Variational approach to shape derivatives

DOI:10.1051/cocv:2008002 www.esaim-cocv.org

VARIATIONAL APPROACH TO SHAPE DERIVATIVES

Kazufumi Ito

, Karl Kunisch

and Gunther H. Peichl

Abstract. A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable.

In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given.

Mathematics Subject Classification.49Q10, 90C31 Received June 27, 2006. Revised December 19, 2006.

Published online February 7, 2008.

1. Introduction

We propose a framework for characterizing the shape derivative for optimization problems of the form minJ(u,Ω,Γ)

subject to E(u,Ω) = 0, (1.1)

where E denotes a partial differential equation depending on a state variable u and a reference domain Ω, and J stands for a cost functional depending, besides uand Ω, on a codimension one manifold Γ, which may constitute part of the boundary of Ω or lie inside of Ω. This topic has been widely analyzed in the past and is covered in the well-know lecture notes [12] and in the monographs [6,9,10,13,16], for example.

The method for computing the shape derivative that we propose is quite elementary and more direct than most common techniques. A widely used approach relies on differentiating the reduced functional ˆJ = J(u(Ω,Γ),Ω,Γ), which is treated as a composite mapping consisting of (Ω,Γ) → u(Ω,Γ) and (u,Ω,Γ) → J(u,Ω,Γ). As a consequence shape differentiability of the state variable is essential in this method. In an al- ternative approach the partial differential equation is realized in a Lagrangian formulation, see [6], for example.

In short, our approach can be described as follows. We pass to the limit with respect to the class of admissible perturbations in an efficiently arranged version of the difference quotient of the costJ with respect

Keywords and phrases. Shape derivative.

∗ K. Kunisch and G.H. Peichl were supported in part by “SFB Mathematical Optimization and Applications in Biomedical Sciences”.

1 Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA;kito@unity.ncsu.edu

2 Institute for Mathematics and Scientific Computing, Karl-Franzens-University Graz, 8010 Graz, Austria;

karl.kunisch@uni-graz.at; gunther.peichl@uni-graz.at

Article published by EDP Sciences c EDP Sciences, SMAI 2008

to the geometry perturbation. The constraintE(u,Ω) = 0 is observed by introducing an appropriately defined adjoint equation. In this process, differentiability of the state with respect to the geometric quantities is not used. In fact, we only require H¨older continuity with exponent greater ¹₂ ofuwith respect to the geometric data.

On a technical level we utilize well-known results from the method of mapping and on the differentiation of functionals with respect to geometric quantities.

For comparison we briefly discuss an example using the “chain rule” approach. Consider the cost functional minΓ J(u,Ω,Γ)≡min

Γ

1 2

Γ

u²dΓ (1.2)

subject to the constraintE(u,Ω) = 0 which is given by the mixed boundary value problem

−Δu=f, in Ω, (1.3)

u= 0, on Γ₀, (1.4)

∂u

∂n=g, on Γ. (1.5)

Here the boundary ∂Ω of the domain Ω is the disjoint union of a fixed part Γ₀ and the unknown part Γ. A formal differentiation leads to the shape derivative of the cost functional

dJ(u,Ω,Γ)h=

Γ

uu_ΩdΓ +1 2

Γ

∂u²

∂n +κu²

h·ndΓ (1.6)

where u_Ω denotes the shape derivative of the solution u of (1.3) at Ω with respect to a deformation field h which realizes the feasible perturbations of the reference domain Ω andκstands for the curvature of Γ. For a thorough discussion of the details we refer to [6,13]. Differentiating formally the constraintE(u,Ω) = 0 with respect to the domain one obtains thatu_Ω satisfies

−Δu_Ω= 0, in Ω,

u_Ω= 0, on Γ₀, (1.7)

∂u_Ω

∂n = div_Γ(h·n∇Γu) +

f+ ∂g

∂n+κg

h·n, on Γ,

where div_Γ,∇Γstand for the tangential divergence, respectively the tangential gradient. Introducing a suitably defined adjoint variable and using (1.7) the first term on the right hand side of (1.6) can be manipulated in such a way thatdJ(u,Ω,Γ)hcan be represented in the form required by the Zolesio-Hadamard structure theorem [6]

dJ(u,Ω,Γ)h=

Γ

Gh·ndΓ. (1.8)

We emphasize that the kernel Gdoes not involve the shape derivative u_Ω any more. Although u_Ω is only an intermediate quantity a rigorous analysis requires to justify the formal steps in the preceding discussion. In addition one has to verify that the solution of (1.7) actually is the shape derivative of uin the sense of the definition in [13]. These in itself are nontrivial tasks. Furthermore, u∈ H²(Ω) is not sufficient to justify the formal calculations rendering (1.6) into (1.8). In our approach, however, we utilize only u ∈ H²(Ω) for the characterization of the shape derivative ofJ(u,Ω,Γ). We return to this example in Section3.3.

In Section3.5we provide an example for the situation where the standard chain rule approach is not applicable due to lack of shape differentiability of the state variable, but our approach allows a rigorous computation of the cost with respect to perturbations of the domain.

In summary, the method that we develop enables us to directly calculate the shape derivative of the cost functional without utilizing the shape derivative of the stateuwith respect to the geometric variable. Its main

ingredients include the weak formulation of the state-equation constraint, the method of mappings and the shape derivatives of the cost functionals. As we shall demonstrate in Section 3 it can readily be applied to a general class of shape optimization problems.

In Section 2 we present the proposed general framework to compute the shape derivative for (1.1). Section 3 contains applications to shape optimization constrained by linear elliptic systems, inverse interface problems, the Bernoulli problem, and shape optimization for the Navier Stokes equations.

2. Shape derivative

Consider the shape optimization problem minJ(u,Ω,Γ)≡

Ω

j₁(u) dx+

Γ

j₂(u) ds+

∂Ω\Γj₃(u) ds (2.1)

subject to the constraint

E(u,Ω) = 0 (2.2)

which represents a partial differential equation posed on a domain Ω with boundary∂Ω. Further Γ is a closed co-dimension one manifold which represents part of∂Ω or is strictly inside Ω. We focus on sensitivity analysis of (2.1)–(2.2) with respect to Ω and Γ. The perturbations of Ω will be such that∂Ω\Γ remains fixed.

To describe the admissible class of geometries, letU ⊂R^dbe a fixed bounded domain withC^1,1-boundary∂U, or convex and Lipschitzean boundary, and letD be a domain withC^1,1-boundary Γ :=∂D, satisfying ¯D⊂U. For the reference domain Ω we admit either of three cases

(i) Ω =D (ii) Ω =U (iii) Ω =U\D.¯ Note that

∂Ω = (∂Ω∩Γ) ˙∪(∂Ω\Γ)⊂U∪˙ ∂U. (2.3) Thus the boundary∂Ω for cases (i)–(iii) is given by

(i)’ ∂Ω = Γ∪ ∅= Γ (ii)’ ∂Ω =∅ ∪∂U=∂U (iii)’ ∂Ω = Γ∪∂U.

To introduce the admissible class of perturbations let h∈C^1,1( ¯U ,R^d) with h|∂U = 0 and define for,t∈R, the mappingsF_t:U →R^dby the perturbation of identity

F_t=id+th. (2.4)

Then there existsτ >0 such that F_t(U) =U and F_t is a diffeomorphism for |t|< τ. Defining the perturbed domains

Ω_t=F_t(Ω) and the perturbed manifolds as

Γ_t=F_t(Γ),

it follows that Γ_tis of classC^1,1and ¯Ω_t⊂U for|t|< τ. Note that sinceh|∂U = 0 the boundary ofU remains fixed astvaries, and hence by (2.3)

(∂Ω)_t\Γ_t=∂Ω\Γ, for |t|< τ.

Alternatively to (2.4) the perturbations could be described as the flow determined by the initial value problem ξ(t) =˙ h(ξ(t))

ξ(0) =x, withF_t(x) =ξ(t; 0, x), i.e. by the velocity method.

The Eulerian derivative ofJ at Ω in the direction of the deformation fieldhis defined as dJ(u,Ω,Γ)h= lim

t→0

1 t

J(u_t,Ω_t,Γ_t)−J(u,Ω,Γ)

whereu_tsatisfies the constraint

E(u_t,Ω_t) = 0. (2.5)

The functional J is called shape differentiable at Ω ifdJ(u,Ω,Γ)hexists for allh∈C^1,1( ¯U ,R^d) and defines a continuous linear functional onC^1,1( ¯U ,R^d). Using the method of mappings one transforms the perturbed state constraint (2.5) to the fixed domain Ω. For this purpose define

u^t=u_t◦F_t.

Thenu^t: Ω→R^lsatisfies an equation on the reference domain Ω which we express as

E(u˜ ^t, t) = 0, |t|< τ. (2.6)

We suppress the dependence of ˜Eonh, becausehwill denote a fixed vector field throughout. Because ofF₀=id one obtainsu⁰=uand

E(u˜ ⁰,0) =E(u,Ω). (2.7)

We axiomatize the above description and impose the following assumptions on ˜E, respectivelyE.

(H1) There is a Hilbert spaceXand aC¹-function ˜E:X×(−τ, τ)→X^∗such thatE(u_t,Ω_t) = 0 is equivalent to

E(u˜ ^t, t) = 0 inX^∗, with ˜E(u,0) =E(u,Ω) for allu∈X.

(H2) There exists 0< τ₀≤τ such that for|t|< τ₀ there exists a unique solutionu^t∈X to ˜E(u^t, t) = 0 and

t→0lim

|u^t−u⁰|X

|t|^1/2 = 0.

(H3) E_u(u,Ω)∈ L(X, X^∗) satisfies

E(v,Ω)−E(u,Ω)−E_u(u,Ω)(v−u), ψX^∗×X=O(|v−u|²_X) for everyψ∈X, whereu, v∈X.

(H4) ˜E andE satisfy limt→0

1

tE(u˜ ^t, t)−E(u, t)˜ −E(u^t,Ω) +E(u,Ω), ψX^∗×X= 0 for everyψ∈X, whereu^tanduare the solutions of (2.6), respectively (2.2).

In applications (H4) typically results in an assumption on the regularity of the coefficients in the partial differ- ential equation and on the vector-fieldh. We assume throughout thatX →L²(Ω,R^l) and, in the case thatj₂, j₃ are non-trivial, that the elements ofX admit traces inL²(Γ,R^l), respectivelyL²(∂Ω\Γ,R^l). TypicallyX will be a subspace ofH¹(Ω,R^l) for somel∈N.

With regards to the cost functionalJ we require:

(H5) j_i ∈C^1,1(R^l,R),i= 1,2,3.

As a consequence of (H1)–(H2) we infer that equation (2.5) has a unique solutionu_t which is given by u_t= u^t◦F_t⁻¹. Condition (H5) implies that j₁(u) ∈ L²(Ω), j₁(u) ∈ L²(Ω)^l, j₂(u) ∈ L²(Γ), j₂(u) ∈ L²(Γ)^l and j₃(u)∈L²(∂Ω\Γ),j₃(u)∈L²(∂Ω\Γ)^l foru∈X. Hence the cost functionalJ(u,Ω,Γ) is well defined for every u∈X.

Lemma 2.1. There is a constant c >0, such that

|ji(v)−j_i(u)−j_i(u)(v−u)|_L¹ ≤c|v−u|²_X hold for all v,u∈X,i= 1,2,3.

Proof. Forj₁ the claim follows from

Ω

|j1(v)−j₁(u)−j₁(u)(v−u)|dx≤

Ω

₁

0

|j₁

u(x) +s(v(x)−u(x))

−j₁(u(x))|ds|v(x)−u(x)|dx

≤ L

2|v−u|²_L2 ≤c|v−u|²_X,

whereL >0 is the Lipschitz constant for j₁. The same argument is valid also forj₂andj₃. Subsequently we use the following notation

I_t= detDF_t, A_t= (DF_t)^−T, w_t=I_t|Atn|,

where DF_t is the Jacobian of F_t and n denotes the outer normal unit vector to Ω. We require additional regularity properties of the transformationF_t. LetI= [−τ0, τ₀] withτ₀sufficiently small.

F₀=id t→F_t∈C(I, C^1,1( ¯U ,R^d)) t→F_t∈C¹(I, C¹( ¯U ,R^d)) t→F_t⁻¹∈C(I, C¹( ¯U ,R^d)) t→I_t∈C¹(I, C( ¯U)) t→A_t∈C(I, C( ¯U ,R^d×d))

t→w_t∈C(I, C(Γ)) (2.8)

d

dtF_t|t=0=h d

dtF_t⁻¹|t=0=−h d

dtDF_t|t=0=Dh d

dtDF_t⁻¹|t=0= d

dt(A_t)^T|t=0=−Dh d

dtI_t|t=0= divh d

dtw_t|t=0= div_Γh.

The limits defining the derivatives att= 0 exist uniformly inx∈U¯. The surface divergence div_Γ is defined for ϕ∈C¹( ¯U ,R^d) by

div_Γϕ= divϕ|Γ−(Dϕ n)·n.

The properties (2.8) are easily verified if F_t is specified by perturbing the identity. As a consequence of (2.8) there exists α >0 such that

I_t(x)≥α, x∈U .¯ (2.9)

We furthermore recall the following transformation theorem where we already utilize (2.9).

Lemma 2.2. (1) Let ϕ_t∈L¹(Ω_t), then ϕ_t◦F_t∈L¹(Ω)and

Ωt

ϕ_tdx_t=

Ω

ϕ_t◦FtdetDF_tdx.

(2) Leth_t∈L¹(Γ_t), thenh_t◦F_t∈L¹(Γ)and

Γt

h_tdΓ_t=

Γ

h_t◦F_tdetDF_t|(DF_t)^−Tn|dΓ.

As the main result of this paper we now formulate the representation of the Eulerian derivative ofJ. Theorem 2.1. Assume that (H1)–(H5)hold, thatF satisfies (2.8)and that the adjoint equation

Eu(u,Ω)ψ, pX^∗×X−(j₁(u), ψ)_Ω−(j₂(u), ψ)_Γ−(j₃(u), ψ)_∂Ω\Γ= 0, ψ∈X, (2.10) admits a unique solutionp∈X, whereuis the solution to (2.2). Then the Eulerian derivative ofJ atΩin the direction h∈C^1,1( ¯U ,R^d)exists and is given by

dJ(u,Ω,Γ)h=−d

dtE(u, t), p˜ X^∗×X|t=0+

Ω

j₁(u) divhdx+

Γ

j₂(u) div_Γhds. (2.11)

Proof. Referring to (H2) letu^t, u∈X satisfy

E(u˜ ^t, t) =E(u,Ω) = 0 (2.12)

for|t|< τ₀. Thenu_t=u^t◦F_tis the solution of (2.5). Utilizing Lemma2.2one therefore obtains 1

t(J(u_t,Ω_t,Γ_t)−J(u,Ω,Γ)) = 1 t

Ω

I_tj₁(u^t)−j₁(u) dx+1

t

Γ

w_tj₂(u^t)−j₂(u) ds +1

t

∂Ω\Γ(j₃(u^t)−j₃(u))ds

= 1 t

Ω

I_t(j₁(u^t)−j₁(u)−j₁(u)(u^t−u)) + (I_t−1)j₁(u)(u^t−u)

+j₁(u)(u^t−u) + (I_t−1)j₁(u) dx +1

t

Γ

w_t(j₂(u^t)−j₂(u)−j₂(u)(u^t−u)) + (w_t−1)j₂(u)(u^t−u)

+j₂(u)(u^t−u) + (w_t−1)j₂(u) ds +1

t

∂Ω\Γ(j₃(u^t)−j₃(u)−j₃(u)(u^t−u))ds +1

t

∂Ω\Γj₃(u)(u^t−u)ds. (2.13)

Lemma2.1 and (2.8) result in the estimates

Ω

I_t(j₁(u^t)−j₁(u)−j₁(u)(u^t−u)) dx

≤c|u^t−u|²_X

Γ

w_t(j₂(u^t)−j₂(u)−j₂(u)(u^t−u)) ds

≤c|u^t−u|²_X,

∂Ω\Γ(j₃(u^t)−j₃(u)−j₃(u)(u^t−u))ds

≤c|u^t−u|²_X (2.14)

wherec >0 does not depend ont. Employing the adjoint statepone obtains (j₁(u), u^t−u)_Ω+ (j₂(u), u^t−u)_Γ+ (j₃(u), u^t−u)_∂Ω\Γ=Eu(u,Ω)(u^t−u), pX^∗×X

=− E(u^t,Ω)−E(u,Ω)−E_u(u,Ω)(u^t−u), pX^∗×X

− E(u˜ ^t, t)−E(u, t)˜ −E(u^t,Ω) +E(u,Ω), pX^∗×X

− E(u, t)˜ −E(u,˜ 0), pX^∗×X, (2.15) where we used (2.12). We estimate the ten additive terms on the right hand side of (2.13). Terms one, five and nine converge to zero by (2.14) and (H2). Terms two and six converge to 0 by (2.8) and (H2). For terms four and eight ones uses (2.8). The claim (2.11) now follows by passing to the limit in terms three, seven and ten

using (2.15), (H3), (H2), (H4) and (H1).

Remark 2.1. The proof of Theorem 2.1 reveals that the assumption (H1) can be considerably weakened. In fact all that is needed is the following.

(H1’) There is a Hilbert spaceX and a function ˜E:X×(−τ, τ)→X^∗ such that (a) E(u_t,Ω_t) = 0 is equivalent to

E(u˜ ^t, t) = 0 inX^∗, with ˜E(u,0) =E(u,Ω) for allu∈X.

(b) The mapping v→ E(v,˜ 0), pX^∗×X is differentiable atv=uandt→ E(u, t), p˜ X^∗×X is differen- tiable att= 0, whereuis the solution ofE(u,Ω) = 0 andpsatisfies the adjoint equation (2.10).

To check (H2) in specific applications the following result will be useful. It relies on

(H6)

⎧⎪

⎨

⎪⎩

the linearized equation

Eu(u,Ω)δu, ψX^∗×X=f, ψX^∗×X, ψ∈X admits a unique solutionδu∈X for everyf ∈X^∗.

Note that this condition is more stringent than the assumption of solvability of the adjoint equation in Theorem2.1which requires solvability only for a specific right hand side.

Proposition 2.1. Assume that (2.2)admits a unique solutionuand that (H6)is satisfied. Then (H2) holds.

Proof. Letu∈X be the unique solution of (2.2). In view of E˜_u(u,0) =E_u(u,Ω)

(H6) implies that ˜E_u(u,0) is bijective. The claim follows from the implicit function theorem.

Computing the derivative _dt^dE(u, t), p˜ X^∗×X|t=0in (2.11) can be facilitated by transforming the expressions E(u, t) and˜ pback toE(u◦F_t⁻¹,Ω_t) andp◦F_t⁻¹, and utilizing the following well known differentiation rules:

Lemma 2.3 [6]. (1)Let f ∈C(I, W^1,1(U))and assume that f_t(0) exists inL¹(U), then d

dt

Ωt

f(t, x) dx|t=0=

Ω

f_t(0, x) dx+

Γ

f(0, x)h·nds.

(2) Letf ∈C(I, W^2,1(U))and assume that f_t(0)exists inW^1,1(U). Then d

dt

Γt

f(t, x) ds|t=0 =

Γ

f_t(0, x) ds+

Γ

∂

∂nf(0, s) +κf(0, s)

h·nds, whereκstands for the additive curvature ofΓ.

The first part of the theorem is valid also for domains Ω with Lipschitz continuous boundary. The addi- tionalC^1,1regularity is used as sufficient condition in [6] for (2).

In the examples belowf(t,·) will be typically given by expressions of the form

μv◦F_t⁻¹, μ∂_i(v◦F_t⁻¹)∂_j(w◦F_t⁻¹), v◦F_t⁻¹w◦F_t⁻¹∂_i(z◦F_t⁻¹)

whereμ∈H¹(U) andv,zandw∈H²(U) are extensions of elements inH²(Ω). The assumptions of Lemma2.3 can be verified using the following result.

Lemma 2.4 [13]. (1) Let u∈L^p(U)thent→u◦F_t⁻¹∈C(I, L^p(U)),1≤p <∞. (2) Letu∈H²(U)then t→u◦F_t⁻¹∈C(I, H²(U)).

(3) Letu∈H²(U)then _dt^d(u◦F_t⁻¹)|t=0 exists inH¹(U) and is given by d

dt(u◦F_t⁻¹)|t=0=−(Du)h.

As a consequence we note that _dt^d∂_i

(u◦F_t⁻¹)

|_t=0exists inL²(U) and is given by d

dt∂_i

(u◦F_t⁻¹)

|_t=0=−∂_i(Du h), i= 1, . . . , d.

In the next section ∇ustands for (Du)^T where uis either a scalar or vector valued function. To enhance readability we use two symbols for the inner product in R^d, (x, y) respectivelyx·y. The latter will only be utilized in the case of nested inner products.

3. Examples

Throughout the examples section it is assumed that (H5) is satisfied and that the regularity assumptions of Section 2 forD,Ω andU hold. IfJ does not depend on Γ we writeJ(u,Ω) in place ofJ(u,Ω,Γ).

3.1. Elliptic Dirichlet boundary value problem

J(u,Ω) =

Ω

j₁(u) dx subject to the constraint

(μ∇u,∇ψ)Ω−(f, ψ)_Ω= 0, (3.1)

where X = H₀¹(Ω), f ∈ H¹(U) and μ ∈ C¹( ¯U ,R^d×d) such that μ(x) is symmetric and uniformly positive definite. Here Ω =D and Γ =∂Ω. ThusE(u,Ω) :X →X^∗ is given by

E(u,Ω), ψX^∗×X= (μ∇u,∇ψ)Ω−(f, ψ)_Ω. The equation on the perturbed domain is determined by

E(ut,Ω_t), ψ_tX_t^∗×Xt =

Ωt

(μ∇ut,∇ψt) dx_t−

Ωt

f ψ_tdx_t

=

Ω

(μ^tA_t∇u^t, A_t∇ψ^t)I_tdx−

Ω

f^tψ^tI_tdx≡ E(u˜ ^t, t), ψ^tX^∗,X, (3.2) for any ψ_t ∈ X_t, with u^t = u_t◦ F_t, μ^t = μ◦F_t, f^t = f◦F_t and X_t = H₀¹(Ω_t). Here we used that

∇ut = (A_t∇u^t)◦F_t⁻¹ and Lemma 2.3. (H1) is a consequence of (2.8), (3.2) and the smoothness of μ and f. Since (3.1) admits a unique solution and (H6) holds, Proposition 2.1 implies (H2). Since ˜E is linear in u assumption (H3) follows. For the verification of (H4) observe that

E(u˜ ^t, t)−E(u, t)˜ −E(u^t,Ω) +E(u,Ω), ψX^∗×X = ((μ^tI_tA_t−μ)∇(u^t−u), A_t∇ψ)Ω

+ (μ∇(u^t−u),(A_t−I)∇ψ).

Hence (H4) follows from differentiability ofμ, (2.8) and (H2).

In view of Theorem2.1we have to compute _dt^dE(u, t), p˜ X^∗×X|t=0for which we use the representation on Ω_t in (3.2). Recall that the solutionuof (3.1) as well as the adjoint statep, defined by

(μ∇p,∇ψ)Ω= (j₁(u), ψ)_Ω, ψ∈H₀¹(Ω) (3.3) belong to H²(Ω)∩H₀¹(Ω). Since Ω∈C^1,1 (actually Lipschitz continuity of the boundary would suffice), uas well as p can be extended to functions in H²(U), which we again denote by the same symbol. Therefore Lemma2.3(1) and Lemma2.4entail that

d

dtE(u, t), p˜ X^∗×X|t=0= _dt^d(

Ωt(μ∇(u◦F_t⁻¹),∇(p◦F_t⁻¹)) dx_t−

Ωtf p◦F_t⁻¹dx_t)|t=0

=

Γ(μ∇u,∇p) (h, n) ds+

Ω

(μ∇(−∇u·h),∇p) +

μ∇u,∇(−∇p·h)

+f(∇p, h) dx.

(3.4)

Note that∇u·has well as ∇p·hdo not belong to H₀¹(Ω) but they are elements ofH¹(Ω). Therefore Green’s theorem implies

Ω

(μ∇(−∇u·h),∇p) + (μ∇u,∇(−∇p·h)) +f(∇p, h) dx=

Ω

div(μ∇p) (∇u, h) dx−

Γ

(μ∇p, n)(∇u, h) ds +

Ω

(div(μ∇u) +f) (∇p, h) dx−

Γ

(μ∇u, n) (∇p, h) ds

=−

Ω

j₁(u) (∇u, h) dx−2

Γ

(μn, n)∂u

∂n

∂p

∂n(h, n) ds. (3.5) Above we used the strong form of (3.1) and (3.3) inL²(Ω) as well as the identities

(μ∇u, n) = (μn, n)∂u

∂n (∇u, h) = ∂u

∂n(h, n)

(together with the ones with uand pinterchanged) which follow from u, p∈ H₀¹(Ω). Applying Theorem2.1 results in

dJ(u,Ω)h=−d

dtE(u, t), p˜ X^∗,X|t=0+

Ω

j₁(u) divhdx

=

Γ

(μn, n)∂u

∂n

∂p

∂n(h, n) ds+

Ω

(j₁(u)(∇u, h) +j₁(u) divh) dx

=

Γ

(μn, n)∂u

∂n

∂p

∂n(h, n) ds+

Ω

div(j₁(u)h) dx, and the Stokes theorem yields the final result

dJ(u,Ω)h=

Γ

(μn, n)∂u

∂n

∂p

∂n+j₁(u)

(h, n) ds.

Remark 3.1. If we were to be content with a representation of the shape variation in terms of volume integrals we could take the expression for _dt^dE(u, t)|˜ t=0 given in (3.4) and bypass the use of Green’s theorem in (3.5).

The regularity requirement on the domain then results fromu∈H²(Ω),p∈H²(Ω). In [1] the shape derivative in terms of the volume integral is referred to as the weak shape derivative, whereas the final form in terms of the boundary integrals is called the strong shape derivative.

3.2. Inverse interface problem

We consider an inverse interface problem which is motivated by electrical impedance tomography. Let U = Ω = (−1,1)×(−1,1) and ∂U = ∂Ω. Further let the domain D = Ω⁻, with ¯Ω⁻ ⊂ U, represent the inhomogeneity of the conducting medium and set Ω⁺ =U \Ω¯⁻. We assume that Ω⁻ is a simply connected domain of classC^1,1with boundary Γ which represents the interface between Ω⁻ and Ω⁺. The inverse problem consists of identifying the unknown interface Γ from measurements z which are taken on the boundary∂U. This can be formulated as

minJ(u,Ω)≡

∂U

(u−z)²ds (3.6)

subject to the constraint

−div(μ∇u) = 0, in Ω⁻∪Ω⁺, [u] = 0,

μ ∂u

∂n⁻

= 0 on Γ, (3.7)

∂u

∂n =g, on∂U,

whereg∈H^1/2(∂U),z∈L²(∂U), with

∂Ug=

∂Uz= 0, with [v] =v⁺−v⁻ on Γ andn^+/− standing for the unit outer normals to Ω^+/−. The conductivityμis given by

μ(x) =

μ⁻ x∈Ω⁻, μ⁺ x∈Ω⁺,

for some positive constantsμ⁻andμ⁺. In the context of the general framework of Section 2 we havej₁=j₂= 0 and j₃ = (u−z)². Clearly (3.7) admits a unique solutionu∈ H¹(U) with

∂Uu= 0. Its restrictions to Ω⁺ and Ω⁻ will be denoted by u⁺ andu⁻, respectively. It turns out that the regularity ofu^± is better than the one of u.

Proposition 3.1. Let ΩandΩ^± be as described above. Then the solution u∈H¹(U)of (3.7)satisfies u^± ∈H²(Ω^±).

Proof. Let Γ_H be the smooth boundary of a domain Ω_H with Ω⁻⊂Ω_H⊂Ω_H⊂U.

Thenu|ΓH ∈H^3/2(Γ_H). The problem

−div(μ⁺∇uH) = 0 in U\ΩH

∂uH

∂n =g on ∂U, u_H =u|ΓH on Γ_H, has a unique solutionu_H∈H²(U\ΩH) withu^H=u⁺|ΩH. Therefore,

b:=u|∂U =u_H|∂U∈H^3/2(∂U).

Then the solutionuto (3.7) coincides with the solution to

⎧⎨

⎩

−div(μ∇u) = 0, in Ω⁻ ∪ Ω⁺, [u] = 0,[μ_∂n^∂u₋] = 0 on Γ,

u=b, on ∂U.

We now argue thatu^±∈H²(Ω^±).Letu_b ∈H²(U) denote the solution to −Δub= 0 in U

u_b=b on∂U.

Definew∈H₀¹(U) as the unique solution to the interface problem

⎧⎪

⎨

⎪⎩

−div(μ∇w) = 0 in Ω

[w] = 0, [μ_∂n^∂w−] =−[μ_∂n^∂u−^b] on Γ.

w= 0 on ∂U.

(3.8)

Thenu_b ∈H²(Ω) implies [μ_∂n^∂u₋^b]∈H^1/2(Γ). By [2] equation (3.8) has a unique solutionw∈H₀¹(U) with the additional regularityw^±∈H²(Ω^±). Consequentlyu=w+u_b satisfiesu|∂Ω =gandu^±∈H²(Ω^±), as desired.

In an analogous wayp^±∈H²(Ω^±).

To consider the inverse problem (3.6), (3.7) within the general framework of Section 2 we set X = {v ∈ H¹(U) : _v

∂U = 0}and define

E(u,Ω), ψX^∗×X= (μ∇u,∇ψ)U−(g, ψ)_∂U, respectively

E˜(u, t), ψX^∗×X= (μ^tA_t∇u, At∇ψIt)_U−(g, ψ)_∂U

= (μ⁺∇(u◦F_t⁻¹),∇(ψ◦F_t⁻¹))_Ω+

t + (μ⁻∇(u◦F_t⁻¹),∇(ψ◦F_t⁻¹))_Ω−

t −(g, ψ)_∂U.

Note that the boundary term is not affected by the transformationF_t since the deformation field hvanishes on∂U. The adjoint state is given by

−div(μ∇p) = 0, in Ω⁻∪Ω⁺, [p] = 0,

μ ∂p

∂n⁻

= 0 on Γ, (3.9)

∂p

∂n= 2(u−z) on∂U, respectively

(μ∇p,∇ψ)U = 2(u−z, ψ)_∂U, forψ∈X. (3.10)

Assumption (H4) requires us to consider 1

t|E(u˜ ^t−u, t)−E(u^t−u,Ω), ψ| ≤1 t

Ω⁺

|(μ⁺I_tA_t∇(u^t−u), A_t∇ψ)−(μ⁺∇(u^t−u),∇ψ)|dx +1

t

Ω⁻

|(μ⁻I_tA_t∇(u^t−u), A_t∇ψ)−(μ⁺∇(u^t−u),∇ψ)|dx

≤μ⁺

Ω⁺

1

t(I_tA_t−I)∇(u^t−u), A_t∇ψ dx +μ⁺

Ω⁺

|(∇(u^t−u),1

t(A_t−I)∇ψ)|dx +μ⁻

Ω⁻

1

t(I_tA_t−I)∇(u^t−u), A_t∇ψ dx +μ⁻

Ω⁻

|(∇(u^t−u),1

t(A_t−I)∇ψ)|dx.

The right hand side of this inequality converges to 0 as t → 0 by (2.8). The remaining assumptions can be verified as in Example3.1and thus Theorem2.1is applicable. By Proposition3.1the restrictionsu^±=u|_Ω^±, p^±=p|_Ω^± satisfyu^±,p^± ∈H²(Ω^±). Using Lemma2.3we find that

d

dtE(u, t), p˜ X^∗×X|t=0=

∂Ω⁺

(μ⁺∇u⁺,∇p⁺)(h, n⁺) ds−

Ω⁺

(μ⁺∇(∇u⁺·h),∇p⁺) dx

−

Ω⁺

(μ⁺∇u⁺,∇(∇p⁺·h)) dx+

∂Ω⁻

(μ⁻∇u⁻,∇p⁻)(h, n⁻) ds

−

Ω⁻

(μ⁻∇(∇u⁻·h),∇p⁻) dx−

Ω⁻

(μ⁻∇u⁻,∇(∇p⁻·h)) dx

=

Γ

[μ∇u,∇p](h, n⁺) ds−

Ω⁺

μ⁺

∇(∇u⁺·h),∇p⁺) + (∇u⁺,∇(∇p⁺·h) dx

−

Ω⁻

μ⁻

∇(∇u⁻·h),∇p⁻) + (∇u⁻,∇(∇p⁻·h) dx.

Applying Green’s formula as in Example 3.1 (observe that (∇u, h), (∇p, h) ∈/ H¹(U)) together with (3.9) results in

−

Ω⁺

(μ⁺(∇(∇u⁺·h),∇p⁺)) dx−

Ω⁻

(μ⁻(∇(∇u⁻·h),∇p⁻)) dx

=

Ω⁺

div(μ⁺∇p⁺)(∇u⁺, h) dx+

Ω⁻

div(μ⁻∇p⁻)(∇u⁻, h) dx

−

∂Ω⁺

(μ⁺∇p⁺, n⁺)(∇u⁺, h) ds−

∂Ω⁻

(μ⁻∇p⁻, n⁻)(∇u⁻, h) ds

=−

Γ

μ ∂p

∂n⁺(∇u, h)

ds.

In the last step we utilizeh= 0 on∂U. Similarly we obtain

−

Ω⁺

(μ⁺(∇u⁺,∇(∇p⁺·h))) dx−

Ω⁻

(μ⁻(∇u⁻,∇(∇p⁻·h))) dx=−

Γ

μ ∂u

∂n⁺(∇p, h)

ds.

Collecting terms results in dJ(u,Ω)h=−

Γ

[μ(∇u,∇p)] (h, n⁺) ds+

Γ

μ ∂p

∂n⁺(∇u, h)

+

μ ∂u

∂n⁺(∇p, h)

ds.

The identity

[ab] = [a]b⁺+a⁻[b] =a⁺[b] + [a]b⁻ implies

[ab] = 0 if [a] = [b] = 0.

Hence the transition conditions

μ ∂u

∂n⁺

= ∂u

∂τ

= 0

μ ∂p

∂n⁺

= ∂p

∂τ

= 0, (3.11)

where _∂τ^∂ stands for the tangential derivative imply

μ ∂p

∂n⁺(∇u, h)

=

μ ∂p

∂n⁺

∂u

∂n⁺(h, n⁺) +μ ∂p

∂n⁺

∂u

∂τ(h, τ)

=

μ ∂p

∂n⁺

∂u

∂n⁺(h, n⁺)

+

μ ∂p

∂n⁺

∂u

∂τ(h, τ)

=

μ ∂p

∂n⁺

∂u

∂n⁺

(h, n⁺), and analogously

μ ∂u

∂n⁺(∇p, h)

=

μ ∂p

∂n⁺

∂u

∂n⁺

(h, n⁺),

which entails

dJ(u,Ω)h=−

Γ

[μ(∇u,∇p)] (h, n⁺) ds+ 2

Γ

μ ∂p

∂n⁺

∂u

∂n⁺

(h, n⁺) ds

=−

Γ

μ∂u

∂τ

∂p

∂τ

(h, n⁺) ds+

Γ

μ ∂u

∂n⁺

∂p

∂n⁺

(h, n⁺) ds

=−

Γ

[μ]∂u

∂τ

∂p

∂τ (h, n⁺) ds+

Γ

μ ∂u

∂n⁺

∂p

∂n⁺

(h, n⁺) ds.

In view of (3.11) this can be rearranged as

−[μ]∂u

∂τ

∂p

∂τ +

μ ∂u

∂n⁺

∂p

∂n⁺

=−μ⁺∂u

∂τ

∂p

∂τ +μ⁻∂u

∂τ

∂p

∂τ +μ⁺∂u⁺

∂n⁺

∂p⁺

∂n⁺ −μ⁻∂u⁻

∂n⁺

∂p⁻

∂n⁺

=−μ⁺ ∂u

∂τ

∂p

∂τ +1 2

∂u⁺

∂n⁺

∂p⁻

∂n⁺ + ∂u⁻

∂n⁺

∂p⁺

∂n⁺

+μ⁻ ∂u

∂τ

∂p

∂τ +1 2

∂u⁻

∂n⁺

∂p⁺

∂n⁺ + ∂u⁺

∂n⁺

∂p⁻

∂n⁺

=−1 2[μ]

(∇u⁺,∇p⁻) + (∇u⁻,∇p⁺) which gives the representation

dJ(u,Ω,Γ)h=−1 2

Γ

[μ]

(∇u⁺,∇p⁻) + (∇u⁻,∇p⁺)

(h, n⁺) ds

=−

Γ

[μ](∇u⁺,∇p⁻) (h, n⁺) ds.

3.3. Elliptic systems

Here we consider a domain Ω =U\D, where ¯D⊂U and the boundaries∂U and Γ =∂D are assumed to be C^1,1regular.

We consider the optimization problem

minJ(u,Ω,Γ)≡

Ω

j₁(u) dx+

Γ

j₂(u) ds whereuis the solution of the elliptic system

E(u,Ω), ψX^∗×X=

Ω

a(x,∇u,∇ψ)−(f, ψ) dx−

Γ

(g, ψ) ds= 0 (3.12)

in X ={v∈H¹(Ω)^l:v|∂U = 0}. Above∇ustands for (Du)^T. We requiref ∈H¹(U)^l and thatg is the trace of a given functionG∈H²(U)^l. Furthermore we assume thata: ¯U×R^d×d×R^d×d satisfies

(1) a(·, ξ, η) is continuously differentiable on ¯U for everyξ,η∈R^d×d.

(2) a(x,·,·) defines a bilinear form onR^d×d×R^d×d which is uniformly bounded in x∈U¯. (3) a(x,·,·) is uniformly coercive for allx∈U¯.

In the case of linear elasticityais given by

a(x,∇u,∇ψ) =λtre(u) tre(ψ) + 2μ e(u) :e(ψ),