from the Cauchy data is one of our aims and the estimation of the Lagrange multiplier which appears by derivation with respect to the domain of the energy of system in an admissible set of domains is another interesting one.In [8], Isakov and Friedman studied the inverse spectral problems. This domain problem was formulated already by Sir A. Shuster who in 1882 introduced spectroscopy as a way to find a shape of a bell by means of the sounds which it is capable of sending out. More rigorously, it has been posed by Bochner in the 1950s and then in the well-known lecture of Kac (see [9]) “Can one hear the shape of a drum?” in 1966. He also studied inverse problem of gravimetry, inverse conductivity problem, tomography, and the inverse seismic problem and indicated their applications.In [10], using conformal mapping technique, Kress studied mathematical modelling of electrostatic or thermal imaging methods in nondestructive testing and evaluation. In these applications, an unknown inclusion within a conducting host medium with constant conductivity is assessed from overdetermined Cauchy data on the accessible exterior boundary of the medium.


## 2. Study of the Shape Optimization Problem

### 2.1. Auxiliary Results

We describe some fundamental properties which will be useful in the following. We consider a fixed and bounded domainD in ℝN which contains all open subsets we used.Definition 1.
LetK1 and K2 be two compact subsets of D. Let(4)dx,K1=miny∈K1dx,y,dx,K2=miny∈K2dx,y.
Note that(5)ρK1,K2=maxx∈K2dx,K1,ρK2,K1=minx∈K1dx,K2.
Let(6)dHK1,K2=maxρK1,K2,ρK2,K1,and we call Hausdorff distance of K1 and K2, the following positive number, denoted by dHK1,K2.
LetΩn be a sequence of open subsets of D and let Ω be an open subset of D. We say that the sequence Ωn converges on Ω in the Hausdorff sense and we denote it by Ωn⟶HΩ if limn⟶+∞dHD¯\Ωn,D¯\Ω=0.
LetΩn be a sequence of open sets of ℝN and let Ω be an open set of ℝN. We say that the sequence Ωn converges on Ω in the sense of Lp, 1≤p<∞ if χΩn converges on χΩ in LlocpℝN,χΩ being the characteristic functions of Ω.Remark 1.
LetKn be a sequence of compact sets included in a fixed and bounded set D of IRN; then there are a compact set K and nk such that Knk converges on K in the sense of Hausdorff.
We have the following lemmas.Lemma 1.
LetΩnn∈IN be a sequence of open set in IRN having the ϵ-cône property with Ω¯n⊂F⊂D, with F being a compact set. Then there exists an open set Ω including F which satisfies the ϵ/2-cône property and a subsequence Ωnk such that(7)χΩnk⟶L1χΩ,Ωnk⟶HΩ,∂Ωnk⟶H∂Ω,Ω¯nk⟶HΩ¯.
For detailed proof, see [11].Lemma 2.
LetΩnn∈ℕ be a sequence of open sets having the ϵ−cône property and converging to Ω in the sense of Hausdorff.
IfvΩn is the solution of problem 2 in Ωn for all n∈ℕ and vΩ is the solution of this problem in Ω, then vΩn converges to v=vΩ.
For detailed proof, see [11].Lemma 3.
LetΩ be a Lipschitz domain with v in W1,pℝℕ; if v=0a.e in Ωc, then v in W01,pΩ.
For detailed proof, see [11].
For allvx,t solution of(8)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ,at time t, the energy of v is defined by(9)Ev,t=12∫Ω∂v∂tx,t2dx+12∫Ω∇vx,t2dx,and it verifies(10)Ev,0−Ev,t=12∫Ωpvx,t2dx−12∫Ωpv0x2dx.
LetJ be the functional defined by(11)Jvw,w=12∫w∂vw∂tx,t2dx+12∫w∇vwx,t2dx+12∫wpvwx,t2dx,t∈0,T,where vwx,t is the solution:(12)∂2v∂t2+pv−Δv=0inw×0,T,vx,t=0on∂w×0,T,vx,0=v0xinw,∂∂tvx,0=v1xinw.
We study the existence of the result of the following optimization problem:infJw,w∈O, where the class of admissible domains is defined by(13)O=ω⊂ℝN,ωdomain uniformly Lipschizian included inD,∫ωdx=Vo,where D is a bounded domain of ℝN containing all Ω and V0 is positive real.Remark 2.
Remark thatω being uniformly Lipschitz means ω satisfies the ϵ-cone property; for details, see [11].


### 2.2. Existence of Solution of Shape Optimization Problem

We study the existence result of the following shape optimization problem.Proposition 1.
“Findω belonging to O such that Jω=minJw,w∈O has a solution.”Proof 1.
For the proof, we take the following:Jw:=Evw,t+1/2∫wpvwx,t2dxJw>0, which implies infJw,w∈O>−∞
Letα=infJw,w∈O. Therefore, there is a minimizing sequence Ωnn∈ℕ∈O such that JΩn converges to α.
The fact that sequenceΩnn∈ℕ∈O is bounded ensures the existence of a subsequence Ωnknk∈ℕ∈O and a domain Ω∈O such that Ωnknk converges to Ω in the sense of Hausdorff according to Lemma 1.
Therefore, we considervΩn,t=v˜n and p˜=p,x∈Ωn,0,x∈ℝn\Ωn,.
Sequencev˜nn∈N is bounded in C0,T;H1D∩C10,T;L2D. If not, JΩn converges to +∞, which is a contradiction.
SpaceH1D∩L2D is reflexive; then there is a subsequence v˜nkk∈N et v∗ such that vnk converges weakly to v∗ in C0,T;H01D∩C10,T;L2D and always according to Lemma 3.(14)∫Ω∂v∗∂tx,t2dx+pv∗x,t2dx+∇v∗2<liminf∫Ωnk∂v˜nk∂tx,t2dx+p˜v˜nkx,t2dx+∇v˜nk2,∀t∈0,T.
Thus, we obtainJΩ≤liminfJΩnk. Therefore, JΩ≤infJw,w∈O; then(15)JΩ=minJw,w∈O.Remark 3.
It is easy to verify thatv∗ equals vΩ according to Lemma 1 and satisfies problem (12). On the other hand, we have a regularity of vΩ solution to problem (12) (see [12, 13]).


## 2.1. Auxiliary Results

We describe some fundamental properties which will be useful in the following. We consider a fixed and bounded domainD in ℝN which contains all open subsets we used.Definition 1.
LetK1 and K2 be two compact subsets of D. Let(4)dx,K1=miny∈K1dx,y,dx,K2=miny∈K2dx,y.
Note that(5)ρK1,K2=maxx∈K2dx,K1,ρK2,K1=minx∈K1dx,K2.
Let(6)dHK1,K2=maxρK1,K2,ρK2,K1,and we call Hausdorff distance of K1 and K2, the following positive number, denoted by dHK1,K2.
LetΩn be a sequence of open subsets of D and let Ω be an open subset of D. We say that the sequence Ωn converges on Ω in the Hausdorff sense and we denote it by Ωn⟶HΩ if limn⟶+∞dHD¯\Ωn,D¯\Ω=0.
LetΩn be a sequence of open sets of ℝN and let Ω be an open set of ℝN. We say that the sequence Ωn converges on Ω in the sense of Lp, 1≤p<∞ if χΩn converges on χΩ in LlocpℝN,χΩ being the characteristic functions of Ω.Remark 1.
LetKn be a sequence of compact sets included in a fixed and bounded set D of IRN; then there are a compact set K and nk such that Knk converges on K in the sense of Hausdorff.
We have the following lemmas.Lemma 1.
LetΩnn∈IN be a sequence of open set in IRN having the ϵ-cône property with Ω¯n⊂F⊂D, with F being a compact set. Then there exists an open set Ω including F which satisfies the ϵ/2-cône property and a subsequence Ωnk such that(7)χΩnk⟶L1χΩ,Ωnk⟶HΩ,∂Ωnk⟶H∂Ω,Ω¯nk⟶HΩ¯.
For detailed proof, see [11].Lemma 2.
LetΩnn∈ℕ be a sequence of open sets having the ϵ−cône property and converging to Ω in the sense of Hausdorff.
IfvΩn is the solution of problem 2 in Ωn for all n∈ℕ and vΩ is the solution of this problem in Ω, then vΩn converges to v=vΩ.
For detailed proof, see [11].Lemma 3.
LetΩ be a Lipschitz domain with v in W1,pℝℕ; if v=0a.e in Ωc, then v in W01,pΩ.
For detailed proof, see [11].
For allvx,t solution of(8)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ,at time t, the energy of v is defined by(9)Ev,t=12∫Ω∂v∂tx,t2dx+12∫Ω∇vx,t2dx,and it verifies(10)Ev,0−Ev,t=12∫Ωpvx,t2dx−12∫Ωpv0x2dx.
LetJ be the functional defined by(11)Jvw,w=12∫w∂vw∂tx,t2dx+12∫w∇vwx,t2dx+12∫wpvwx,t2dx,t∈0,T,where vwx,t is the solution:(12)∂2v∂t2+pv−Δv=0inw×0,T,vx,t=0on∂w×0,T,vx,0=v0xinw,∂∂tvx,0=v1xinw.
We study the existence of the result of the following optimization problem:infJw,w∈O, where the class of admissible domains is defined by(13)O=ω⊂ℝN,ωdomain uniformly Lipschizian included inD,∫ωdx=Vo,where D is a bounded domain of ℝN containing all Ω and V0 is positive real.Remark 2.
Remark thatω being uniformly Lipschitz means ω satisfies the ϵ-cone property; for details, see [11].


## 2.2. Existence of Solution of Shape Optimization Problem

We study the existence result of the following shape optimization problem.Proposition 1.
“Findω belonging to O such that Jω=minJw,w∈O has a solution.”Proof 1.
For the proof, we take the following:Jw:=Evw,t+1/2∫wpvwx,t2dxJw>0, which implies infJw,w∈O>−∞
Letα=infJw,w∈O. Therefore, there is a minimizing sequence Ωnn∈ℕ∈O such that JΩn converges to α.
The fact that sequenceΩnn∈ℕ∈O is bounded ensures the existence of a subsequence Ωnknk∈ℕ∈O and a domain Ω∈O such that Ωnknk converges to Ω in the sense of Hausdorff according to Lemma 1.
Therefore, we considervΩn,t=v˜n and p˜=p,x∈Ωn,0,x∈ℝn\Ωn,.
Sequencev˜nn∈N is bounded in C0,T;H1D∩C10,T;L2D. If not, JΩn converges to +∞, which is a contradiction.
SpaceH1D∩L2D is reflexive; then there is a subsequence v˜nkk∈N et v∗ such that vnk converges weakly to v∗ in C0,T;H01D∩C10,T;L2D and always according to Lemma 3.(14)∫Ω∂v∗∂tx,t2dx+pv∗x,t2dx+∇v∗2<liminf∫Ωnk∂v˜nk∂tx,t2dx+p˜v˜nkx,t2dx+∇v˜nk2,∀t∈0,T.
Thus, we obtainJΩ≤liminfJΩnk. Therefore, JΩ≤infJw,w∈O; then(15)JΩ=minJw,w∈O.Remark 3.
It is easy to verify thatv∗ equals vΩ according to Lemma 1 and satisfies problem (12). On the other hand, we have a regularity of vΩ solution to problem (12) (see [12, 13]).


## 3. Derivation with respect to the Domain

These results would allow us to assume regularityC2 on Ω solution of the shape optimization problem to proceed with the derivation with respect to the domain and to show the result of monotony.LetE:O⟶X, with O being the set of domains having the ϵ−cône property and X being a normal vector space.Let us considerθ⟶ξθ=EI+θΩ, where θ varies around 0 in a normalized vector space Θ of applications from ℝN to ℝN. We can introduce the differentiability in the classic sense of Frechet for the application θ∈Θ⟶ξθ∈X. This is efficient in proving the regularity properties of the shape functional, in using the derivation calculations, and in clearly identifying the derivatives structures so-called “of shape”.We will use a numerical variable to be comfortable in the calculations.Let us choose functionϕ:y↦I+yθ with θ being a regular vector field from ℝN to ℝN; y⟶ϕy∈Θ. We analyse the derivative of ϕy and the expression of y∈0,Y⟶EΩy, where Ωy=ϕyΩ.Let us considervy=vΩy. The question is, how can we derive the function y↦vy∈H1Ωy where Ωy is a variable domain?(1)
We know thatvy is extended by 0 because vy∈H01Ωy(2)
We know that functionuy=vy°ϕy is always defined on the fixed domain Ω; it belongs to space H1ΩTo derive functionvy, it suffices to “transport” it by ϕy, because y↦uy has more regularity than y↦vy. Therefore, it is more strategic to study this problem.We fixΩ⊂ℝN as measurable. It is easy to verify that Ωy is measurable and that if Ω is open, then Ωy is also measurable.


### 3.1. Notations

LetW1,∞ℝN,ℝN denote the bounded space, and Lipschitzian applications from ℝN in itself provide with the norm(16)∀θ∈W1,∞,θ1,∞=supy,y^∈ℝN,y≠y^θy+θy−θy^y−y^,where ℝN is provided with the Euclidean norm ∥. Let I denote the identity of ℝN.We recall that this space is identified with the subspace ofL∞ℝN, whose partial derivatives in the sense of distributions are functions of L∞ℝN. In addition, the functions of W1,∞ are a.e. differentiable, and we have(17)∀θ∈W1,∞,θ1,∞=θ∞+esssupy∈ℝNDyθy,where the norms of differential are understood as linear operators of ℝN. However, the consideration of W1,∞ is interesting for deformations of the Lipschitzian domain.Ifθ1,∞<1, by the theorem of fixed point, 1+θ is inversible such that 1+θ−1∈W1,∞ and we have (see [11])(18)I+θ−1−I1,∞≤θ1,∞I−θ1,∞−1,I+θ−1−I+θ∞≤θ1,∞I−I+θ−1∞.Thus,θ∈W1,∞⟶I+θ−1∈W1,∞ is continuous in 0θ∈W1,∞⟶I+θ−1∈L∞ is differentiable in 0, and its differentiability is the opposite of identityLet us considerϕ:y∈0;Y⟶W1,∞ℝN derivable in 0 with(19)ϕ0=I,ϕ′0=V.Becauseϕy is close to the identity in W1,∞ℝN for y close to 0, it is inversible and even if it decreases Y according to 7.(20)t∈0;Y⟶ϕy−1∈W1,∞is continuous in0,t∈0;Y⟶ϕy−1∈L∞is derivable in0,with derivative−V.We write independentlyϕyx or ϕy,x (same for all other functions). Let Jy,x=detDxϕyx be the Jacobian of ϕy in x (which is, therefore, a.e. defined in x∈ℝN).For ally∈0;Y, we take fy,.∈L1Ωy and consider y∈0;Y⟶Iy=∫Ωyfy,xdx=∫Ωfy,ϕy,xJy,xdx.Subsequently, we note∫Ωyfy=∫Ωfy,ϕyJy.


### 3.2. Derivation Formula

To calculate derivation, we use the following theorem.Theorem 3.1.
Let us considerϕ verifying (19). We suppose that(21)y∈0;Y⟶fy∈L1ℝNis derivable in0,of derivativef′0,f0∈W1,1ℝN.
Then,t⟶Iy=∫Ωyfy is derivable in 0, and we have(22)I′0=∫Ωf′0+divf0V.
If, in addition,Ω is a Lipschitzian domain, then(23)I′0=∫Ωf′0+∫∂Ωf0n.V.
For detailed proof, see [11].


### 3.3. Optimality Conditions of the Problem

Again,y↦ϕy verifying (19) , Ω⊂ℝN is open bounded Lipschitzian domain, and Ωy=ϕyΩ. If y is close to 0, vy is a solution of the problem defined by its variational formulation (3). As in (11), we are interested in the following functional:(24)Jvy,Ωy=12∫Ωy∂vy∂tx,t2dx+12∫Ωy∇vyx,t2dx+12∫Ωpvx,t2dx.We set(i)
Ωy=Id+yθΩ with θ∈W1,∞ℝd,ℝd, a vector field with compact support and y sufficiently small such that Id+yθ defines a diffeomorphism(ii)
Iy=JΩy=JvΩy,ΩyWe look for the derivative ofJ with respect to domain Ω in direction θ, that is, I′0.As regards derivative ofJ, for a choice of θ, as mentioned above, we deform only the boundary part ∂Ω\Γ0 that we will denote by Γ.To calculate the derivativey↦Iy, it is useful to derive y↦vy in the appropriate direction vy∈H01Ωy.For the functional derivative, we have(25)I′0=<dJvΩ,Ω,θ>=∫Ω∂v∂t.∂v′∂tdx+∫Ω∇v.∇v′dx+∫Ωpv.v′dx,+12∫∂Ω∂v∂tx,t2θ.ndσ+12∫∂Ω∇vx,t2θ.ndσ+12∫∂Ωpvx,t2θ.ndσ.This gives, according to Hadamard and the boundary conditions,(26)I′0=<dJvΩ,Ω,θ>=−12∫∂Ω∂v∂n2θ.ndσ+12∫Ω∂2∂t2vv′dx+∫Ωv′∂2v∂t2dx+∫Ω∇v.∇v′dx+∫Ωpv.v′dx,+12∫∂Ω∂v∂tx,t2θ.ndx+12∫∂Ω∇vx,t2θ.ndx+12∫∂Ωpvx,t2θ.ndx.Therefore,(27)I′0=<dJvΩ,Ω,θ>=−∫Γ∇v2θ.n.We suppose that it is possible to estimate the normal derivative ofvΩx,t on Γ0; that is, there exists h:ℝN⟶ℝ−∗ such that x↦∂vx/∂n=hx on Γ0, where n is the exterior normal unit vector defined on Γ0. We have the following necessary conditions of optimality.Proposition 2.
IfΩ is the solution of the shape optimization problem minJw,w∈O, there exists a Lagrange multiplier kΩ<0 not depending on t such that ∇v=−∂vΩ/∂n=−kΩ1/2 on Γ.Proof 2.
AsJΩ=infJω,ω∈O, using the derivative with respect to domain, in the direction of vector field, we show that there exists kΩ Lagrange multiplier such that(28)dJΩ,θ=kΩdJ1Ω,θ,where J1Ω=∫Ωdx−V0. Note that we perturb Ω only on ∂Ω\Γ0=Γ,Γ0 is fixed.(29)dJΩ,θ=kΩdJ1Ω,θ=kΩ∫Γθ.n=−∫Γ∇v2θ.n⇔∇vΩ2=−kΩonΓ.
This gives us the following relationship according to (27):(30)kΩ=−∇v2.
For more details on the expression in (29), see [14].
Let us takevΩ=v. To estimate kΩ, it suffices to recognize Ω and if we suppose that ∂Ω\Γ0=Γ is of class C2, since v=0onΓ, then we have(31)∇v=−∂v∂n,onΓ,where n is the outer normal vector. Note that ∇v=−kΩ1/2 is an optimality condition and if we situate Γ, we will be able to estimate ∇v on Γ. Therefore, we deduce an approximation of Lagrange multiplier kΩ.


## 3.1. Notations

LetW1,∞ℝN,ℝN denote the bounded space, and Lipschitzian applications from ℝN in itself provide with the norm(16)∀θ∈W1,∞,θ1,∞=supy,y^∈ℝN,y≠y^θy+θy−θy^y−y^,where ℝN is provided with the Euclidean norm ∥. Let I denote the identity of ℝN.We recall that this space is identified with the subspace ofL∞ℝN, whose partial derivatives in the sense of distributions are functions of L∞ℝN. In addition, the functions of W1,∞ are a.e. differentiable, and we have(17)∀θ∈W1,∞,θ1,∞=θ∞+esssupy∈ℝNDyθy,where the norms of differential are understood as linear operators of ℝN. However, the consideration of W1,∞ is interesting for deformations of the Lipschitzian domain.Ifθ1,∞<1, by the theorem of fixed point, 1+θ is inversible such that 1+θ−1∈W1,∞ and we have (see [11])(18)I+θ−1−I1,∞≤θ1,∞I−θ1,∞−1,I+θ−1−I+θ∞≤θ1,∞I−I+θ−1∞.Thus,θ∈W1,∞⟶I+θ−1∈W1,∞ is continuous in 0θ∈W1,∞⟶I+θ−1∈L∞ is differentiable in 0, and its differentiability is the opposite of identityLet us considerϕ:y∈0;Y⟶W1,∞ℝN derivable in 0 with(19)ϕ0=I,ϕ′0=V.Becauseϕy is close to the identity in W1,∞ℝN for y close to 0, it is inversible and even if it decreases Y according to 7.(20)t∈0;Y⟶ϕy−1∈W1,∞is continuous in0,t∈0;Y⟶ϕy−1∈L∞is derivable in0,with derivative−V.We write independentlyϕyx or ϕy,x (same for all other functions). Let Jy,x=detDxϕyx be the Jacobian of ϕy in x (which is, therefore, a.e. defined in x∈ℝN).For ally∈0;Y, we take fy,.∈L1Ωy and consider y∈0;Y⟶Iy=∫Ωyfy,xdx=∫Ωfy,ϕy,xJy,xdx.Subsequently, we note∫Ωyfy=∫Ωfy,ϕyJy.


## 3.2. Derivation Formula

To calculate derivation, we use the following theorem.Theorem 3.1.
Let us considerϕ verifying (19). We suppose that(21)y∈0;Y⟶fy∈L1ℝNis derivable in0,of derivativef′0,f0∈W1,1ℝN.
Then,t⟶Iy=∫Ωyfy is derivable in 0, and we have(22)I′0=∫Ωf′0+divf0V.
If, in addition,Ω is a Lipschitzian domain, then(23)I′0=∫Ωf′0+∫∂Ωf0n.V.
For detailed proof, see [11].


## 3.3. Optimality Conditions of the Problem

Again,y↦ϕy verifying (19) , Ω⊂ℝN is open bounded Lipschitzian domain, and Ωy=ϕyΩ. If y is close to 0, vy is a solution of the problem defined by its variational formulation (3). As in (11), we are interested in the following functional:(24)Jvy,Ωy=12∫Ωy∂vy∂tx,t2dx+12∫Ωy∇vyx,t2dx+12∫Ωpvx,t2dx.We set(i)
Ωy=Id+yθΩ with θ∈W1,∞ℝd,ℝd, a vector field with compact support and y sufficiently small such that Id+yθ defines a diffeomorphism(ii)
Iy=JΩy=JvΩy,ΩyWe look for the derivative ofJ with respect to domain Ω in direction θ, that is, I′0.As regards derivative ofJ, for a choice of θ, as mentioned above, we deform only the boundary part ∂Ω\Γ0 that we will denote by Γ.To calculate the derivativey↦Iy, it is useful to derive y↦vy in the appropriate direction vy∈H01Ωy.For the functional derivative, we have(25)I′0=<dJvΩ,Ω,θ>=∫Ω∂v∂t.∂v′∂tdx+∫Ω∇v.∇v′dx+∫Ωpv.v′dx,+12∫∂Ω∂v∂tx,t2θ.ndσ+12∫∂Ω∇vx,t2θ.ndσ+12∫∂Ωpvx,t2θ.ndσ.This gives, according to Hadamard and the boundary conditions,(26)I′0=<dJvΩ,Ω,θ>=−12∫∂Ω∂v∂n2θ.ndσ+12∫Ω∂2∂t2vv′dx+∫Ωv′∂2v∂t2dx+∫Ω∇v.∇v′dx+∫Ωpv.v′dx,+12∫∂Ω∂v∂tx,t2θ.ndx+12∫∂Ω∇vx,t2θ.ndx+12∫∂Ωpvx,t2θ.ndx.Therefore,(27)I′0=<dJvΩ,Ω,θ>=−∫Γ∇v2θ.n.We suppose that it is possible to estimate the normal derivative ofvΩx,t on Γ0; that is, there exists h:ℝN⟶ℝ−∗ such that x↦∂vx/∂n=hx on Γ0, where n is the exterior normal unit vector defined on Γ0. We have the following necessary conditions of optimality.Proposition 2.
IfΩ is the solution of the shape optimization problem minJw,w∈O, there exists a Lagrange multiplier kΩ<0 not depending on t such that ∇v=−∂vΩ/∂n=−kΩ1/2 on Γ.Proof 2.
AsJΩ=infJω,ω∈O, using the derivative with respect to domain, in the direction of vector field, we show that there exists kΩ Lagrange multiplier such that(28)dJΩ,θ=kΩdJ1Ω,θ,where J1Ω=∫Ωdx−V0. Note that we perturb Ω only on ∂Ω\Γ0=Γ,Γ0 is fixed.(29)dJΩ,θ=kΩdJ1Ω,θ=kΩ∫Γθ.n=−∫Γ∇v2θ.n⇔∇vΩ2=−kΩonΓ.
This gives us the following relationship according to (27):(30)kΩ=−∇v2.
For more details on the expression in (29), see [14].
Let us takevΩ=v. To estimate kΩ, it suffices to recognize Ω and if we suppose that ∂Ω\Γ0=Γ is of class C2, since v=0onΓ, then we have(31)∇v=−∂v∂n,onΓ,where n is the outer normal vector. Note that ∇v=−kΩ1/2 is an optimality condition and if we situate Γ, we will be able to estimate ∇v on Γ. Therefore, we deduce an approximation of Lagrange multiplier kΩ.


## 4. Auxiliary Lemmas

In this section, we sum up some fundamental lemmas for the algorithm which we will present in the next sections. These lemmas are based only on maximum principle theory in wave equation in high dimension. We assume also thatui∈C2Ω∩CΩ¯\Γi,i=1,2.In sequel, we need some hypothesis for the operator and the initial value problem in order to apply maximum principle for hyperbolic problem to obtain additional information about functions which satisfy(32)ℋ∂2v∂t2+pv−Δv≥0inΩ×0,T,vx,t=0on∂Ω×0,T=Γ0×0,T,vx,0=v0x≤0inΩ,∂∂tvx,0=v1x≤0inΩ.If the hypothesisℋ holds, then the solution v satisfies v≤0,for allt>0; see [3], page 234.Lemma 4.
LetΩ,Ω′⊂IRN be two open sets such that Ω′⊂Ω and Γ0⊂∂Ω∩∂Ω′. We consider ℋ and(33)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T=Γ0×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ,∂2v′∂t2+pv′−Δv′=0inΩ′×0,T,v′x,t=0on∂Ω′×0,T=Γ0×0,T,v′x,0=v0′xinΩ′,∂∂tv′x,0=v1′xinΩ′,where(34)Γ¯0∪Γ¯=∂Ω,Γ0∩Γ=∅,Γ¯0∪Γ′¯=∂Ω′,Γ0∩Γ′=∅.
Then(35)∂v∂ns>∂v′∂nsfor alls∈Γ0.Proof 3.
Consideringv−v′, we have(36)∂2v−v′∂t2+pv−v′−Δv−v′=0inΩ′×0,T,v−v′x,t=0on∂Ω×0,T=Γ0×0,T,v−v′x,0=v0−v0′xinΩ′,∂∂tv−v′x,0=v1−v1′xinΩ′.
By maximum principle (see [3]), v≤0 in Ω′.
Then, using again the maximum principle (see [3]), v−v′x≤0 for all x∈Ω′. Let x0∈Γ0, and x=x0−nτ∈Ω where n is exterior normal on Γ0 and τ>0. We suppose that v−v′x<v−v′x0=0; in this case, by the Hopf lemma, we have(37)∂v∂nx0>∂v′∂nx0.Lemma 5.
Ifℋ is satisfied and Ω,vandΩ′,v′ are two solutions of the free boundary problem(38)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ,∂v∂nx,t=kΩonΓ0×0,T,such thatkΩ is the Lagrangian multiplierΩ′⊂Ω∂Ω′∩∂Ω\Γ0≠∅
ThenkΩ′<kΩ.Proof 4.
We have(39)∂2v−v′∂t2+pv−v′−Δv−v′=0inΩ′×0,T,v−v′x,t=0on∂Ω′∩∂Ω×0,T=Γ0×0,T,v−v′x,0=v0−v0′xinΩ′,∂∂tv−v′x,0=v1−v1′xinΩ′.
Thanks to maximum principle (see [3]), v≤0 on Ω′ and v−v′≤0 on Ω′. Let x0∈∂Ω′∩∂Ω; then v−v′x0=0 and x=x0−nτ∈Ω′ where n is exterior normal on Γ0 and τ>0. We suppose that v−v′x<v−v′x0=0; in this case, by the Hopf lemma, we have(40)∂v∂nx0>∂v′∂nx0this implies thatkΩ>kΩ′.


## 5. Uniqueness and Convergence Result

In this section, using results established in Section4, we show the uniqueness of the domain Ω under some hypothesis, by following the methods of I. Ly et al. (see [15]). Most of the time, in the inverse problems, it is a great challenge to get uniqueness results. Now we are not able to produce a uniqueness result in the case where the unknown domain Ω is supposed to be star-shaped with respect to a fixed point x0,∈Ω. We think that it would be interesting to investigate this question. Our uniqueness result is obtained for any domains belonging to Bh, a class of geometrical sets in the Beurling sense of admissible domains satisfying an inequality constraint on the accessible boundary Γ0 (see [11, 16]). Another important hypothesis for our uniqueness result is the inclusion property in the following sense: one assumes that there are two domains in class Bh and one of the two domains is included in the other. This is an interesting problem to weaken the inclusion property’s hypothesis. The uniqueness result is started in Proposition 3.Using results in Section4, we show, under geometrical assumptions, that the domain Ω is unique. We have already assumed that one is able to estimate the normal derivative of vΩonΓ0,i.e.,, there exists(41)h:IRN⟶IR−∗such that∂vΩ∂n=honΓ0,where n is the exterior unit normal vector field defined on Γ0.Let us define a class of geometrical sets in the Beurling sense (see, e.g., [15–17]).LetΩ be a bounded domain which is uniformly Lipschitz and let Γ0 be a subset of IRN−1C2 that is regular. Let us take(42)Bh=Ω⊂IRN,Γ0⊂∂Ω,∂vΩ∂n≤honΓ0,and vΩ is solution to the following problem:(43)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ.By construction,Bh is a nonempty set. In fact, this is because of Proposition 1 and measure assumption. We use set Bh in the proof of Proposition 3. At first, we get the following lemma.Lemma 6.
Ifℋ is satisfied, let Ω,Ω′∈Bh; then Ω∩Ω′∈Bh.Proof 5.
Letw be solution of(44)∂2w∂t2+pw−Δw=0inΩ∩Ω′×0,T,wx,t=0on∂Ω∩Ω′×0,T,wx,0=v0xinΩ∩Ω′,∂∂twx,0=v1xinΩ∩Ω′.
We have(45)∂2v−v′∂t2+pv−v′−Δv−v′=0inΩ∩Ω′×0,T,v−v′x,t=0on∂Ω×0,T=Γ0×0,T,v−v′x,0=v0−v0′xinΩ∩Ω′,∂∂tv−v′x,0=v1−v1′xinΩ∩Ω′.
By the maximum principle (see [3]), we show that(46)v≤0andv−w≤0.
Letx0∈Γ0, and we get(47)vx0−nτ−vx0τ≤v−wx0−nτ−v−wx0τ,passing to the limit we getlimτ⟶0vx0−nh−vx0τ≤limτ⟶0v−wx0−nτ−v−wx0τ,this implies that−∂v∂n≤−∂v−w∂n.
Then,(48)∂v−w∂nx0≤∂v∂nx0≤hfor allx0∈Γ0.
Then we getΩ∩Ω′∈Bh.Proposition 3.
LetΩ be a bounded domain and let Γ0,h,Γ be defined as in the Introduction. Consider the following Cauchy problem:(49)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ,∂v∂nx,t=hxonΓ0×0,T.
Assume also that there are two domainsΩandΩ′,such thatΩ′⊂Ω for which (49) is verified. Then, we have Ω=Ω′.Proof 6.
ConsiderΩ,Ω′ such that(50)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T=Γ0×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ,∂2v′∂t2+pv′−Δv′=0inΩ′×0,T,v′x,t=0on∂Ω′×0,T=Γ0×0,T,v′x,0=v0′xinΩ′,∂∂tv′x,0=v1′xinΩ′.
AsΩandΩ′∈Bh, we obtain Ω∩Ω′∈Bh.
SinceΩ′⊂Ω, we have Ω3=Ω∩Ω′=Ω′, and, by Lemma 6, we have ∂v′/∂n≤honΓ0.
Consider the following problem:(51)∂2v−v′∂t2+pv−v′−Δv−v′=0inΩ∩Ω′×0,T,v−v′x,t=0on∂Ω×0,T=Γ0×0,T,v−v′x,0=v0−v0′xinΩ∩Ω′,∂∂tv−v′x,0=v1−v1′xinΩ∩Ω′.
We have (see [3])(52)v≤0andv−v′≤0.
AsΓ0 satisfies the interior sphere condition, supposing that v−v′<0, by the Hopf lemma, we get(53)∂v∂n>∂v′∂nonΓ0i.ehx>hx∀x∈Γ0,which is false; then Ω=Ω′.


## 6. Stability of the Inverse Problem

In this section, we establish the stability of solutionΩ under some hypothesis too. The stability issue for this inverse problem seems important and difficult to be ignored. Recall the following lemma solving the Cauchy problem for the usual wave equation and giving energy and trace estimates of the solution.Lemma 7.
LetΩ be uniformly Lipschitzian included in ℝN, and a final time T>0.
v0,v1∈H01Ω×L2Ω is given. Let v∈C0,T;H01Ω∩C10,T;L2Ω be the unique weak solution of the wave equation(54)∂2v∂t2+pv−Δv=0inΩ×0,T,vx,t=0on∂Ω×0,T=Γ0×0,T,vx,0=v0xinΩ,∂∂tvx,0=v1xinΩ.
Then there exists a constantCΩ,T>0 (which depends only on Ω and T) such that, for all t∈0,T,(55)∫Ω∂v∂tx,t2dx+∫Ω∇vx,t2dx≤C∫Ωv1x2dx+∫Ω∇v0x2dx.
The normal derivative∂v/∂n belongs to L20,T;L2∂Ω and verifies(56)∂v∂nL20,T;L2∂Ω≤Cv0H01Ω+v1H01Ω.
This result is very classical; estimate (20) can be formally deduced from the multiplication of (19) by∂v/∂t and then by part integrations of this equality on 0,T×Ω. Concerning estimate (56), we refer to [18]. This is a hidden regularity result that can be proved using the method of multipliers.Proposition 4.
LetΩ be a bounded uniformly Lipschitz domain and let Γ0,Γ0′ be subsets of IRN−1C2 that is regular such that(57)Γ¯0∪Γ¯=∂Ω,Γ0∩Γ=∅,Γ0′¯∪Γ′¯=∂Ω,Γ0′∩Γ′=∅.
Assume that (49) is verified. Then(58)there isη>0such thatdH∂Ω\Γ,∂Ω\Γ′<η∂v∂n|Γ′−∂v∂n|ΓL20,T;L2∂Ω.
This inequality (58) describes the Lipschitz stability of the inverse problem.Proof.
From (56), ∂v/∂nL20,T;L2∂Ω is bounded; then there exists η>0 such that
∂v/∂nL20,T;L2∂Ω≥η. Then we have(59)∂v∂n|Γ′−∂v∂n|ΓL20,T;L2∂Ω≥|χΓ′∂v∂nL20,T;L2∂Ω−χΓ∂v∂nL20,T;L2∂Ω|,≥ηΓ′△Γ≥ηdH∂Ω\Γ′,∂Ω\Γ.


## 7. Numerical Simulations

In this part, we solve our inverse problem numerically using polar coordinates. Leth be a function as defined in the Introduction; we seek to determine Γ of class C2 under the constraints given. We choose function h defined by(60)hr,θ=∑n=0Ne−n+λntsinθcosnrsinθchnrcosθ+cosθsinnrsinθshnrcosθ,with N∈ℕ,λn∈ℝ+,chthe hyperbolic cosine function andshthe hyperbolic sine function. The integer n takes values from 0 to N and λn is a multiple for r depending on n. We consider that h is the normal derivative of a function u unique solution of problem 1 in a domain Ω of which a part Γ of the boundary is unknown. We seek to determine its optimal shape of Γ. With Matlab, by varying the parameters N,λn,θandt, we determine Γ of class C2 which we describe as a trajectory. Thus, we choose arbitrary values for r and λn and a maximum value for time t. By varying the angle θ and N, we obtain the geometrical optimal shapes of Γ in the below figures. For each given unit of time, we vary N and θ. Thus, for each fixed time, we obtain an optimal shape when N=300 and θ is equal to the value indicated in the legend.In Figure1 where the observation time is 5 seconds for a maximum angle equal to 300×π/28, the length of Γ is equal to 5.43 kilometers.Figure 1
T = 5, N = 300, and theta = pi/28.In Figure2 where the observation time is 8 seconds for a maximum angle equal to 300×π/18, the length of Γ is equal to 23.79 kilometers.Figure 2
T = 8, N = 300, and theta = pi/18.In Figure3 where the observation time is 10 seconds for a maximum angle equal to 10π, the length of Γ is equal to 195.15 kilometers.Figure 3
T = 10, N = 300, and theta = pi/30.We notice that if the observation time is small, then the optimal shape ofΓ looks like an arc of circle. But if it is larger, the optimal shape of Γ is a part of a hyperbola.


## 8. Conclusion

In this paper, we prove the existence result of the solution for our inverse problem by determining the optimal shape in Section2 and we prove the existence of a Lagrange multiplier, which appears in the optimality condition of the problem. By maximum principle for hyperbolic equations, we prove the uniqueness and the Lipschitz stability of the solution for our inverse problem. We make some numerical simulations with Matlab to illustrate the theoretical results and then identify the optimal shape of Γ . It would be interesting to study the problem with nonsmooth boundaries; topological optimization is the technique to use in future researches.



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*Source: 1716027-2021-10-18.xml*