# On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

Authors: Jerico B. Bacani; Julius Fergy T. Rabago
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.


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## Body

## 1. Introduction

Shape optimization is a key research topic with many applications in various fields of pure and applied sciences, especially in biomechanics and engineering (cf. [1, 2] for applications in structural mechanics, [3] for some applications in fluid mechanics or aerodynamics, and [4] for other applications). A typical problem in this line of research is to find a domain, for instance, Ω, in a set of admissible domains A such that an objective functional J achieves a minimum (or maximum) on it [3]. For instance, suppose, among all three-dimensional shapes of given volume, that we wish to find the one which has a minimal surface area. In this particular case, the problem can be described mathematically as finding the minimum of JΩ=Area(∂Ω) with the constraint E(Ω)=Volume(Ω)-constant. Obviously, the answer to this question would be the sphere. In general and in most cases of greater interest, shape optimization problems can be described mathematically as(1)minu,ΩJu,Ωs.t.Eu,Ω=0,Ω∈A,where the state u is the solution to a partial differential equation (PDE) E on the domain Ω. For an extensive introduction to shape optimization problems, we refer to the book of Delfour and Zolésio [5] (see also [6]).Recently, there has been an increasing interest in the applications of shape optimization in the study of Bernoulli problems. Abda et al. [7] rephrased the Bernoulli problem into a shape optimization problem and explicitly determined the shape derivative of the cost functional being studied. In [8], a framework for calculating theshape Hessian for the domain optimization problem with a PDE as the constraint was presented. In [9], a similar approach as in [8] was applied in solving a shape optimization problem.Another way to approach the solutions of shape optimization problems is through iterative methods. For the past few decades, several numerical methods have been developed to solve the two-dimensional Bernoulli problem (see, e.g., [10–13]). These strategies were also developed based on reformulating the Bernoulli problem as a shape optimization problem. This reformulation can be achieved in several ways. For instance, for a given domain, one can choose one of the boundary conditions on the free boundary to obtain a well-posed state equation. The domain is determined by the requirement that the other condition on the free boundary is satisfied in a least square sense (see [13–15]).Many authors have also studied the second variation of a cost functional for linear PDEs. Building on the shape optimization setting that is based on theperturbation of the identity method introduced by Murat and Simon (cf. [16, 17]), Fujii [18] used a second-order perturbation of the identity along the normal of the boundary for second-order elliptic problems in 1986. Simon [19] computed the second variation via the first-order perturbation of the identity in 1988. A general approach via the velocity method (Figure 2) was systematically characterized by Delfour and Zolésio [20, 21], and they computed the shape Hessian for a simple Neumann problem in [20] and a nonhomogeneous Dirichlet problem in [21].However, a standard approach in dealing with the solution to (1) requires some information on gradients. So shape derivatives are essential in understanding the problem.The recent paper focuses on the exterior Bernoulli free boundary problem (FBP). As far as the authors are concerned, the same functional was first studied by Eppler and Harbrecht and published in [22] wherein the first-order shape derivative, or equivalently the shape gradient, was derived for arbitrary variations in terms of the perturbation of the identity. Moreover, the second-order shape derivative, or equivalently the shape Hessian, has been computed and analyzed for the special cases of star-like domains. As a main result, by analyzing the shape Hessian at the optimal domain, Eppler and Harbrecht found out that the optimization problem is algebraically ill posed. In the present paper, the same functional is studied again but we focus on the application of velocity method in dealing with shape optimization problem. It would be a challenging research in the near future to study the ill-posedness of the shape optimization problem for general domains, as well as the comparison of the shape Hessians in this paper from [22] for the former uses Cartesian coordinates, while the latter used spherical/polar coordinates. The nice thing in the present paper is that the results attest to classical results in shape optimization problems.Now, the exterior Bernoulli FBP is formulated as follows.Given a bounded and connected domainA⊂R2 with a fixed boundary ∂A≔Γ, we need to find a bounded connected domain B with a free boundary Σ that contains the closure of A, A¯, and an associated real-valued (state) function u defined on Ω (where Ω is the annulus formed by B and A; refer to Figure 1) such that both unknowns u and Ω satisfy the following boundary value problem:BP-Δu=0inΩ,u=0onΣ,u=1onΓ,∂u∂n=λonΣ,where λ<0.Figure 1
The exterior Bernoulli free boundary problem.Figure 2
Transport ofΩ by velocity method.In recent papers, Bacani and Peichl employed shape optimization methods to study the exterior Bernoulli FBP by reformulating it into Kohn-Vogelius-type cost functionalJKV, which is defined asKVJΩ≔JKV=12∫Ω∇uD-uN2dx,and minimizing this functional over a class of admissible domains, where the two state functions uD and uN solve a homogenous Dirichlet D and nonhomogenous Neumann N problem, respectively:D-ΔuD=0inΩ,uD=1onΓ,uD=0onΣ,andN-ΔuN=0inΩ,uN=1onΓ,∂uN∂n=λonΣ.Bacani and Peichl presented two strategies in computing the first-order shape derivative of the Kohn-Vogelius objective functional. One is by using the Hölder continuity of the two state variables involved [9], and the other one is by using the shape derivatives of states [23]. The authors also computed its second-order shape derivative for general domains via the boundary differentiation scheme and via Tiihonen’s approach [24]. The computation is found in [25].In this recent paper, we are going to solve the shape optimization problem using velocity method, wherein we consider nonautonomous velocity fields. The study is important since it confirms a classical result of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and the perturbation of identity technique (cf. [26]).In the next section (Section2), we present an overview of some concepts necessary for the understanding of the present study. The section also includes some important results of methods of shape optimization which will be useful in our investigation. We formally present our main contribution which pertains to the form of the second-order shape derivative of the Kohn-Vogelius objective functional in Section 3. Finally, we end our paper by summarizing our results in Section 4.


## 2. Preliminaries

In this section we provide some important (but not exhaustive) background of some shape optimization techniques. We give an overview of these concepts to understand the present study.


### 2.1. Perturbation of the Identity

LetΩ and the universal or hold-all domain U be smooth subdomains of R2, such that Ω¯⊆U. A class of perturbations Ωt of the domain Ω obtained from the perturbation of identity operator Tt is defined as (2)Tt=I+tV:U¯⟶R2,Ttx=x+tVx,x∈U¯,where the deformation field V is in Θ1 defined as(3)Θ1=V∈C1,1U¯,R2:VΓ∪∂U=0.Then, for sufficiently small t, (i) Tt:U¯→U¯ is a homeomorphism, (ii) Tt:U→U is a C1,1 diffeomorphism and, in particular, Tt:Ω→Ωt is a C1,1 diffeomorphism, (iii) Γt=Tt(Γ)=Γ, and ∂Ωt=Γ∪Σt, and (iv) ∂Ωt=Γ∪Σt (cf. [9, Theorem 7]). For convenience, we will use the following notations throughout the discussion:(4)Itx=det⁡DTtx,Mtx=DTtx-T,Atx=ItxMtTxMtx,x∈U¯,wtx=ItxDTtx-Tnx,x∈Σ. The next lemma provides some properties of the transformation Tt which are useful in accomplishing our main objective.Lemma 1 (see [13, 27]).
Consider a fixed vector fieldV∈Θ1 and the transformation Tt (the perturbation of identity operator). Then, we can find a constant tV>0 such that the functions defined above restricted to the interval IV=(-tV,tV) have the following regularities and properties: (1)
t↦Tt∈C1(IV,C1(U¯)),(2)
t↦It∈C1(IV,C(Ω¯)),(3)
t↦wt∈C1(IV,C(Σ¯)),(4)
t↦Tt-1∈C(IV,C1(U¯)),(5)
t↦At∈C1(IV,C1(Ω¯)),(6)
It=1+tdivV+t2det⁡DV,(7)
there are positive constantsα1, α2, and β such that 0<α1≤It(x)≤α2 and At(x)≥βI for x∈Ω,(8)
(d/dt)DTtt=0=-(d/dt)DTt-1t=0=DV,(9)
(d/dt)Ttt=0=V,(10)
(d/dt)Itt=0=divV,(11)
limt↓0wt=1,(12)
(d/dt)Att=0=A, where A=(divV)I-(DV+(DV)T),(13)
(d/dt)wtt=0=divΣV, where the surface divergence divΣ is defined by(5)divΣV=divVΣ-DVn·n.


### 2.2. The Velocity (Speed) Method (See [5, 20, 21])

In this paper, we are interested in solving the second-order shape derivative of the Kohn-Vogelius objective functional via the velocity method with nonautonomous velocity fields. Detailed discussions about this method can be seen in [5, 20, 21]. We present some of the details here. As our primary interest is focused on nonautonomous velocity fields, we will use the notation V^ to denote time-dependent velocity fields in contrast to velocity fields denoted by V which are not time-dependent.LetV^:[0,tV]×R2→R2 be a given velocity field for some fixed tV>0. The map V^ can be viewed as a family {V^(t)} ofnonautonomous velocity fields on R2 defined by (6)x⟼V^tx≔V^t,x:R2⟼R2.Let V^ satisfy (V1)∀x∈R2,V^(·,x)∈C([0,tV);R2),(V2)∃c>0, ∀x,y∈R2, V^(·,y)-V^(·,x)C([0,tV);R2)≤cy-x,where V^(·,x) denotes the function t↦V^(t,x). Associate with V^ the solution x(t;X) of the ODE:(7)dxdtt=V^t,xt,t∈0,τ,x0=X∈R2.That is, we suppose thatV^ is continuous in t and at the same time Lipschitz in spatial variables. We remark that, in the case ofautonomous velocity fields, the condition to be satisfied can be simplified as(V2′)∃c>0,   ∀x,y∈R2, V(x)-V(y)≤cx-y.Remark 2.
The above statement can be described equivalently as follows: LetV^ belong to Ek=C([0,tV);Ck(R2,R2)) for some integer k≥2 and a small real number tV>0. Let Ω⊂R2 be a smooth bounded domain with boundary Γ which is at least twice differentiable. The field (8)V^tx=V^t,x,x∈R2is an element of Ck(R2,R2) which may depend on t≥0. It generates the transformations (9)TtV^X≔TtX=xt;X,t≥0,X∈R2through the differential equation(10)dxdtt;X=V^t,xt;X,x0;X=X.As already described in the previous section, we denote the transformed domainTt(Ω) at t≥0 by Ωt.Now, the following theorem describes the relation between the perturbation of the identity method and the velocity method. The theorem basically tells us that we can start from either a family of velocity fields{V^(t)} on RN or a family of transformations {Tt} of RN provided that the map V^,V^(t,x)=V^(t)(x), verifies (V1) and (V2) or the map T,T(t,X)=Tt(X), verifies assumptions (T1), (T2) and (T3) given below.Theorem 3 (see [21]).
(i) Under assumptions (V1) and (V2) on the mapV^, the maps Tt defined previously have the following properties: (T1)
∀X∈RN,T(·,X)∈C1([0,tV];RN),∃c>0,∀X,Y∈RN,T(·,Y)-T(·,X)C1([0,tV];RN)≤c|Y-X|,(T2)
∀t∈[0,tV],X↦Tt(X)=T(t,X):RN→RN is bijective,(T3)
∀x∈RN,T-1(·,x)∈C([0,tV);R2),∃c>0,∀x,y∈RN,T-1(·,y)-T-1(·,x)C([0,tV);R2)≤c|y-x|.
(ii) If there exists a real numbertV>0 and a map T:[0,tV]×RN→RN verifying assumptions (T1), (T2), and (T3), then the map(11)t,x⟼V^t,x=∂T∂tt,T-1x:0,tV×RN⟶RNverifies assumptions (V1) and (V2), where Tt-1 is the inverse of X↦Tt(X).In the above discussion, we see that the solution to the differential equation (10), V^=V∈Θ1, is the perturbation of the identity operator Tt. Conversely, if Tt is the perturbation of the identity operator, then (10) is satisfied. Hence, we consider the special case, where x(t)=I+tV, and determine the relationship of the autonomous field V∈Θ1 and the nonautonomous V^ defined in (10). Differentiating x(t)=I+tV with respect to t, we get(12)dxdtt;X=VX=V^t,xt;X,X∈Ω.This simply implies that V(X)=V^(t,(I+tV)(X)). Now, replacing X by (I+tV)-1(X), we get V(Tt-1(X))=V^(t,X) or, equivalently, V^(t)=V∘Tt-1. This is why many results obtained by the perturbation of identity technique can be acquired as well through the velocity method using nonautonomous velocity fields. As an immediate consequence, we note that V^(0)=V. The relation V^(t)=V∘Tt-1 also implies that(13)∂V^∂tt=DVTt-1∂Tt-1∂t.So, in particular, we have(14)∂V^0∂t≔V^˙0=-DVV.


### 2.3. Domain and Boundary Transformation

We recall the following theorems on domain and boundary transformations.Lemma 4 (see [28]).
We have the following important transformations which will also be central to our investigation:(1)
Letφt∈L1(Ωt). Then φt∘Tt∈L1(Ω) and(15)∫Ωtφtdxt=∫Ωφt∘TtItdx.(2)
Letφt∈L1(Σt). Then φt∘Tt∈L1(Σ) and(16)∫Σtφtdst=∫Σφt∘Ttwtds.


### 2.4. Material Derivatives

The material and shape derivatives of the state variables are defined as follows (see [4, 24]).Letu be defined in [0,tV]×U. An element u˙∈Hk(Ω), called thematerial derivative of u, is defined as (17)u˙≔u˙0,x≔limt↓0⁡ut,Ttx-u0,xt=ddtut,x+tVxt=0if the limit exists in Hk(Ω).Remark 5.
As pointed out in [9], the material derivative of the state function u can be written as(18)u˙x=limt↓0⁡ut∘Ttx-uxt=ddtut∘Ttxt=0 and it, in fact, characterizes the behavior of the function u at x∈Ω⊂U in the direction V(x).Now, on the other hand, an elementu′∈Hk(Ω) is called theshape derivative of u at Ω at the direction of V, if the following limit exists in Hk(Ω): (19)u′≔u′0,x≔limt↓0⁡ut,x-u0,xt≔∂∂tutxt=0.Remark 6.
We note that ifu˙ and ∇u·V exist in Hk(Ω), then the shape derivative can be expressed as(20)u′x=u˙x-∇u·Vx.In general, if u˙(x) and ∇u·V(x) both exist in Wm,p(Ω), then so does u′(x).


### 2.5. The First- and Second-Order Eulerian Shape Derivatives

We first recall the definition ofdirectional Eulerian shape derivative or simplyshape derivative of a shape functional.Suppose that the shape functionalJ:Ω→R is well-defined. Given the deformation field V^, thedirectional Eulerian shape derivative of J at Ω in the direction of deformation field V^ is defined as(21)dJΩ;V^≔limt↓0⁡JΩt-JΩtif the limit exists. The objective functional J is shape differentiable at Ω provided that dJ(Ω;V^) exists for all V^ and if dJ(Ω;V^) is linear and continuous with respect to V^. On the other hand, thesecond-order Eulerian shape derivative of a well-defined shape functional J at Ω in the direction of the deformation fields V^ and W^ is defined to be(22)d2JΩ;V^,W^≔lims↓0⁡dJΩsW^;V^-dJΩ;V^s=ddsdJΩs;V^s=0if the limit exists. The functional is said to be twice shape differentiable if, for all V^ and W^, d2J(Ω;V^;W^) exists and if d2J(Ω;V^;W^) is bilinear and continuous with respect to V^ and W^. Following these definitions, the second-order shape derivative of the functional being studied can also be computed as follows:(23)d2JΩ;V^;W^=∂∂s∂∂tdJΩt,st=0s=0.In this case, the transported domainΩt,s which is a result of two deformation fields V^ and W^ is illustrated in Figure 3.Figure 3
Transport ofΩ after two deformations.Remark 7.
Under appropriate hypothesis on the mapV^↦dJ(Ω;V^(0)), one can show that dJ(Ω;V^)=dJ(Ω;V^(0)) (cf. [21]). Therefore, if V^ is associated with the deformation field V in the perturbation of the identity, then dJ(Ω;V^(0)) coincides with dJ(Ω;V^) by V^(0)=V. Hence, the first-order shape derivative of a cost functional obtained via the velocity method coincides with the one obtained from the perturbation of the identity technique.


## 2.1. Perturbation of the Identity

LetΩ and the universal or hold-all domain U be smooth subdomains of R2, such that Ω¯⊆U. A class of perturbations Ωt of the domain Ω obtained from the perturbation of identity operator Tt is defined as (2)Tt=I+tV:U¯⟶R2,Ttx=x+tVx,x∈U¯,where the deformation field V is in Θ1 defined as(3)Θ1=V∈C1,1U¯,R2:VΓ∪∂U=0.Then, for sufficiently small t, (i) Tt:U¯→U¯ is a homeomorphism, (ii) Tt:U→U is a C1,1 diffeomorphism and, in particular, Tt:Ω→Ωt is a C1,1 diffeomorphism, (iii) Γt=Tt(Γ)=Γ, and ∂Ωt=Γ∪Σt, and (iv) ∂Ωt=Γ∪Σt (cf. [9, Theorem 7]). For convenience, we will use the following notations throughout the discussion:(4)Itx=det⁡DTtx,Mtx=DTtx-T,Atx=ItxMtTxMtx,x∈U¯,wtx=ItxDTtx-Tnx,x∈Σ. The next lemma provides some properties of the transformation Tt which are useful in accomplishing our main objective.Lemma 1 (see [13, 27]).
Consider a fixed vector fieldV∈Θ1 and the transformation Tt (the perturbation of identity operator). Then, we can find a constant tV>0 such that the functions defined above restricted to the interval IV=(-tV,tV) have the following regularities and properties: (1)
t↦Tt∈C1(IV,C1(U¯)),(2)
t↦It∈C1(IV,C(Ω¯)),(3)
t↦wt∈C1(IV,C(Σ¯)),(4)
t↦Tt-1∈C(IV,C1(U¯)),(5)
t↦At∈C1(IV,C1(Ω¯)),(6)
It=1+tdivV+t2det⁡DV,(7)
there are positive constantsα1, α2, and β such that 0<α1≤It(x)≤α2 and At(x)≥βI for x∈Ω,(8)
(d/dt)DTtt=0=-(d/dt)DTt-1t=0=DV,(9)
(d/dt)Ttt=0=V,(10)
(d/dt)Itt=0=divV,(11)
limt↓0wt=1,(12)
(d/dt)Att=0=A, where A=(divV)I-(DV+(DV)T),(13)
(d/dt)wtt=0=divΣV, where the surface divergence divΣ is defined by(5)divΣV=divVΣ-DVn·n.


## 2.2. The Velocity (Speed) Method (See [5, 20, 21])

In this paper, we are interested in solving the second-order shape derivative of the Kohn-Vogelius objective functional via the velocity method with nonautonomous velocity fields. Detailed discussions about this method can be seen in [5, 20, 21]. We present some of the details here. As our primary interest is focused on nonautonomous velocity fields, we will use the notation V^ to denote time-dependent velocity fields in contrast to velocity fields denoted by V which are not time-dependent.LetV^:[0,tV]×R2→R2 be a given velocity field for some fixed tV>0. The map V^ can be viewed as a family {V^(t)} ofnonautonomous velocity fields on R2 defined by (6)x⟼V^tx≔V^t,x:R2⟼R2.Let V^ satisfy (V1)∀x∈R2,V^(·,x)∈C([0,tV);R2),(V2)∃c>0, ∀x,y∈R2, V^(·,y)-V^(·,x)C([0,tV);R2)≤cy-x,where V^(·,x) denotes the function t↦V^(t,x). Associate with V^ the solution x(t;X) of the ODE:(7)dxdtt=V^t,xt,t∈0,τ,x0=X∈R2.That is, we suppose thatV^ is continuous in t and at the same time Lipschitz in spatial variables. We remark that, in the case ofautonomous velocity fields, the condition to be satisfied can be simplified as(V2′)∃c>0,   ∀x,y∈R2, V(x)-V(y)≤cx-y.Remark 2.
The above statement can be described equivalently as follows: LetV^ belong to Ek=C([0,tV);Ck(R2,R2)) for some integer k≥2 and a small real number tV>0. Let Ω⊂R2 be a smooth bounded domain with boundary Γ which is at least twice differentiable. The field (8)V^tx=V^t,x,x∈R2is an element of Ck(R2,R2) which may depend on t≥0. It generates the transformations (9)TtV^X≔TtX=xt;X,t≥0,X∈R2through the differential equation(10)dxdtt;X=V^t,xt;X,x0;X=X.As already described in the previous section, we denote the transformed domainTt(Ω) at t≥0 by Ωt.Now, the following theorem describes the relation between the perturbation of the identity method and the velocity method. The theorem basically tells us that we can start from either a family of velocity fields{V^(t)} on RN or a family of transformations {Tt} of RN provided that the map V^,V^(t,x)=V^(t)(x), verifies (V1) and (V2) or the map T,T(t,X)=Tt(X), verifies assumptions (T1), (T2) and (T3) given below.Theorem 3 (see [21]).
(i) Under assumptions (V1) and (V2) on the mapV^, the maps Tt defined previously have the following properties: (T1)
∀X∈RN,T(·,X)∈C1([0,tV];RN),∃c>0,∀X,Y∈RN,T(·,Y)-T(·,X)C1([0,tV];RN)≤c|Y-X|,(T2)
∀t∈[0,tV],X↦Tt(X)=T(t,X):RN→RN is bijective,(T3)
∀x∈RN,T-1(·,x)∈C([0,tV);R2),∃c>0,∀x,y∈RN,T-1(·,y)-T-1(·,x)C([0,tV);R2)≤c|y-x|.
(ii) If there exists a real numbertV>0 and a map T:[0,tV]×RN→RN verifying assumptions (T1), (T2), and (T3), then the map(11)t,x⟼V^t,x=∂T∂tt,T-1x:0,tV×RN⟶RNverifies assumptions (V1) and (V2), where Tt-1 is the inverse of X↦Tt(X).In the above discussion, we see that the solution to the differential equation (10), V^=V∈Θ1, is the perturbation of the identity operator Tt. Conversely, if Tt is the perturbation of the identity operator, then (10) is satisfied. Hence, we consider the special case, where x(t)=I+tV, and determine the relationship of the autonomous field V∈Θ1 and the nonautonomous V^ defined in (10). Differentiating x(t)=I+tV with respect to t, we get(12)dxdtt;X=VX=V^t,xt;X,X∈Ω.This simply implies that V(X)=V^(t,(I+tV)(X)). Now, replacing X by (I+tV)-1(X), we get V(Tt-1(X))=V^(t,X) or, equivalently, V^(t)=V∘Tt-1. This is why many results obtained by the perturbation of identity technique can be acquired as well through the velocity method using nonautonomous velocity fields. As an immediate consequence, we note that V^(0)=V. The relation V^(t)=V∘Tt-1 also implies that(13)∂V^∂tt=DVTt-1∂Tt-1∂t.So, in particular, we have(14)∂V^0∂t≔V^˙0=-DVV.


## 2.3. Domain and Boundary Transformation

We recall the following theorems on domain and boundary transformations.Lemma 4 (see [28]).
We have the following important transformations which will also be central to our investigation:(1)
Letφt∈L1(Ωt). Then φt∘Tt∈L1(Ω) and(15)∫Ωtφtdxt=∫Ωφt∘TtItdx.(2)
Letφt∈L1(Σt). Then φt∘Tt∈L1(Σ) and(16)∫Σtφtdst=∫Σφt∘Ttwtds.


## 2.4. Material Derivatives

The material and shape derivatives of the state variables are defined as follows (see [4, 24]).Letu be defined in [0,tV]×U. An element u˙∈Hk(Ω), called thematerial derivative of u, is defined as (17)u˙≔u˙0,x≔limt↓0⁡ut,Ttx-u0,xt=ddtut,x+tVxt=0if the limit exists in Hk(Ω).Remark 5.
As pointed out in [9], the material derivative of the state function u can be written as(18)u˙x=limt↓0⁡ut∘Ttx-uxt=ddtut∘Ttxt=0 and it, in fact, characterizes the behavior of the function u at x∈Ω⊂U in the direction V(x).Now, on the other hand, an elementu′∈Hk(Ω) is called theshape derivative of u at Ω at the direction of V, if the following limit exists in Hk(Ω): (19)u′≔u′0,x≔limt↓0⁡ut,x-u0,xt≔∂∂tutxt=0.Remark 6.
We note that ifu˙ and ∇u·V exist in Hk(Ω), then the shape derivative can be expressed as(20)u′x=u˙x-∇u·Vx.In general, if u˙(x) and ∇u·V(x) both exist in Wm,p(Ω), then so does u′(x).


## 2.5. The First- and Second-Order Eulerian Shape Derivatives

We first recall the definition ofdirectional Eulerian shape derivative or simplyshape derivative of a shape functional.Suppose that the shape functionalJ:Ω→R is well-defined. Given the deformation field V^, thedirectional Eulerian shape derivative of J at Ω in the direction of deformation field V^ is defined as(21)dJΩ;V^≔limt↓0⁡JΩt-JΩtif the limit exists. The objective functional J is shape differentiable at Ω provided that dJ(Ω;V^) exists for all V^ and if dJ(Ω;V^) is linear and continuous with respect to V^. On the other hand, thesecond-order Eulerian shape derivative of a well-defined shape functional J at Ω in the direction of the deformation fields V^ and W^ is defined to be(22)d2JΩ;V^,W^≔lims↓0⁡dJΩsW^;V^-dJΩ;V^s=ddsdJΩs;V^s=0if the limit exists. The functional is said to be twice shape differentiable if, for all V^ and W^, d2J(Ω;V^;W^) exists and if d2J(Ω;V^;W^) is bilinear and continuous with respect to V^ and W^. Following these definitions, the second-order shape derivative of the functional being studied can also be computed as follows:(23)d2JΩ;V^;W^=∂∂s∂∂tdJΩt,st=0s=0.In this case, the transported domainΩt,s which is a result of two deformation fields V^ and W^ is illustrated in Figure 3.Figure 3
Transport ofΩ after two deformations.Remark 7.
Under appropriate hypothesis on the mapV^↦dJ(Ω;V^(0)), one can show that dJ(Ω;V^)=dJ(Ω;V^(0)) (cf. [21]). Therefore, if V^ is associated with the deformation field V in the perturbation of the identity, then dJ(Ω;V^(0)) coincides with dJ(Ω;V^) by V^(0)=V. Hence, the first-order shape derivative of a cost functional obtained via the velocity method coincides with the one obtained from the perturbation of the identity technique.


## 3. Analysis for the Nonautonomous Case

In this section, we will see how important is the expressionV∘Ts-1 in considering nonautonomous velocity fields V^. Of course if V is autonomous, V∘Ts-1 is no other than V^(s). Since we are interested in the second-order shape derivative of the Kohn-Vogelius objective functional, we should first recall (without proof) the first-order shape derivative of the Kohn-Vogelius objective functional which is stated in the following theorem.Theorem 8 (see [9, 22, 23]).
For aC1,1-bounded domain, the first-order shape derivative of the Kohn-Vogelius KV cost functional in the direction of a perturbation field V^∈Θ, where V^∈Θ=V^∈C1,1(U¯,R2):V^Γ∪∂U=0 and the state functions uD and uN satisfy the Dirichlet problem D and the Neumann problem N, respectively, is given byJ1dJΩ;V^=12∫Σλ2-∇uN·τ2-∇uD·n2+2λuNκV·nds.where n is the unit exterior normal vector to Σ, τ is a unit tangent vector to Σ, and κ is the mean curvature of Σ.The above result was first proven by Eppler and Harbrecht in [22]. Other proofs were also given by Bacani and Peichl [9, 23], by using two different approaches.Now, we give our main result of the second-order shape derivative of the Kohn-Vogelius objective functional via velocity method.Theorem 9.
DenoteF=λ2-(∇uD·n)2+2λκuN-(∇uN·τ)2 and let V^ and W^ be any two velocity fields from the set(24)Θ2=V^∈C2,1U¯,R2:V^Γ∪∂U=0.Assume that, for some sufficiently small t, dJ(Ωt(W^);V^(t)) exists at Ωt(W^)=Tt(W^)(Ω) in the direction V^(t). Then, for a C2,1-bounded domain, the second-order shape derivative of the Kohn-Vogelius KV cost functional in the directions of the two perturbation fields V^ and W^ and the state functions uD and uN which satisfy the Dirichlet problem D and the Neumann problem N, respectively, is given by (25)d2JΩ;V^;W^=∫ΣF˙Wvn+∂F∂n+κFvnwndΣ-∫ΣFvΣ·DΣnwΣ+n·DΣvwΣ+n·DΣwvΣdΣ+∫ΣFDVW·n+FV′0·ndΣ,where n is the unit exterior normal vector to Σ, τ is a unit tangent vector to Σ, and κ is the mean curvature of Σ.Before we proceed in the formal computation of the second-order shape derivative of the Kohn-Vogelius objective functional, we first prove the following auxilliary result.Lemma 10.
LetF and F˙W be, respectively, defined as follows: (26)F≔λ2-∇uD·n2+2λκuN-∇uN·τ2F˙W≔-2∇uD·n-DWT∇uD·n+∇uD,W′·n+∇∇uD·W·n-2∇uN·τ-DWT∇uN+∇uN,W′·τ+∇∇uN·W·τ+∇uN·τ˙W+2λκ˙WuN+κuN,W′+∇uN·W.Then, we have (27)∂∂sFs∘Tswss=0=F˙W+FdivΣW.Proof.
We first note the following:(28)wss=0=1,Fs∘Tss=0=F,∂∂swss=0=divΣW.Now, for the expression (∂/∂s)(Fs∘Ts)s=0, we note that(29)Fs∘Ts=λ2∘Ts-∇uD,s∘Ts·ns∘Ts2+2λ∘Tsκs∘TsuN,s∘Ts-∇uN,s∘Ts·τs∘Ts2,and since λ is constant, λ∘Ts=λ. Hence, we get (30)∂∂sFs∘Tswss=0=∂∂sFs∘Tsws+Fs∘Ts∂∂swss=0,by applying product rule. Furthermore, by substitution and distributing the (partial) differential operator, we obtain(31)∂∂sFs∘Tswss=0=∂∂sλ2s=0-∂∂s∇uD,s∘Ts·ns∘Ts2s=0+∂∂s2λ∘Tsκs∘TsuN,s∘Tss=0-∂∂s∇uN,s∘Ts·τs∘Ts2s=0.Using the chain rule and again the product rule twice, we have(32)∂∂sFs∘Tswss=0=-2∇uD·n∂∂s∇uD,s∘Ts·ns∘Tss=0+2λ∂∂sκs∘TsuN,s∘Tss=0-∂∂s∇uN,s∘Ts·τs∘Ts2s=0=-2∇uD·n∂∂sDTs-T∇uD,s∘Ts·ns∘Tss=0+2λ∂∂sκs∘TsuN,s∘Tss=0-2∇uN·τ∂∂sDTs-T∇uN,s∘Ts·τs∘Tss=0≕E1+E2+E3.Now, we simplify each of the expressions E1, E2, and E3. In the sequel, we will be needing the material and shape derivatives of the vectors n and τ and the mean curvature κ. For their corresponding forms, we refer the readers to [25, Theorems 4, 5, and 6] which are proven in [5, 28]. So we proceed as follows. First, using the product rule twice and then the chain rule, we get(33)E1=-2∇uD·n∂∂sDTs-T∇uD,s∘Ts·ns∘Tss=0=-2∇uD·n∂∂sDTs-T∇uD,s∘Tss=0·ns∘Tss=0-2∇uD·nDTs-T∇uD,s∘Tss=0·∂∂sns∘Tss=0=-2∇uD·n∂∂sDTs-T∇uD,s∘Tss=0·n+∇uD·-DWTnΣ=-2∇uD·n∂∂sDTs-Ts=0∇uD,s∘Tss=0·n-2∇uD·nDTs-Ts=0∂∂s∇uD,s∘Tss=0·n-2∇uD·n∇uD·-DWTnΣ.
Interchanging the gradient and the differential operator and upon evaluation ofs at 0, we get(34)E1=-2∇uD·n-DWT∇uD+∇∂∂suD,s∘Tss=0·n-2∇uD·n∇uD·-DWTnΣ=-2∇uD·n-DWT∇uD·n+∇u˙D,W·n-2∇uD·n∇uD·DWn·nn-DWTn.
Here we note thatu˙D,W is the material derivative of uD in the direction W. By definition, u˙D,W can be written in terms of the shape derivative uD,W′: u˙D,W=uD,W′+∇uD·W, where uD,W′ satisfies(35)-ΔuD,W′=0inΩ,uD,W′=0onΓ,uD,W′=-∂uD∂nW·nonΣ.
Now, for the second expressionE2, we have(36)E2=2λ∂∂sκs∘TsuN,s∘Tss=0=2λ∂∂sκs∘TsuN,s∘Ts+κs∘Ts∂∂suN,s∘Tss=0=2λκ˙WuN+κu˙N,W.
Here we stress thatu˙N,W is the material derivative of uN in the direction W. Also, u˙N,W=uN,W′+∇uN·W, where uN,W′ satisfies(37)-ΔuN,W′=0inΩ,uN,W′=0onΓ,∂uN,W′∂n=divΣW·n∇ΣuN+κλW·nonΣ.The expression κ˙W represents the material derivative of the mean curvature κ and it can be shown that κ˙W=-Tr[DnDW]-(D[(DWn·n)n-(DW)Tn])n·n.
Finally, for the last expressionE3, we have the following: (38)E3=-2∇uN·τ∂∂sDTs-T∇uN,s∘Ts·τs∘Tss=0=-2∇uN·τ∂∂sDTs-T∇uN,s∘Ts·τs∘Tss=0-2∇uN·τ∇uN,s∘Ts·∂∂sτs∘Tss=0=-2∇uN·τ-DWT∇uN+∇u˙N,W·τ+∇uN·τ˙W,where τ˙W is the material derivative of τ in the direction W and is given by τ˙=[(DW)Tn·τ]n.
Combining all of these simplification expressions and by the relations in (28), we get the desired result.Now, we are in the position to prove our main result. In the sequel, we supposeV^ and W^ to be nonautonomous velocity fields and proceed for the computation as follows. Letting Fs=λ2-(∇uD,s·ns)2+2λκsuN,s-(∇uN,s·τs)2, we have(39)d2JΩ;V^;W^=lims↓0⁡dJΩsW^;V^-dJΩ;V^s=∂∂s∂∂tdJΩt,st=0s=0=∂∂s∫ΣFsV∘Ts-1·ns∘Tsdet⁡DTsDTs-TndΣs=0=∂∂s∫ΣFsV∘Ts-1·ns∘TswsdΣs=0=∫Σ∂∂sFs∘TsV·ns∘Tswss=0dΣ=∫Σ∂∂sFs∘Tswss=0V·ns∘Tss=0dΣ+∫ΣFs∘Tswss=0∂∂sV·ns∘Tss=0dΣ≕∫ΣΞ1Ξ2dΣ+∫ΣΞ3Ξ4dΣ.Note that we already have a simplified form forΞ1 from Lemma 10, so we only need to simplify the expressions Ξ2, Ξ3, and Ξ4. For Ξ2, we have need for Ξ2=V·(ns∘Ts)s=V·n and, for Ξ3, we easily get Ξ3=(Fs∘Ts)wss=F, where F is the same with what is given in Lemma 10. Now, for the last expression Ξ4, we have(40)Ξ4=∂∂sV·ns∘Tss=0=V·∂∂sns∘Tss=0+∂∂sVs=0·n=V·DWn·nn-DWTn+DWn·nvn-DWV·n+V′0·n+DVW·n. Here, we stress that (∂V/∂s)s=0=(∂V(0)/∂s)s=0 and that the relation V=V(s)∘Ts implies the identity(41)∂V∂ss=0=∂∂sVs∘Tss=0=V˙0=V′0+DVW,where V′(0)(x)=limt↓0(1/t)V(t,x)-V(0,x).With these identities, we now have(42)d2JΩ;V^;W^=∫ΣF˙W+FdivΣWV·n+FDWn·nvn-DWV·n+V′0+DVW·ndΣ=∫ΣF˙Wvn+FdivW-DWn·nvn+FDWn·nvn-FDWV·n+V′0+DVW·ndΣ=∫ΣF˙Wvn+FdivWvn-FDWV·ndΣ+∫ΣFDVW·n+FV′0·ndΣ=∫ΣF˙Wvn+∂F∂n+κFvnwndΣ-∫ΣFvΣ·DΣnwΣ+n·DΣvwΣ+n·DΣwvΣdΣ+∫ΣFDVW·n+FV′0·ndΣ.Now, we note that, for autonomous velocity fields, we have the relationV^(0)=V (i.e., for V^(s)=V∘Ts-1, we have V^(0)=V). Thus, we now have(43)d2JΩ;V0;W0=d2JΩ;V^;W^=∫ΣF˙Wvn+FdivWvn-FDWV·n+FDVW·ndΣ, from which we see that(44)d2JΩ;V^;W^=d2JΩ;V0;W0+dJΩ;V′0 and here we remark that this result coincides with the one presented by Delfour and Zolésio in [5, page 1420, Equation (28)]. This proves the main Theorem 9.


### 3.1. On the Boundary Transformation Approach

Now, in this section we compute for the second-order shape derivative of the Kohn-Vogelius objective functional through boundary transformation approach. Again, we assumeV^ and W^ to be nonautonomous velocity fields. So we proceed as follows.First, using the definition of the second-order shape derivative and by the relationV^=V∘Tt-1, we have (45)d2JΩ;V^;W^=∂∂s∂∂tdJΩt,st=0s=0=∂∂s∫ΣsFsV^·NsdΣss=0=∂∂s∫ΣsFsV∘Ts-1·NsdΣss=0.Invoking the boundary differentiation rule, we get (46)d2JΩ;V^;W^=∫Σ∂∂nFsV∘Ts-1·Ns+κsFsV∘Ts-1·NsW·ns=0dΣ+∫Σ∂∂sFsV∘Ts-1·Nss=0dΣ=∫ΣFss=0∂∂sV∘Ts-1·Nss=0+∂∂nFV·n+κFV·nW·ndΣ+∫Σ∂∂sFss=0V∘Ts-1·Nss=0dΣ=∫ΣF∂∂sVss=0·Nss=0+V∘Ts-1s=0·∂∂sNss=0dΣ+∫Σ∂∂nFV·n+κFV·nW·ndΣ+∫ΣFW′V·ndΣ=∫ΣFDV-W·n+V·N′+∂∂nFV·n+κFV·nW·n+FW′vndΣ=∫ΣFW′vn+FV·N′+∂∂nFV·n+κFV·nW·n+FV˙0·ndΣ=∫ΣFW′vn+∂F∂n+κFvnwn-FvΣ·DΣnwΣ+n·DΣvwΣ+n·DΣwvΣdΣ+∫ΣFDVW·n+FV˙0·ndΣ=d2JΩ;V0;W0+dJΩ;V˙0.Remark 11.
Here,d2J(Ω;V(0);W(0)) is the bilinear term which Delfour and Zolésio refer to as theshape Hessian.Remark 12.
We remark that, as compared to the form of the shape Hessian presented in [22], the result we have established here in Theorem 9 clearly shows the relation pointed in [26] by Delfour and Zolésio about the form of shape Hessians obtained through nonautonomous velocity fields.


## 3.1. On the Boundary Transformation Approach

Now, in this section we compute for the second-order shape derivative of the Kohn-Vogelius objective functional through boundary transformation approach. Again, we assumeV^ and W^ to be nonautonomous velocity fields. So we proceed as follows.First, using the definition of the second-order shape derivative and by the relationV^=V∘Tt-1, we have (45)d2JΩ;V^;W^=∂∂s∂∂tdJΩt,st=0s=0=∂∂s∫ΣsFsV^·NsdΣss=0=∂∂s∫ΣsFsV∘Ts-1·NsdΣss=0.Invoking the boundary differentiation rule, we get (46)d2JΩ;V^;W^=∫Σ∂∂nFsV∘Ts-1·Ns+κsFsV∘Ts-1·NsW·ns=0dΣ+∫Σ∂∂sFsV∘Ts-1·Nss=0dΣ=∫ΣFss=0∂∂sV∘Ts-1·Nss=0+∂∂nFV·n+κFV·nW·ndΣ+∫Σ∂∂sFss=0V∘Ts-1·Nss=0dΣ=∫ΣF∂∂sVss=0·Nss=0+V∘Ts-1s=0·∂∂sNss=0dΣ+∫Σ∂∂nFV·n+κFV·nW·ndΣ+∫ΣFW′V·ndΣ=∫ΣFDV-W·n+V·N′+∂∂nFV·n+κFV·nW·n+FW′vndΣ=∫ΣFW′vn+FV·N′+∂∂nFV·n+κFV·nW·n+FV˙0·ndΣ=∫ΣFW′vn+∂F∂n+κFvnwn-FvΣ·DΣnwΣ+n·DΣvwΣ+n·DΣwvΣdΣ+∫ΣFDVW·n+FV˙0·ndΣ=d2JΩ;V0;W0+dJΩ;V˙0.Remark 11.
Here,d2J(Ω;V(0);W(0)) is the bilinear term which Delfour and Zolésio refer to as theshape Hessian.Remark 12.
We remark that, as compared to the form of the shape Hessian presented in [22], the result we have established here in Theorem 9 clearly shows the relation pointed in [26] by Delfour and Zolésio about the form of shape Hessians obtained through nonautonomous velocity fields.


## 4. Conclusion

We have computed the second-order shape derivatived2J(Ω;V^;W^) of the Kohn-Vogelius objective functional JKV in the direction of two (nonautonomous) deformation fields V^ and W^ via velocity method. The computed expression for d2J(Ω;V^;W^) is given by (47)d2JΩ;V^;W^=∫ΣF˙Wvn+∂F∂n+κFvnwndΣ-∫ΣFvΣ·DΣnwΣ+n·DΣvwΣ+n·DΣwvΣdΣ+∫ΣFDVW·n+FV′0·ndΣ, where F=λ2-(∇uD·n)2+2λκuN-(∇uN·τ)2, n is the unit exterior normal vector to Σ, τ is a unit tangent vector to Σ, and κ is the mean curvature of Σ. We stress out that the expression computed for the second-order shape derivative of JKV is not yet in its explicit form. The explicit form, however, could easily be obtained by substituting all those computed values for each of the expressions found on the integrand or the interested reader may refer directly to [29] for the explicit form of d2J(Ω;V^;W^). It is apparent in our first result that the computed expression for the shape derivative coincides with the one presented by Delfour and Zolésio in [5]. Interestingly, the result obtained through boundary transformation approach which again uses nonautonomous velocity fields also verifies the result found by Delfour and Zolésio in [5].



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