Abstract and Applied Analysis

Volume 2013 (2013), Article ID 384320, 19 pages

http://dx.doi.org/10.1155/2013/384320

Research Article

On the First-Order Shape Derivative of the Kohn-Vogelius Cost Functional of the Bernoulli Problem

Jerico B. Bacani1,2 and Gunther Peichl²

¹Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, Baguio City 2600, Philippines

²Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz, Austria

Received 18 August 2013; Accepted 1 November 2013

Academic Editor: Sergei V. Pereverzyev

Copyright © 2013 Jerico B. Bacani and Gunther Peichl. 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.

Abstract.

The exterior Bernoulli free boundary problem is being considered. The solution to the problem is studied via shape optimization techniques. The goal is to determine a domain having a specific regularity that gives a minimum value for the Kohn-Vogelius-type cost functional while simultaneously solving two PDE constraints: a pure Dirichlet boundary value problem and a Neumann boundary value problem. This paper focuses on the rigorous computation of the first-order shape derivative of the cost functional using the Hölder continuity of the state variables and not the usual approach which uses the shape derivatives of states.

1. Introduction

The Bernoulli problem is the prototype of a stationary free boundary problem. It arises in various applications such as electrochemical machining, potential flow in fluid mechanics, tumor growth, optimal insulation, molecular diffusion, and steel and glass production [1–6]. A characteristic feature of free boundary problems is that not only the state variable is unknown but also the domain on which the state equation is posed. This represents a significant theoretical as well as numerical challenge. One can characterize the Bernoulli problem, at least along general lines, by finding a connected domain as well as a function which is harmonic on this domain. One component on the boundary is known. The other one is determined by a set of overdetermined boundary conditions (a Dirichlet condition and a Neumann condition) for the state. If the free boundary component is strictly exterior to the fixed part of the boundary, the problem is called exterior Bernoulli problem and interior Bernoulli problem otherwise. For more discussions related to interior and exterior Bernoulli problems, we refer the reader to [1, 4, 7–10].

Recent strategies to compute a numerical solution are based on reformulating the Bernoulli problem as a shape optimization problem. This can be achieved in several ways. 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 squares sense (cf. [11–13]). Alternatively, one can compute on a given domain two auxiliary states: which satisfies the Dirichlet condition and which satisfies the Neumann condition on the free boundary. The underlying domain is selected such that the difference is as small as possible. In fact, if for a domain then and is a solution of the Bernoulli problem. Sometimes is called Kohn-Vogelius functional since Kohn and Vogelius were among the first who used such a functional in the context of inverse problems [14]. Standard algorithms to minimize require some gradient information. So in this paper, the first-order sensitivity analysis is carried out for the functional for the exterior Bernoulli problem. The main contribution in this paper is the application of a shape optimization technique that leads to the explicit expression for the shape derivative of the cost functional. This is done through variational means similar to the techniques developed in [9, 10, 13], wherein we use the Hölder continuity of the state variables satisfying the Dirichlet and Neumann problems but we do not introduce any adjoint variables. In our approach, we also bypass the use of the material derivatives of the states (which was done in [1]) and the use of states’ shape derivatives.

The rest of the paper is structured as follows. Section 2 presents the Bernoulli free boundary problem and its shape optimization formulations. Section 3 provides a list of shape optimization tools that are needed in the analysis for the shape derivatives of the Kohn-Vogelius cost functional . Section 4 presents an exhaustive discussion on the first-order shape derivative of . Finally, Section 5 draws conclusion and observation.

2. The Bernoulli Problem

The exterior Bernoulli free boundary problem is formulated as follows. Given a bounded and connected domain with a fixed boundary and a constant , one needs to find a bounded connected domain with a free boundary , containing the closure of , and an associated state function , where , such that the overdetermined conditions are satisfied:

On the other hand, the interior Bernoulli free boundary problem has the following formulation. Given a bounded and connected domain with a fixed boundary and a constant , one determines a bounded connected domain with a free boundary and an associated state function , where , subject to the following constraints:

In both problems is the outward unit normal vector to . The difference in the domains of these two types of Bernoulli problems is depicted in Figure 1.

(a)

(b)

Figure 1: The domain for the interior Bernoulli problem (a) and exterior problem (b).

Methods of shape optimization can be employed in solving the exterior Bernoulli free boundary problem (1). As we observe, this boundary problem is ill-posed due to the fact that we have overdetermined conditions on the free boundary . So to overcome the difficulty of solving it, one can reformulate it as one of the following shape optimization problems which involves now a well-posed state equation.

(1)Tracking Neumann data [11, 12] as

where the state function is the solution to the Dirichlet problem

(2)Tracking Dirichlet data [11, 13] as

where the state function is the solution to the Neumann problem

(3)Minimizing the Kohn-Vogelius type cost functional [12, 15] as

where state functions and satisfy (4) and (6), respectively.

In this paper, we are just interested in the study of minimizing the Kohn-Vogelius functional .

3. Tools in Shape Optimization

3.1. Feasible Domain

In this work, we are interested in -domains, where . Aside from being we also assume that these are bounded and connected subsets of a bigger set which is also a bounded connected domain. This is called the universal or the hold-all domain. The smoothness of these domains can be defined in the following sense (cf. [16]).

Consider the standard unit orthonormal basis in . For a point , let so as to write . Consider the unit ball and introduce the subsets

Definition 1. A domain with a nonempty boundary is called a -domain, where ,  , if for every there exists a neighborhood of and a diffeomorphism such that , , and .

To illustrate this for and , see Figure 2.

Figure 2: A -domain , where is a diffeomorphism from the neighborhood to the ball .

Note that if is a bounded, open, connected set with a boundary, then . This was given in [17] and we prove it as follows.

Theorem 2 (see [17]). If is a bounded open connected subset of with Lipschitz continuous boundary, then .

Proof. The interior of is the largest open set contained in the set . Moreover, . It follows that . Next, we show that . Clearly, . We now show that if , then .

Suppose and . We need to show that any open set containing contains an element not in . We first note that by definition of domain, there exists a neighborhood of and a diffeomorphism . Let be an open set containing with . It follows that is an open set containing and this set is contained in . Hence, there exists such that . This implies that . Thus, contains an element not in , which is a contradiction. Therefore, . We have proven that if , then . Taking the contrapositive of this statement we get that if , then . Since but , we conclude . Thus, . We have shown that and . Therefore, .

3.2. The Perturbation of Identity Technique

Given bounded connected domains and of , where , and a linear space of vector fields , one can deform via the perturbation of identity operator

where . For a given we denote the deformed domain to be , which is the image of under .

Throughout the paper, we use the usual infinity norms in the spaces , , and , where is a compact subset of . In addition to this, we also denote the Frobenius norm of to be

This norm and the infinity norm of the matrix can be related as

This can be shown easily. One can also show that if and , then the vector is bounded in . In fact,

and the proof is trivial. Finally, the symbols or will refer to the usual Euclidean norm.

The Perturbed Domain . The domains that are considered in this work are of annulus type with boundary , which is the union of two disjoint sets and , referred to as the fixed and free boundaries, respectively. These domains are obtained through the operator defined in (9), where belongs to , which is defined as

For , we obtain the reference domain , with a fixed boundary and a free boundary .

The main objective in this subsection is to show that is a diffeomorphism from to for sufficiently small . To verify this, we need the following results, which are given and proven in [17].

Theorem 3. If is a bounded, open, connected set in such that and is a continuous injective mapping from to , then

Theorem 4. Suppose

(1) is a bounded, open, connected set in such that ,(2) where is injective,(3) such that

Then

(i) is a homeomorphism (i.e., is a bijection, is continuous, and is continuous),(ii) is a -diffeomorphism (i.e., is a bijection, , ),(iii), .

We also consider the following property of a domain, which is also found in [17, page 52].

Lemma 5. If is a bounded, open, connected subset of having a Lipschitz continuous boundary, then there is a number such that, for any given points , one can find a finite sequence of points , , satisfying the following properties:

(a), for , ,(b) for ,(c).

We also recall the useful property of the determinant of the Jacobian of which is given in the next lemma. Here we use the notation

Lemma 6 (see [9, 13]). Consider the operator defined by (9), where , which is described by (13). Then

(i),(ii)there exist such that , for , .

Proof. In general, for -dimensional case, the Jacobian of is given by , where if, , and . By definition of the determinant, we can write as

where refers to the set of all permutations of , is the identity permutation, , and is either (if the number of inversions is even) or (if the number of inversions is odd). We observe that the expression can be written as , where . We also observe that, for , each term of the expression has at least 2 factors that are of the form , . Hence we can write , where is in . All terms of have factors of the form ,  , and thus we have , which can be written as , where . Combining , , and , we get

with . In particular, for , the determinant is computed as follows:

This verifies . To show we first get the lower bound for . Take

For , we obtain

On the other hand, by triangle inequality we have

Hence, we have shown that there are positive constants and such that for .

Considering the theorems and lemmas presented beforehand, we are now ready to prove the following theorem.

Theorem 7. Let and be nonempty bounded open connected subsets of with Lipschitz continuous boundaries, such that , and is the union of two disjoint boundaries and . Let be defined as in (9) where belongs to , defined as (13).

Then for sufficiently small ,

(1) is a homeomorphism,(2) is a diffeomorphism, and in particular, is a diffeomorphism,(3),(4).

Proof. First, because is a domain, it follows that by Theorem 2. Second, , and it is injective. Third, it is evident that is because is . For , because vanishes on . For , the determinant of the Jacobian of the perturbation of identity operator is given by (19). By Lemma 6, there exists a , given by (20), such that for all and for . Hence, by applying Theorem 4, we conclude that and for all , and is a homeomorphism. Furthermore, by Theorem 4, we find that is a diffeomorphism. To show that is a diffeomorphism, we are left to show that is Lipschitz continuous. To verify this we use Lemma 5.

Given any two points we choose such that properties (a)–(c) of Lemma 5 are satisfied. For fixed , differentiating the identities and will lead to

for all . Thus,

This implies

Applying the infinity norm in the space we have

Since is Lipschitz continuous, we have

where is the maximum of all Lipschitz constants of for all . Then finally, using the mean value theorem and property (c) in Lemma 5, we obtain

Hence is Lipschitz continuous which shows that is a diffeomorphism for sufficiently small . Restricting to , this proves that is a diffeomorphism. (2) is clear because the fixed boundary is invariant under ; that is, since vanishes on . Lastly, using Theorem 3, definition of , (1), and (2), we obtain (3).

Corollary 8. Let and be two domains of with boundary. Then for , where is given by (20), the perturbed domain is also of class .

Proof. Given , we let . Then there exists a neighborhood of and a diffeomorphism such that ,  , and . We have also shown that defined in Theorem 7 is a diffeomorphism. Since is continuous, is a neighborhood of in . Define . This is bijective because and are bijective. because (hence ) and . Also, .

Next, we note that . Since is injective, we have . Thus by definition of we get

We also observe the following:

This shows that is indeed of class .

Remark 9. Theorem 7 and Corollary 8 tell us that the reference and the perturbed domain have the same topological structure and regularity under the perturbation of identity operator for sufficiently small . See Figure 3 for illustration.

Figure 3: The action of on a -domain.

Properties of . In addition to (16) we also use the following notations throughout the work:

Remark 10. We note the following observations for fixed, sufficiently small .

(1).(2).(3).(4).(5) implies that and are both finite.

We now provide several properties of .

Lemma 11 (see [9, 13, 16, 18]). Consider the transformation , where the fixed vector field belongs to , defined in (13). Then there exists such that and the functions in (16) and (31) restricted to the interval have the following regularity and properties.

(1).(2).(3).(4).(5).(6)There is such that for .(7).(8).(9).(10).(11).(12).(13).(14),

where the surface divergence is defined by

We provide proofs for properties (3) and (8). The rest can be seen in [19].

Proof. (3) Suppose , , and . Then

Using Lemma 5, we connect and by a chain , , satisfying

(i), for , ,(ii) for ,(iii),

and then we get

Thus,

By reducing if necessary, we can assume without loss of generality that for . This allows us to represent as a Neumann series:

and its norm is estimated as follows:

This shows uniform convergence in and . Hence, for every one can choose a which implies that, for every , whenever . In other words, .

To show that is continuous from to , we only need to show that for every , whenever and . Let . Using (37), estimate as follows:

Using the definition of Jacobian of a transformation and the regularity of , we further simplify (38) as follows:

where is the Lipshitz constant for and is upper bound for . Taking the maximum of both sides of the inequality for all and using (34) we get

where . Thus, for any , we choose , so that if , then . Therefore, .

Proof of property in Lemma 11 is as follows. Given , we have . This implies that

Manipulating the left hand side of (41), we get

We first work on . Applying the definition of , we get

Thus,

Similarly, we can write as follows:

Hence, we have

Suppose is a coordinate function of . By the mean value theorem, we observe that

where is a point on the segment joining and , and as tends to infinity, (47) tends to . Thus,

Combining (44) and (48), we get

which implies that

Evaluating (50) at , we get .

3.3. The Method of Mapping

If is defined in and is defined in , then the direct comparison of with is generally not possible since the functions are defined on different domains. To overcome this difficulty, one maps back to by composing it with ; that is, one defines . With this new mapping one can define the material and the shape derivatives of states, the domain and boundary integral transformations, and derivatives of integrals, as well as the Eulerian derivative of the shape functional. This technique is called the method of mapping.

Material and Shape Derivatives. The material and shape derivatives of state variables are defined as follows [20, 21].

Definition 12. Let be defined in . An element , called the material derivative of , is defined as

if the limit exists in ().

Remark 13. The material derivative can be written as

It characterizes the behavior of the function at in the direction .

Definition 14. Let be defined in . An element is called the shape derivative of at in the direction , if the following limit exists in :

Remark 15. The shape derivative of is also defined as follows:

We note that if and exist in , then the shape derivative can be written as

In general, if and both exist in , then also exists in that space.

Domain and Boundary Transformations

Lemma 16 (see [18]). Let . Then and

Let . Then and

where and are defined in (31).

Proofs can be found in [13, 18].

Domain and Boundary Differentiation. We recall some results concerning the derivative of integrals with respect to the domain of integration. For the first theorem, it is sufficient to have domains while the second theorem requires domains. For proofs, see [18].

Theorem 17 (domain differentiation formula). Let and suppose exists in . Then

Theorem 18 (boundary differentiation formula). Let be defined in a neighborhood of . If and , then

where is the mean curvature of the free boundary .

The First-Order Eulerian Derivative

Definition 19. The Eulerian derivative of the shape functional defined in (7) at the domain in the direction of the deformation field is given by

if the limit exists.

Remark 20. is said to be shape differentiable at if exists for all and is linear and continuous with respect to .

4. Main Result

In this section we derive in a rigorous manner the first-order shape derivative of the Kohn-Vogelius functional , defined by (7), subject to the Dirichlet and Neumann boundary value problems (BVPs) (4) and (6), respectively. Our strategy bypasses the material or shape derivatives of states. In the derivation, we have employed techniques used in [9, 10, 13] but there is no need to use adjoint variables.

This section discusses the variational forms of the PDEs, the state variables in the perturbed domains, the Hölder continuity of the state variables, and the higher regularity of the solutions to the BVPs. The rest of the proof is presented in the last part of this section.

4.1. Variational Forms of the Dirichlet and Neumann Problems

We recall that we are considering the shape optimization problem (7) where solves the pure Dirichlet problem (4) and solves the Neumann problem (6). As in [13], we consider the Hilbert space

which is endowed with the norm

and a linear manifold defined by

for .

First, we determine the variational equations for the Dirichlet and the Neumann problems. The variational form of the Dirichlet problem (4) is given by the following.

Find such that

Equation (64) can be shown to have a unique solution using Theorem of [22]. Similarly, the variational form of the Neumann problem (6) is formulated as follows.

Find such that

It is also well known that (65) has a unique solution.

4.2. Analysis of State Variables in Deformed Domains

We now consider the class of perturbed problems:

where solves the pure Dirichlet problem

and solves the Neumann problem

Here, is the outward unit normal to the deformed free boundary . The variational form of (67) is formulated as follows.

Find such that

It is known that (69) has a unique solution.

Remark 21. The function can be referred to as the reference domain by composing with ; that is,

and by chain rule of differentiation, we get

Let be the solution of (69). Applying Lemma 16 for all we have

Applying (71) and noting that because , we obtain

where . Hence, if solves the variational equation (69), then satisfies the variational equation,

for all ,   on , and on .

Now we show that is the unique solution of (74) in . First, we show that is the unique solution to

for all . The bilinear form defined by

is continuous, because

The bilinear form is also coercive. To show this we recall that uniformly on . This is equivalent to the statement

Let . So for sufficiently small , , and

So is coercive.

Next, we show that the functional is bounded:

Therefore, by the Lax-Milgram lemma, is the unique solution to the variational equation

This implies the existence of a unique solution of (74) as verified below.

Let . Using (81) we obtain

Thus (74) is satisfied. The boundary conditions are also satisfied because on , and and on , both and are zero. To show uniqueness, we let and be solutions of (74). This implies that there exist and such that and , where and are solutions to (81). Taking the difference of and and considering that solution to (81) is unique, we get .

Next, we consider (68) whose variational form is formulated as follows.

Find such that

Similarly, if solves the variational problem (83), then solves the variational equation

where on .

As shown before, the bilinear form defined by

is coercive and continuous. The linear functional defined by is continuous on because

By the Lax-Milgram lemma, is the unique solution in of

Let . Then, by (87), we get

Since , on . Uniqueness of follows from the uniqueness of . Therefore, is the unique solution of the variational problem (84) in .

4.3. Hölder Continuity of the States

We show that and are Hölder continuous on .

Theorem 22 (see [13]). The solutions of (74) are uniformly bounded in for and

where is the weak solution of (4).

Proof. We first prove the uniform boundedness of in for . Since , by using coercivity of we get

Also, by applying (81), we have

Therefore,

Now we take the difference between the weak form of (4) and the variational equation (74), to get

Note that . This implies that is in . Note also that . Let . So for sufficiently small , . Now choosing as a test function and by the uniform coercivity of one obtains

If , then the inequality above holds. For we have

Hence,

Squaring and multiplying on both sides of the inequality give us

Since is uniformly bounded in by Lemma 11,

follows.

Theorem 23. The solutions of (84) are uniformly bounded in for and

where is the solution of (65).

Proof. Subtracting (65) from (84) for all we get

Hence

Note that belongs to . Hence

By Cauchy-Schwarz inequality, we obtain

Furthermore, by trace theorem we have . Therefore,

holds, where . This implies

which entails

We now show that is bounded uniformly in in . Since , we have

Consequently,

and this shows that is uniformly bounded in because is bounded. In addition, and are differentiable at by Lemma 11. Therefore,

4.4. Higher Regularity of the Solutions

In this section we will show that the solutions to the PDEs (4) and (6) have higher regularity. We begin by considering the state variable . For domains, we show that these solutions also exist in and more generally in if domains are of class , .

To prove higher regularity of , we require the following two theorems, which are proven in [22].

Theorem 24 (see [22, page 124]). Let be a bounded open subset of with a boundary. Consider the Dirichlet boundary value problem:

where

Let be uniformly Lipschitz functions and let be bounded measurable functions such that , , and that there exists with

for all and for almost every . Assume in addition that either

(i),    and a.e. or(ii) a.e.

Then for every and every , there exists a unique that solves (110).

Theorem 25 (see [22, page 128]). Let be a bounded open subset of with a boundary. Consider the operator defined by (110) with , and assume that there exists such that (112) holds for all and for every . Also, consider a real boundary operator which is either the identity operator or with , and everywhere on . Furthermore, assume that satisfies and . Then .

We will also justify the higher regularity of . We use the following results whose proofs are given in the corresponding texts.

Theorem 26 (see [23, page 316]). Let be a bounded open subset of . Suppose is a weak solution of the PDE

where

Assume furthermore that for , and . Then .

Theorem 27 (see [24, page 12]). Let be a bounded domain with boundary for some nonnegative integer . Suppose the data and of the problem

are in and , respectively, for some real number with . Then .

For proof, see [22].

Theorem 28 (see [22, page 84]). Let be a bounded domain with boundary and . Consider the Neumann problem

If , , and , then the weak solution to (116) exists in .

For proof, see [22].

Using the theorems presented above, we will now prove our claim that the solutions to the PDEs (4) and (6) have indeed higher regularity. This result is given in the following theorem.

Theorem 29. Let be a bounded domain with boundary of class . Let be weak solutions of the (4) and (6), respectively. Then and also belong to . More generally, if is of class , where is a nonnegative integer, then and are elements of .

Proof. We first consider the solution to the Dirichlet problem (4). We use Theorem 25 to show that is an element of . Here, (110) is applied with the following settings.

We consider . The domain is of class . , and hence for and for , with . We also observe that , for all . Thus . Furthermore, we have the following data: ,  , . Therefore, by using Theorem 24, there exists a unique , which is a solution to (4).

For higher regularity of we apply Theorem 25. At first we consider -domains. In this case, . We have for and for , . The operator is the identity operator, thus of order . From the first consequence, it is known that satisfies and on , where , . Therefore, by applying Theorem 25, we have . In general, for smoother domains with boundaries, solutions to (4) are elements of .

Next, we recall that, for domain, there is a weak solution to the boundary value problem (6). We also show that the solution actually lies in and if the domain is more regular, then so is the solution. More precisely, we want to show that if is a domain whose boundary is of class , then is in , where is a nonnegative integer. For this purpose we need Theorem 26 which implies .

Choose a bounded connected domain with boundary such that and , where and are the domains described in Section 2. Let be the annulus having boundaries and , and let be the other annulus with boundaries and . First, we consider the following elliptic problem on :

Since is bounded with compact boundaries, we have . So by applying Theorem 27, we get . Since also solves (117), then by uniqueness we have . If is a domain with boundary , then by Theorem 27 we have . Moreover, if is a -domain, then is in .

Second, we consider the following boundary value problem:

Because , it follows that . We have also shown that . This implies that . Hence, . Also, . Since is a domain of class , then by Theorem 28, we infer that (118) has a unique solution . Note, however, that also solves (118). So by uniqueness of the solution, we get . Therefore, . Now, if domain is of class and , we get by applying Theorem 25 and so is . Doing this recursively, we end up with .

Hence, for -domains , if we combine and , we get . Moreover, is in a neighborhood of because . Therefore, .

Remark 30. In the computation of the first-order shape derivative, since we are dealing with -domains, we may consider -regularity for the solutions and , as justified by Theorem 29.

4.5. The Shape Derivative of

First, we state and prove the following lemma.

Lemma 31. Let be a bounded Lipschitz domain. Then the following equation:

is valid for vector field and scalar function having the following regularity:

(i) and ;(ii) and .

Proof. First we recall the Gauss’ divergence theorem in saying that if a domain is a bounded Lipschitz domain, then we have

for a vector field . Second we take the divergence of the product of a scalar function and the vector field to get

Then, integrating both sides over and applying the divergence theorem to the vector field we obtain (119).

If and , then ,  , and the integral is bounded. Hence, (119) is well defined. Note that the formula

holds for and . We write

where is a trace operator. Let and . By density, we pick and such that in and in . By (122),

Note that and . Also in implies that in . Moreover, in implies in . Furthermore, since , and in . Therefore, (119) holds for and .

If and , then and . Note that (cf. [25, page 316]), where refers to the trace of on , hence . Therefore, (119) is also well defined for this case. Using similar arguments as above and using the density of in we can show that (119) is also valid for this case.

Now we apply Lemma 31 to prove the next lemma.

Lemma 32 (see [1]). Let and belonging to satisfy the Dirichlet problem (4) and the Neumann problem (6), respectively. Then

where is given by property of Lemma 11.

Proof. From Lemma 11, we recall the expression for , which is given by . Our first goal is to derive an expression for for . We begin by writing as follows:

We manipulate each term on the right-hand side of (127). First, because , we have . Hence we can use (119) by taking and by choosing . In addition, we take into account that vanishes on the fixed boundary . This leads to

The other two terms on the right hand side of (127) are manipulated as follows. The term is written as

where represents the Hessian of . Because Hessian is symmetric, we obtain

Substituting (130) into (128) we get

Next, we expand the expression using (121) as

Integrating both sides of (132) over , applying Stoke’s theorem, and considering on we end up with

or equivalently

Interchanging and we get

Also, because , we obtain

Thus,

Adding (131), (135), and (137) altogether, we express (127) as

Set in (138). The first two integrals on the right hand side of (138) vanish because in . Moreover, since on we have . Thus, we can write (138) as follows:

Therefore, (125) is satisfied.

On the other hand, by replacing both and by and by considering that in and on , we derive (126) as

Now, we derive the explicit form of the first-order shape derivative of .

Theorem 33. For bounded domain , the first-order shape derivative of the Kohn-Vogelius cost functional

in the direction of a perturbation field , where is defined by (13) and the state functions and satisfy the Dirichlet problem (4) and the Neumann problem (6), respectively, is given by

where is the unit exterior normal vector to , is a unit tangent vector to , and is the mean curvature of .

Proof. First we consider the functionals defined on the reference domain and perturbed domains

Let and . Note that if , then we have . Hence

By this together with Lemma 16, we can write as follows:

Then we write as

Using Lemma 11, the symmetry of , and noting that as we have

Hence,

To manipulate we use the identity as

is manipulated as follows:

Applying Theorems 22 and 23 yields . is treated as follows:

Since , the variational equation for and implies

Choosing as test function and applying (65) and (84), can be written as

From this we obtain

and therefore,

Combining and we get

We know from the previous section that and exist in since is of class . Using this smoothness we can now apply Lemma 32 and write (156) as follows:

Since

and , we obtain the first-order shape derivative of :

5. Conclusion

In this paper we derived the explicit form of the first-order Eulerian shape derivative of the Kohn-Vogelius cost functional given by (7) in a rigorous manner. As seen in the presentation, we can avoid working on the shape derivatives of the states and apply their Hölder continuity instead. We employed techniques similar to [9, 13] but it was not necessary to introduce adjoint variables. For the shape derivative of the cost functional to be well defined we observe that we can consider domains with boundaries and we need regularity for the state variables.

Rewriting the first-order shape derivative as , where

we conclude that is shape differentiable at . This is because exists for all and the mapping is linear and continuous with respect to since

We also observe that the shape derivative of which is formulated from the Bernoulli free boundary problem depends on the normal component of the deformation field at the free boundary ; that is, there exists a function defined on the free boundary such that

This agrees with the Hadamard structure theorem [26, 27].

Theorem 34 (see [26, page 318]). Let be a domain with boundary for some integer . Assuming that at a shape gradient of exists. Then there exists a scalar distribution in such that

where is the space of functions from to , and is the normal component of on .

Proof. See [26].

Acknowledgments

The paper is partially supported by the ÖAD—Austrian Agency for International Cooperation in Education and Research for the Technologiestipendien Südostasien (Doktorat) scholarship in the frame of the ASEA-Uninet; the SFB Research Center ‘‘Mathematical Optimization and Applications in Biomedical Sciences” SFB F32; the University of the Philippines Baguio. The first author is thankful to the Institute for Mathematics and Scientific Computing, University of Graz, for funding a presentation of partial results during the 7th European Conference on Elliptic and Parabolic Problems held on May 21–25, 2012 at Gaeta, Italy. He is likewise grateful to Professor Gilbert Peralta for his helpful insights and suggestions. Moreover, he is thanking UP Baguio for giving a Ph.D. incentive grant for writing this paper. Last but not least, the authors would like to thank the referees for devoting time to review this material.

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