Topological sensitivities via a Lagrangian approach for semi-linear problems

In this paper we present a methodology that allows the efficient computation of the topological derivative for semilinear elliptic problems within the averaged adjoint Lagrangian framework. The generality of our approach should also allow the extension to evolutionary problems and other nonlinear problems. Our strategy relies on a rescaled differential quotient of the averaged adjoint state variable which we show converges weakly to a function satisfying an equation defined in the whole space. A unique feature and advantage of this framework is that we only need to work with weakly converging subsequences of the differential quotient. This allows the computation of the topological sensitivity within a simple functional analytic framework under mild assumptions.


Introduction
Shape functions (also called shape functionals) are real valued functions defined on sets of subsets of the Euclidean space R d . The field of mathematics dealing with the minimisation of shape functions that are constrained by a partial differential equation is called PDE constrained shape optimisation. Numerous applications in the engineering and life sciences, such as the aircraft and car design or electrical impedance/magnetic induction tomography, underline its importance; [24,25]. Among other approaches [9,12,16,31,35] the topological derivative approach [10,19,34] constitutes an important tool to solve shape optimisation problems for which the final topology of the shape is unknown. We refer to the monograph [31] and references therein for applications of this approach.
The idea of the topological derivative is to study the local behaviour of a shape function J with respect to a family of singular perturbations (Ω ǫ ). Two important singular perturbations are obtained by translating and scaling of an inclusion ω which contains the origin by ω ǫ (z) := z + ǫω; then the singular perturbations are given by Ω ǫ := Ω ∪ ω ǫ (z) for z ∈ Ω c and Ω ǫ := Ω \ ω ǫ (z) for z ∈ Ω. Both singular perturbations are examples of the class of perturbations K. Sturm called dilatations that are considered in [13]. The topological derivative of a shape function J with respect to perturbations (Ω ǫ ) is defined by where ℓ : [0, τ] → R, τ > 0, is an appropriate function depending on the perturbation chosen.
If Ω is perturbed by a family of transformations Φ ǫ := Id+ǫV : R d → R d generated by a Lipschitz vector field V : R d → R d , that is, Ω ǫ := Φ ǫ (Ω), then we can choose ℓ(ǫ) = ǫ and (1.1) reduces to the definition of the shape derivative [35]. So the topological derivative can be seen as an generalisation of the shape derivative. In some cases, notably when shape functions are constrained by elliptic partial differential equations, the topological derivative can be obtained as the singular limit of the shape derivative as presented in the monograph [31, pp. 12]. While the shape derivative can be interpreted as the Lie derivative on a manifold, the topological derivative is merely a semi-differential defined on a cone, which makes its computation a challenging topic; see [13]. The goal of this paper is to give a coincide way to compute topological sensitivities for the following class of semilinear problems. Given a bounded domain D ⊂ R d , d ∈ {2, 3}, with Lipschitz boundary ∂ D we want to find the topological derivative of the objective function in an open set Ω ⊂ D subject to u = u Ω solves the semilinear transmission problem − div(β 1 ∇u + ) + ̺ 1 (u + ) = f 1 in Ω where u + , u − denote the restriction of u to Ω and D \ Ω, respectively. The function ν denotes the outward pointing unit normal field along ∂ Ω. The technical assumptions for the matrix valued functions β 1 , β 2 and the scalar functions j, ̺ 1 , ̺ 2 , f 1 , f 2 will be introduced in Section 4. A related work is [31, Ch. 10, pp. 277], which is based on the research article [26], where a semilinear problem without transmission conditions in a Hölder space setting is studied. There are two main approaches to compute topological derivatives for PDE constrained shape functions. A typical and general strategy to obtain the topological sensitivity is to derive the asymptotic expansion of the partial differential equation with respect to the singular perturbation of the shape [29,30]. For our problem above this would amount to prove that an expansion of the form (see [31, p. 280]) u ǫ (x) = u(x) + ǫK 1 (ǫ −1 x) + ǫ 2 (K 2 (ǫ −1 x) + u ′ (x)) + ǫ (x) (1.2) exists. Here K 1 , K 2 are so-called called boundary layer correctors, which solve certain exterior boundary value problems and u ′ is called regular corrector and solves a linearised system. The function u ǫ denotes the solution to (S) for the singular perturbed domain Ω ǫ and ǫ (x) is an appropriate remainder. However, the proof of an expansion like (1.2) can technically involved and depends very much on the problem; [26].
A second approach to compute the topological derivative is presented in [5] and based on a perturbed adjoint equation, see also [5,6,11,22,23] and [28]. A key of this method is to prove where K 1 is the same as in (1.2), Q is the solution to an exterior problem, and 1 ǫ , 2 ǫ are appropriate remainder that have to go to zero in some function space. Here p ǫ is the solution to a certain perturbed adjoint equation depending on the derivative of J; see [5]. As a by-product of this approach one obtains the topological sensitivity for non-transmission type problems where Neumann boundary conditions on the inclusion are imposed. However, the proof of the expansions (1.3), particularly for nonlinear problems, can be technically involved and necessitate knowledge of the asymptotic behaviour of Q and K 1 at infinity.
In this paper we will show that neither the expansion (1.2) nor (1.3) are necessary to obtain the topological sensitivity for (S). For this purpose, we use a Lagrangian approach which uses the averaged adjoint variable q ǫ [15,36,37]. The key ingredient, which leads to the existence of the topological derivative of (C), is the convergence property where Q is the same function as in (1.3). The averaged adjoint variable reduces to the usual adjoint in the unperturbed situation, that is, q 0 = q = p = p 0 . We emphasise that the weak convergence property (1.4) is a relaxation of (1.2) and (1.3), since no remainder estimates are necessary. In addition no further knowledge about the asymptotic behaviour of Q at infinity is needed. We will demonstrate that the proof of (1.4) is constructive in that it reveals the equation Q must satisfy. This is particularly important when dealing with other more complicated nonlinear equations, e.g., quasilinear equations. We will show that our strategy also allows, with minor changes, to treat the extremal case where β 1 , ̺ 1 , f 1 = 0, i.e., the transmission problem (S) reduces to a semilinear equation with homogeneous Neumann boundary conditions on ∂ Ω. Compared to previous works we can prove the existence of the topological derivative under milder assumptions on the regularity of the inclusion.

Notation and definitions
Notation for derivatives Let (ǫ, u, q) → G(ǫ, u, q) : [0, τ]×X ×Y → R be a function defined on real normed vector spaces X , Y , and τ > 0. When the limits exist we use the following notation: (1.6) The notation t ց 0 means that t goes to 0 by strictly positive values.

Miscellaneous notation
the Sobolev conjugate of p. Given a normed vector space V we denote by (V, R) the space of linear and continuous functions on V . We denote by B δ (x) the ball centred at x with radius δ > 0 and setB δ (x) := B δ (x).

Lagrangians and infimum
The following material can be found in [15]. We begin with the definition of a Lagrangian function.
Definition 2.1. Let X and Y be vector spaces and τ > 0. A parametrised Lagrangian (or short Lagrangian) is a function (2.1) The next definition formalises the notion of state and perturbed state variable associated with G.
The key ingredient is the so-called averaged adjoint equation. The definition of the averaged adjoint equation requires that the set of states is nonempty: Before we can introduce the averaged adjoint equation we need the following hypothesis.
This follows at once by applying the fundamental theorem of calculus to s → G(ǫ, The following gives the definition of the averaged adjoint equation; see [38]. Definition 2.5. Given ǫ ∈ [0, τ] and (u 0 , u ǫ ) ∈ X (0) × X (ǫ), the averaged adjoint state equation is defined as follows: find q ǫ ∈ X , such that For every triplet (ǫ, u 0 , u ǫ ) the set of solutions of (2.8) is denoted by Y (ǫ, u 0 , u ǫ ) and its elements are referred to as adjoint states for ǫ = 0 and averaged adjoint states for ǫ > 0.
Notice that Y (0, u 0 ) := Y (0, u 0 , u 0 ) is the usual set of adjoint states associated with u 0 , An important consequence of the introduction of the averaged adjoint state is the following identity: for all ǫ ∈ [0, τ], (u 0 , u ǫ ) ∈ X (0) × X (ǫ) and q ǫ ∈ Y (ǫ, u 0 , u ǫ ), This is readily seen by substituting q ǫ into equation (2.7). The following result is an extension of [15,Thm. 3.1]. We refer the reader to [14,38] for further results on the averaged adjoint approach and [36] for more examples involving the shape derivative.

Linear elliptic equations in R d
In preparation for the study of the semilinear problem (S), we first recall existence and uniqueness results for the following exterior problem. Let ω ⊂ R d be an open and bounded set, and let ζ ∈ R d be a vector. Given a suitable vector space V of functions R d → R we consider: find Here A : R d → R d×d is a measurable, uniformly coercive (not necessarily symmetric) matrixvalued functions, that is, there are constants M 1 , M 2 > 0, such that The well-posedness of (3.1) can be achieved by several choices of V . The most popular ones are weighted Sobolev spaces; see [17]. In the next section we discuss a more straight forward choice for V .

Solution in the Beppo-Levi space
Then the Beppo-Levi space is defined bẏ where /R means that we quotient out the constant functions. We denote by [u] the equivalence classes ofḂ L(R d ). The Beppo-Levi space is equipped with the norm The Beppo-Levi space is a Hilbert space (see [17,32] and also [8]) and Proof. This is a direct consequence of the theorem of Lax-Milgram.
As shown in [32], in dimension d ≥ 3 every equivalence class [u] ofḂ L(R d ) contains an element u 0 ∈ [u] that is in turn contained in the Banach space equipped with the norm and from the Gagliardo-Nirenberg-Sobolev inequality; see [32]. As a result for d ≥ 3 we can replace the Beppo-Levi space by E 2 (R d ) and can even consider a more general problem. (3.9) Proof. A proof can be found in the appendix.

Relation to weighted Sobolev spaces
Since the exterior equation (3.1) is, as we will see later, of paramount importance for the first topological derivative we review here an alternative choice for the space V , namely, a weighted Sobolev Hilbert space. We follow the presentation of [7], where a more general situation than the following is considered. For this purpose we introduce the weight function w ∈ L 1 (R d ) defined by where γ d := d 2 + δ and δ ∈ (0, 1/2) is arbitrary, but fixed. Since the weight satisfies |w| p ≤ |w| The weight w is chosen in such a way that the set of constant functions on R d are contained in and equip this space, as in [7], with the quotient norm . Therefore existence of a solution to (3.1) follows directly from the theorem of Lax-Milgram.
In the following lemma let us a agree that the Sobolev conjugate of 2 in dimension two is given by Therefore the Hölder conjugate of 2 * /2 is given by 2 * 2 * −2 = d/2 and Hölder's inequality yieldŝ Since d ≥ 2 we deduce wu ∈ L 2 (R d ) and since by definition also ∇u ∈ L 2 (R d ) d we deduce and the continuity of the embedding follows from (3.15). In case d = 2 we have 2 * = ∞ and thus Hölder's inequality directly gives (3.15) and thus the continuous embedding.

The topological derivative via Lagrangian
In this section we show how Theorem 2.6 of Section 2 can be used to compute the topological derivative for a semilinear transmission problem. Our approach is related to the one of [5] (see also [4]), where also a perturbed adjoint equation is used, too. However the main difference here is that we only need to work with weakly converging subsequences and do not need to know any asymptotic behaviour of the limiting function.

Weak formulation and apriori estimates
In the following exposition we restrict ourselves to the shape function and similarly ̺ Ω is defined by It can be checked that the following proofs remain true when the shape function (4.1) is replaced by (C) from the introduction under the assumption that j is sufficiently smooth. However, in favour of a clearer presentation we use the simplified cost function (4.1). The functions β i , ̺ i and f i are specified in the following assumption. The extremal case where β 1 , ̺ 1 , f 1 are zero will be discussed in the last section.
, ̺ i (0) = 0 and the monotonicity condition Notice that since for x ∈ D the matrix β Ω (x) is either equal to β 1 (x) or β 2 (x) and in view of the bound (4.5), we have Similarly, in view of the monotonicity property (4.6) and ̺ i (0) = 0, we get Proof. (i) Our assumptions imply that we can apply [39, Theorem 4.5] which gives the existence of a solution to (4.2) and also the apriori bound (4.9). As pointed out in this reference the constant C is independent of the nonlinearity ̺ Ω .  Although we restrict ourselves to Dirichlet boundary conditions in (S) other boundary conditions, e.g., Neumann boundary conditions, can be considered as well. This only requires minimal changes in the following analysis and we will make remarks at the relevant places.

The parametrised Lagrangian
From now on we fix: • an open and bounded set ω • an open set Ω ⋐ D and a point z ∈ D \ Ω, To simplify notation we will often write ω ǫ instead of ω ǫ (z). Let X = Y = H 1 0 (D) and introduce the Lagrangian G : where we use the abbreviations We are now going to verify that Hypotheses (H0)-(H4) are satisfied with ℓ(ǫ) = |ω ǫ |. Moreover, we will determine the explicit form of R(u, p).

Remark 4.3 (Removing an inclusion).
We only treat the case of "adding" a hole here, i.e., Ω ǫ := The second case of "removing" a hole, i.e., Ω ǫ := Ω \ ω ǫ (z) for z ∈ Ω can be dealt with in the same way.

Analysis of the perturbed state equation
The perturbed state equation reads: find (4.14) Henceforth we write u := u 0 to simplify notation. Since (4.14) is precisely (4.2) with Ω = Ω ǫ , we infer from Lemma 4.1 that (4.14) admits a unique solution. This means that E(ǫ) = {u ǫ } is a singleton and thus E(ǫ) = X (ǫ) and Hypothesis (H0) is satisfied. From this and Assumption 1 we also infer that Hypothesis (H1) is satisfied. We proceed by shoing a Hölder-type estimate for (u ǫ ).

Analysis of the averaged adjoint equation
We introduce for ǫ ∈ [0, τ] the (not necessarily symmetric) bilinear form b ǫ : Then the averaged adjoint equation (2.8) for the Lagrangian G given by (4.11) reads: find for all ψ ∈ H 1 0 (D). In view of Assumption 1 we have ̺ ′ ǫ ≥ 0 and β ǫ ≥ β m I and thus b ǫ is coercive, As for the state equation, we use the notation q := q 0 . (ii) We find for every z ∈ D \ Ω a number δ > 0, such that Proof. (i) Since b ǫ is coercive and continuous on H 1 0 (D), the theorem of Lax-Milgram shows that (4.20) admits a unique solution.
(ii) The proof is the same as the one for item (ii) of Lemma 4.1 and therefore omitted.
Let us finish this section with the verification of Hypothesis (H3).

Variation of the averaged adjoint equation and its weak limit
The goal of this section is to verify Hypothesis (H4), that is, to show that exists and, if possible, to determine its explicit form. In contrast to previous works we consider the variation of the averaged adjoint state variable which we will show converges weakly to a function Q defined on the whole space R d . For this purpose we need the following definition. Notice that ∪ ǫ>0 D ǫ = R d and that ǫ → D ǫ is monotonically decreasing, that is,  In the same way we extend u ǫ to a functionũ ǫ : R d → R. Notice thatũ ǫ ,q ǫ ∈ H 1 (R d ) for all ǫ > 0. We will use the notation q ǫ :=q ǫ • T ǫ .
Notice that for every ǫ > 0 we have Q ǫ ∈ H 1 (R d ).
Our next task is to show that (Q ǫ ) converges inḂ L(R d ) to a equivalence class of functions [Q] and determine an equation for it. The first step is to prove the following apriori estimates.
Notice that for ǫ > 0 the function Q ǫ belongs to H 1 (R d ), but it is not bounded with respect to ǫ. However, the bound (4.32) is sufficient to show the following key theorem.
Let us now show that ǫQ ǫ 0 in H 1 (R d ) as ǫ ց 0. From the first part of the proof it is clear that ∇(ǫQ ǫ ) 0 in L 2 (R d ) d . To see the weak convergence of (ǫQ ǫ ) in L 2 (R d ) we fix r > 0. Then Poincaré's inequality for a ball yields where (ǫQ ǫ ) r := ffl B r (0) ǫQ ǫ d x denotes the average over the ball B r (0). Since the gradient ∇Q ǫ L 2 (R d ) d is uniformly bounded (see Lemma 4.13), the right hand side of (4.44) goes to zero as ǫ ց 0. But also ǫQ ǫ is bounded in L 2 (R d ) and therefore we find for any null-sequence (ǫ n ) a subsequence (ǫ n k ) andQ ∈ L 2 (R d ), such that ǫ n k Q ǫ n k Q in L 2 (R d ). It is clear that Therefore we obtain from (4.44) together with the weak lower semi-continuity of the L 2 -norm This shows thatQ = (Q) r a.e. on B r (0) and thusQ is constant on B r (0). Since r > 0 was arbitrary, Q must be constant on R d . FurtherQ ∈ L 2 (R d ) impliesQ = 0 and this finishes the proof.
We are now ready to compute R(u, q) and thereby verify the second part of Hypothesis (H2).

Lemma 4.15. We have
where [Q] is the solution to (4.35).

Topological derivative and polarisation matrix
Topological derivative Now we are in a position to formulate our main result. In the previous sections we have checked that Hypotheses (H0)-(H4) of Theorem 2.6 are satisfied for the Lagrangian G given by (4.11). Therefore Theorem 4.16 can be applied and we obtain the following result.
Theorem 4.16. The topological derivative of J at Ω in z ∈ D \ Ω is given by where and where Q depends on z and is the solution to (4.35).

K. Sturm
Next we rewrite the term R(u, q) with the help of the so-called polarisation matrix. For this purpose we fix z ∈ D \ Ω in the following and denote by [Q ζ ], ζ ∈ R d , the solution to (3.6) with A := A ω := β 1 (z)χ ω + β 2 (z)χ R d \ω . Also we denote by Q ζ an arbitrary representative of [Q ζ ].
Definition 4.17. The matrix representing the linear averaging operator is called weak polarisation matrix and will be denoted z ∈ R d×d . Notice that this matrix depends on β 1 (z) and β 2 (z).
We use the term weak polarisation matrix here, because it is defined via the weak formulation (3.6) and therefore does not require any regularity assumptions on ∂ ω or Ω. We give another definition of a polarisation matrix later and relate it to the weak polarisation matrix. We also refer to [33]
Further properties of the polarisation matrix Next we derive further properties of the polarisation matrix. Furthermore we relate our polarisation matrix to previous definitions. We refer the reader to [2] for further information on polarisation matrices. Lemma 4.19. If β 2 (z) = β ⊤ 2 (z) and β 1 (z) = β ⊤ 1 (z), then the polarisation matrix is symmetric, that is, z = ⊤ z .
Proof. We compute for the (i, j)-entry of the polarisation matrix:  This shows the symmetry.
The polarisation matrix is also positive definite (even in the nonsymmetric case).
Lemma 4.20. The matrix z is positive definite.
Proof. Let w = (w 1 , . . . , w d ) ∈ R d be an arbitrary vector. Set W := d i=1 w i Q e i . Then we readily check using (4.55), (4.56) This shows that z is positive semidefinite. Suppose now w is such that w · z w = 0. Then, in view of (4.56), we must have  Suppose from now on β 1 = γ 1 I and β 2 = γ 2 I for γ 1 , γ 2 > 0. We select Q ζ ∈ [Q ζ ] and suppose that it can be represented by a single layer potential: there is a function h ζ ∈ C(∂ ω), such that where E denotes the fundamental solution of u → −∆u; [21,Chap. 3]. It is readily checked using (4.58) that |Q ζ (x)| = O(|x| 1−d ).

Definition 4.21.
The strong polarisation matrix is the matrix˜ z = (˜ z ) i j ∈ R d×d with entries (4.59) The strong and weak polarisation matrices are related as shown in the following lemma.

Lemma 4.22.
Assume that ∂ ω is C 2 . Then we have Proof. At first we obtain by partial integration, noting that e i = ∇x i , , in view of (4.35) d x. (4.61) Next we express ∂ ν Q e j in terms of h e j . For this recall (see, e.g., [21]) that the jump condition is satisfied. In addition we get from (4.35), Combining (4.62) and (4.63) we obtain Inserting this expression into (4.61) yields This is equivalent to formula (4.60), since by Gauss's divergence theorem Remark 4.23. In some cases, see, e.g., [3,5,27], the polarisation matrix can be computed explicitly: for instance when β 1 = γ 1 I, β 2 = γ 2 I, β 1 , β 2 > 0, and ω is a circle or more generally an ellipse. However for general inclusions ω the exterior equation (4.35) has to be solved numerically in order to evaluate formula (4.50).

The extremal case of void material
In this last section we discuss the extremal situation in which β 1 = 0, ̺ 1 = 0 and f 1 = 0 in (4.2). This case corresponds to the insertion of a hole with Neumann boundary conditions imposed on the inclusion; see [26]. Since the extremal case is similar to the considerations from the previous section, we will only work out the main differences in detail.

Problems setting
We suppose as before that D ⊂ R d is a bounded Lipschitz domain. For a simply connected domain Ω ⋐ D with Lipschitz boundary ∂ Ω, we consider the shape function where H 1 This setting corresponds to the limiting case of (4.2) in which β 1 = 0, ̺ 1 = 0 and f 1 = 0. The rest of this section is dedicated to the computation of the topological sensitivity of J at Ω = with respect to the inclusion ω (which will be specified below), i.e., We will see that almost all steps are the same as in the last section with two main differences. The first main difference being that X (ǫ) is not a singleton and that we have to introduce a new equation on the inclusion, which requires a more detailed explanation and a thorough analysis. The second difference concerns the required assumptions on the regularity of the inclusion ω. While in the previous section it was sufficient to assume that ω is merely an open set, here we strengthen the assumption and assume that ω is a simply connected Lipschitz domain.

Assumption 2.
We assume that either (a) β 2 ∈ R d×d is symmetric, positive definite and ̺ 2 satisfies Assumption 1, (b) and it is bounded, or (b) β 2 satisfies Assumption (1), (a) and ̺ 2 satisfies Assumption 1, (b) and additionally ̺ ′ 2 > λ for some λ > 0, is satisfied. In both cases we assume Under these assumptions we can prove, using similar arguments as in Lemma 4.1, that (5.2) admits a unique solution and that there is a constant C > 0 (depending on Ω), such that for r > d/2 close enough to d/2. Moreover, for every z ∈ D \ Ω, we find δ > 0, such that u Ω ∈ H 3 (B δ (z)).

Lemma 5.1.
There is a constant C > 0, such that for all small ǫ > 0, Proof. We first establish an estimate for u ǫ − u on ω ǫ . For this purpose we fix a bounded domain S ⊂ D containing ω. We note that the difference e ǫ (x) := u ǫ (T ǫ (x))−u(T ǫ (x)) satisfies − div(β 2 • T ǫ ∇e ǫ ) = 0 on ω and e ǫ = u ǫ (T ǫ (x)) − u(T ǫ (x)) on ∂ ω. Hence by standard elliptic regularity and the trace theorem we find for all λ ∈ R. Since the quotient norms on the spaces H 1 (ω)/R and H 1 (S \ ω) are equivalent to the seminorms |v| H 1 (ω) := ∇v L 2 (ω) d and |v| H 1 (S\ω) := ∇v L 2 (S\ω) d , respectively, we conclude Therefore estimating the right hand side and changing variables shows A fortiori using (5.13) and a similar argument shows that (5.12) implies This finishes the first step of the proof. We now establish an estimate for the right hand side of (5.13). Following the steps of Lemma 4.4 we find for all ϕ ∈ H 1 0 (D). Let us first assume that Assumption 2, (a) holds. Fixǭ > 0 and let 0 < ǫ <ǭ. Changing variables in (5.15) yields (recalling that we denote byũ ǫ the extension of u ǫ to R d ) Sinceǭ > 0 is arbitrary, this shows that K ǫ 0 weakly inḂL(R d \ ω). But this means that K ǫ must be bounded inḂL(R d \ ω) and hence we find C > 0, such that ∇K ǫ L 2 (R d \ω) d ≤ C or equivalently after changing variables ∇(u ǫ −u) L 2 (D\ω ǫ ) ≤ Cǫ d/2 . Combining this estimate with (5.13) and using Poincaré's inequality gives (5.11).

Analysis of the averaged adjoint equation
We introduce for ǫ ∈ [0, τ] the (not necessarily symmetric) bilinear form b ǫ : Then the averaged adjoint equation (2.8) for the Lagrangian G given by (5.5) reads: for for all ψ ∈ H 1 0 (D). In view of Assumption 1 we have ̺ ′ 2 ≥ 0 and β 2 ≥ β m I and thus b ǫ satisfies, for all ψ ∈ H 1 0 (D) and ǫ ∈ [0, τ]. As for the state equation, we use the notation q := q 0 .
The previous lemma shows that Y (ǫ, u, u ǫ ) = {q ∈ H 1 0 (D) : q = q ǫ a.e. on D \ ω ǫ } and thus Hypothesis (H2) is satisfied. In the same way as done in (5.9) we extend the restriction q ǫ | D\ω ǫ (which is unique) to a function q ǫ ∈ H 1 0 (D) by solving the following Dirichlet problem: find (5.20) With this function we define again It is clear that q ǫ ∈ Y (ǫ, u, u ǫ ). We proceed with a Hölder-type estimate for the extension ǫ → q ǫ .

Lemma 5.3.
There is a constant C > 0, such that for all small ǫ > 0, Proof. The proof is the same as the one of Lemma 4.6 and therefore omitted.
It is readily checked that Hypothesis (H3) is satisfied, too. Proof. Since f 2 ∈ C(B δ (z)) and u, q ∈ C 1 (B δ (z)) for a small δ > 0, we can repeat the steps of the proof of Lemma 4.8.

Variation of the averaged adjoint equation and its weak limit
The next step is to consider the variation of the averaged adjoint state. The variation of the averaged adjoint variable, denoted Q ǫ , is defined as in Definition 4.12.
The following is the analogue of Lemma 4.13 with the main difference that we have an additional equation which gives information of Q inside the inclusion ω.
Lemma 5.5. There is a constant C > 0, such that for all small ǫ > 0, Proof. We follow the steps of Lemma 4.13, but use Lemmas 5.3,5.1 instead Lemmas 4.6,4.4.
Proof. It follows from Lemma 5.5 that for every null-sequence (ǫ n ) there is a subsequence (indexed the same) and Q ∈ BL(R d ) such that (5.25a) and (5.25b) holds for this subsequence. Now using the same arguments as in the proof of Theorem 4.14 shows that Q satisfies (5.26a). The uniqueness of Q| R d \ω follows directly from (5.26a). To prove (5.26b) note that Q ǫ n satisfieŝ ω β(T ǫ n (x))∇ψ · ∇Q ǫ n d x = 0 for all ψ ∈ H 1 0 (ω). (5.27) Using (5.25a) and (5.25b) we may pass to the limit n → ∞ which shows that Q satisfies (5.26b).
Since Q| ∂ ω is uniquely determined from (5.26a) also (5.26b) admits a unique solution, because it is a Dirichlet problem with boundary values Q| ∂ ω .
We are now ready to compute R(u, q).
where [Q] is the solution to (5.26b).
Proof. The proof follows the lines of Lemma 4.15 and Lemma 5.1.
Collecting all previous results we see that Theorem 2.6 can be applied and we obtain the following result.

K. Sturm
Theorem 5.8. The topological derivative of J given by (5.1) in z ∈ D is given by and and Q depends on z and is the unique solution to (5.26a).
Let Q ζ denote the solution to (5.26a)-(5.26b) for fixed z ∈ D and for ζ ∈ R d . Since Q ζ depends linearly on ζ we can proceed as in Subsection 4.6 and introduce a polarisation matrix ∈ R d×d (depending on β 2 (z)) such that ζ = ffl ω ∇Q ζ d x to simplify (5.29). Finally in the same way done as in Lemmas 4.19,4.20 we can show that is symmetric if β 2 is symmetric and that it is always positive definite. Since the considerations are almost identical with the ones of Subsection 4.6 the details are left to the reader.

Concluding remarks
In this paper we showed that the Lagrangian averaged adjoint framework of [15] provides an efficient and fairly simple tool to compute topological derivatives for semilinear problems. We illustrated that using standard apriori estimates for the perturbed states and averaged adjoint variables are sufficient to obtain the topological sensitivity under comparatively mild assumptions on the inclusion. Our work also provides a second examples (the first was given by [14]) for which the R term in [15,Thm. 3.1] is not equal to zero and thus underlines the flexibility of this theorem.
There are several problems that remain open for further research. Firstly, it would be interesting to consider quasilinear equations, but also other types of equations, such as Maxwell's equation. Secondly, an important point we have not addressed here is the topological derivative when Dirichlet boundary conditions are imposed on the inclusion. This case is know to be much different from the Neumann case and needs further investigations.
Using the NSG inequality we can estimate as follows where C := min{ 1 4 C 2 N M 1 , 1 4 M 1 }. On the other hand using again the NSG inequality yields Combining (6.10) and (6.11) yields (6.7) with C 1 = C/C and C 2 = F (E p ,R) .

K. Sturm
Recall the Gagliardo-Nirenberg-Sobolev inequality (short NSG inequality) u L p * (R d ) ≤ C N ∇u L p (R d ) (6.12) valid for all u ∈ C ∞ c (R d ). The constant C N does not depend on the support of the function u. Notice also that for p = d the inequality fails. Thanks to Lemma 6.2 we know that C ∞ c (R d ) is dense in E p (R d ). Hence it follows that (6.12) holds for all test functions u ∈ E p (R d ). For instance for d = 3 and E 2 (R 3 ) we have u L 6 (R 3 ) ≤ C ∇u L 2 (R 3 ) . (6.13) Lemma 6.2. For all 1 < p < d the space (E p (R d ), · E p ) is a Banach space. For every sequence (u n ) in E p (R d ) we find a subsequence (u n k ) and an element u ∈ E p (R d ), such that u n k u weakly in L p * (R d ) as n → ∞, ∇u n k ∇u weakly in L p (R d ) d as n → ∞. (6.14) Moreover, C ∞ c (R d ) is dense in E p (R d ). Proof. Let (u n ) be a bounded sequence in E p (R d ). Since the L p (R d )-spaces are reflexive for all p ∈ (1, ∞), we find elements η ∈ L p * (R d ) and ζ ∈ L p (R d ) d and a subsequence (u n k ), such that u n k η weakly in L p * (R d ) as n → ∞, ∇u n k ζ weakly in L p (R d ) d as n → ∞. for all ϕ ∈ C ∞ c (R d ). Now we pass to the limit in (6.16) and obtain for all ϕ ∈ C ∞ c (R d ), which proves the claim. Since a linear and continuous functional on a Banach space is continuous if and only if it is weakly continuous the claim follows.
To prove the completeness of E p (R d ) let (u n ) be a Cauchy sequence in E p (R d ). Then (u n ) is a Cauchy sequence in L p * (R d ) and (∇u n ) is a Cauchy sequence in L p (R d ) d . Since (u n ) is a Cauchy sequence in L p * (R d ) and (∇u n ) is a Cauchy sequence in L p (R d ) we find elements η ∈ L p * (R d ) and ζ ∈ L p (R d ) d so that u n → ζ strongly in L 2 * (R d ) and ∇u n → ζ strongly in L p (R d ) d . Now we can follow the previous argumentation to show that ∇η = ζ which shows that η ∈ E p (R d ) and thus shows that E p (R d ) is complete.
Let us now show the density of C ∞ c (R d ) in E p (R d ). As shown in [1, Thm. 3.22, p. 68] it suffices to show every u ∈ E p (R d ) with bounded support can be approximated by function in C ∞ c (R d ). Suppose that the support of u is compact. Denote by u ǫ := ̺ ǫ * u the standard mollification of u with a mollifier ̺ ǫ ∈ C ∞ (R d ), ǫ > 0; see [1, pp. 36]. Then u ǫ ∈ L p * (R d ) and ∂ i u ǫ = ̺ ǫ * ∂ i u ∈ L 2 (R d ). Moreover according to [1,Thm. 2.29,p. 36] we have lim ǫց0 u ǫ − u L 2 * (R d ) = 0 and lim ǫց0 ∂ x i (u ǫ − u) L 2 (R d ) = 0. This finishes the proof.