# Shape derivatives of boundary integral operators in electromagnetic scattering. Part II : Application to scattering by a homogeneous dielectric obstacle

**ABSTRACT** We develop the shape derivative analysis of solutions to the problem of

scattering of time-harmonic electromagnetic waves by a bounded penetrable

obstacle. Since boundary integral equations are a classical tool to solve

electromagnetic scattering problems, we study the shape differentiability

properties of the standard electromagnetic boundary integral operators. The

latter are typically bounded on the space of tangential vector fields of mixed

regularity $TH\sp{-1/2}(\Div_{\Gamma},\Gamma)$. Using Helmholtz decomposition,

we can base their analysis on the study of pseudo-differential integral

operators in standard Sobolev spaces, but we then have to study the G\^ateaux

differentiability of surface differential operators. We prove that the

electromagnetic boundary integral operators are infinitely differentiable

without loss of regularity. We also give a characterization of the first shape

derivative of the solution of the dielectric scattering problem as a solution

of a new electromagnetic scattering problem.

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**ABSTRACT:**We construct of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. We show regularity properties in the frame of Schauder spaces for the corresponding single layer potentials.Integral Equations and Operator Theory 06/2012; 76(1). · 0.71 Impact Factor - SourceAvailable from: Martin Costabel[Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators.Integral Equations and Operator Theory 05/2011; · 0.71 Impact Factor

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arXiv:1002.1541v2 [math.NA] 22 Feb 2010

Shape derivatives of boundary integral operators in

electromagnetic scattering

M. Costabel1, F. Le Louër2

1IRMAR, University of Rennes 1

2POEMS, INRIA-ENSTA, Paris

Abstract

We develop the shape derivative analysis of solutions to the problem of scattering

of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since

boundary integral equations are a classical tool to solve electromagnetic scattering

problems, we study the shape differentiability properties of the standard electro-

magnetic boundary integral operators.To this end, we start with the Gâteaux

differentiability analysis with respect to deformations of the obstacle of boundary

integral operators with pseudo-homogeneous kernels acting between Sobolev spaces.

The boundary integral operators of electromagnetism are typically bounded on the

space of tangential vector fields of mixed regularity TH−1

decomposition, we can base their analysis on the study of scalar integral operators

in standard Sobolev spaces, but we then have to study the Gâteaux differentiability

of surface differential operators. We prove that the electromagnetic boundary in-

tegral operators are infinitely differentiable without loss of regularity and that the

solutions of the scattering problem are infinitely shape differentiable away from the

boundary of the obstacle, whereas their derivatives lose regularity on the boundary.

We also give a characterization of the first shape derivative as a solution of a new

electromagnetic scattering problem.

2(divΓ,Γ). Using Helmholtz

Keywords :

operators, shape derivatives, Helmholtz decomposition.

Maxwell’s equations, boundary integral operators, surface differential

Contents

1 The dielectric scattering problem4

2 Boundary integral operators and main properties5

3 Some remarks on shape derivatives10

4 Gâteaux differentiability of pseudo-homogeneous kernels13

5 Shape differentiability of the solution 22

1

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Introduction

Consider the scattering of time-harmonic electromagnetic waves by a bounded obstacle

Ω in R3with a smooth and simply connected boundary Γ filled with an homogeneous

dielectric material. This problem is described by the system of Maxwell’s equations,

valid in the sense of distributions, with two transmission conditions on the boundary or

the obstacle guaranteeing the continuity of the tangential components of the electric and

magnetic fields across the interface. The transmission problem is completed by the Silver-

Müller radiation condition at infinity (see [23] and [24]). Boundary integral equations are

an efficient method to solve such problems for low and high frequencies. The dielectric

scattering problem is usually reduced to a system of two boundary integral equations for

two unknown tangential vector fields on the interface (see [6] and [24]). We refer to [9] and

[10] for methods developed by the authors to solve this problem using a single boundary

integral equation.

Optimal shape design with the modulus of the far field pattern of the dielectric scat-

tering problem as goal is of practical interest in some important fields of applied mathe-

matics, as for example telecommunication systems and radars. The utilization of shape

optimization methods requires the analysis of the dependency of the solution on the shape

of the dielectric scatterer. An explicit form of the shape derivatives is required in view

of their implementation in a shape optimization algorithms such as gradient methods or

Newton’s method.

In this paper, we present a complete analysis of the shape differentiability of the

solution of the dielectric scattering problem using an integral representation. Even if nu-

merous works exist on the calculus of shape variations [14, 25, 26, 31, 32], in the framework

of boundary integral equations the scientific literature is not extensive. However, one can

cite the papers [27], [29] and [28], where R. Potthast has considered the question, starting

with his PhD thesis [30], for the Helmholtz equation with Dirichlet or Neumann boundary

conditions and the perfect conductor problem, in spaces of continuous and Hölder con-

tinuous functions. Using the integral representation of the solution, one is lead to study

the Gâteaux differentiability of boundary integral operators and potential operators with

weakly and strongly singular kernels.

The natural space of distributions (energy space) which occurs in the electromagnetic

potential theory is TH−1

are in the Sobolev space H−1

main difficulties: On one hand, to be able to construct shape derivatives of the solution –

which is given in terms of products of boundary integral operators and their inverses – it is

imperative to prove that the derivatives are bounded operators between the same spaces

as the boundary integral operators themselves. On the other hand, the very definition of

shape differentiability of operators defined on TH−1

Our approach consists in using the Helmholtz decomposition of this Hilbert space. In this

way, we split the analysis in two main steps: First the Gâteaux differentiability analysis

of scalar boundary integral operators and potential operators with pseudo-homogeneous

kernels, and second the study of derivatives with respect to smooth deformations of the

obstacle of surface differential operators in the classical Sobolev spaces.

2(divΓ,Γ), the set of tangential vector fields whose components

2(Γ) and whose surface divergence is in H−1

2(Γ). We face two

2(divΓ,Γ) poses non-trivial problems.

2

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This work contains results from the thesis [21] where this analysis has been used to

develop a shape optimization algorithm of dielectric lenses in order to obtain a prescribed

radiation pattern.

The paper is organized as follows:

In section 1 we define the scattering problem of time-harmonic electromagnetic waves

at a dielectric interface and the appropriate spaces. In section 2 we recall some results

about trace mappings and boundary integral operators in electromagnetism, following the

notation of [6, 24]. We then give an integral representation of the solution following [9].

In section 3, we introduce the notion of shape derivative and its connection to Gâteaux

derivatives. We also recall elementary results about differentiability in Fréchet spaces.

The section 4 is dedicated to the Gâteaux differentiability analysis of a class of bound-

ary integral operators with respect to deformations of the boundary. We generalize the

results proved in [27, 29] for the standard acoustic boundary integral operators, to the

class of integral operators with pseudo-homogenous kernels. We also give higher order

Gâteaux derivatives of coefficient functions such as the Jacobian of the change of variables

associated with the deformation, or the components of the unit normal vector. These re-

sults are new and allow us to obtain explicit forms of the derivatives of the integral

operators.

The last section contains the main results of this paper: the shape differentiability

properties of the solution of the dielectric scattering problem. We begin by discussing the

difficulties of defining the shape dependency of operators defined on the shape-dependent

space TH−1

(see [11]) on the boundary of smooth domains. We then analyze the differentiability of

a family of surface differential operators. Again we prove their infinite Gâteaux differen-

tiability and give an explicit expression of their derivatives. These results are new and

important for the numerical implementation of the shape derivatives. Using the chain

rule, we deduce the infinite shape differentiability of the solution of the scattering prob-

lem away from the boundary and an expression of the shape derivatives. More precisely,

we prove that the boundary integral operators are infinitely Gâteaux differentiable with-

out loss of regularity, whereas previous results allowed such a loss [28], and we prove that

the shape derivatives of the potentials are smooth far from the boundary but they lose

regularity in the neighborhood of the boundary.

These new results generalize existing results: In the acoustic case, using the variational

formulation, a characterization of the first Gâteaux derivative was given by A. Kirsch in

[20] for the Dirichlet problem and then for a transmission problem by F. Hettlich in

[15, 16]. R. Potthast used the integral equation method to obtain a characterisation of

the first shape derivative of the solution of the perfect conductor scattering problem.

We end the paper by formulating a characterization of the first shape derivative as

the solution of a new electromagnetic scattering problem. We show that both by directly

deriving the boundary values and by using the integral representation of the solution, we

obtain the same characterization.

2(divΓ,Γ), and we present an altervative using the Helmholtz decomposition

3

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1 The dielectric scattering problem

Let Ω denote a bounded domain in R3and let Ωcdenote the exterior domain R3\Ω. In

this paper, we will assume that the boundary Γ of Ω is a smooth and simply connected

closed surface, so that Ω is diffeomorphic to a ball. Let n denote the outer unit normal

vector on the boundary Γ.

In Ω (resp. Ωc) the electric permittivity ǫi(resp. ǫe) and the magnetic permeability

µi(resp. µe) are positive constants. The frequency ω is the same in Ω and in Ωc. The

interior wave number κiand the exterior wave number κeare complex constants of non

negative imaginary part.

Notation:

For a domain G ⊂ R3we denote by Hs(G) the usual L2-based Sobolev

space of order s ∈ R, and by Hs

loc(G) the space of functions whose restrictions to any

bounded subdomain B of G belong to Hs(B). Spaces of vector functions will be denoted

by boldface letters, thus

Hs(G) = (Hs(G))3.

If D is a differential operator, we write:

Hs(D,Ω)

Hs

loc(D,Ωc)

={u ∈ Hs(Ω) : Du ∈ Hs(Ω)}

{u ∈ Hs

=

loc(Ωc) : Du ∈ Hs

loc(Ωc)}

The space Hs(D,Ω) is endowed with the natural graph norm. When s = 0, this defines

in particular the Hilbert spaces H(curl,Ω) and H(curlcurl,Ω). We denote the L2scalar

product on Γ by ?·,·?Γ.

The time-harmonic Maxwell’s sytem can be reduced to second order equations for the

electric field only. The time-harmonic dielectric scattering problem is then formulated as

follows.

The dielectric scattering problem : Given an incident field Einc∈ Hloc(curl,R3)

that satisfies curlcurlEinc− κ2

Ei∈ H(curl,Ω) and Es∈ Hloc(curl,Ωc) satisfying the time-harmonic Maxwell equations

eEinc= 0 in a neighborhood of Ω, we seek two fields

curlcurlEi− κ2

curlcurlEs− κ2

iEi

eEs

= 0

in Ω,

in Ωc,

(1.1)

= 0

(1.2)

the two transmission conditions,

n × Ei= n × (Es+ Einc)

i(n × curlEi) = µ−1

on Γ

(1.3)

µ−1

en × curl(Es+ Einc)

on Γ

(1.4)

and the Silver-Müller radiation condition:

lim

|x|→+∞|x|

????curlEs(x) ×

x

|x|− iκeEs(x)

????= 0.

(1.5)

4

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The interior and exterior magnetic fields are then given by Hi=

1

iωµiEiand Hs=

1

iωµeEs.

It is well known that this problem admit a unique solution for any positive real values of

the exterior wave number [6, 21, 24].

An important quantity, which is of interest in many shape optimization problems, is

the far field pattern of the electric solution, defined on the unit sphere of R3, by

E∞(ˆ x) = lim

|x|→∞4π|x|Es(x)

eiκe|x|,

with

x

|x|= ˆ x.

2 Boundary integral operators and main properties

2.1 Traces and tangential differential calculus

We use surface differential operators and traces. More details can be found in [8, 24].

For a vector function v ∈ (Ck(R3))qwith k,q ∈ N∗, we note [∇v] the matrix whose

the i-th column is the gradient of the i-th component of v and we set [Dv] =T[∇v]. The

tangential gradient of any scalar function u ∈ Ck(Γ) is defined by

∇Γu = ∇˜ u|Γ−?∇˜ u|Γ· n?n,

and the tangential vector curl by

(2.1)

curlΓu = ∇˜ u|Γ× n,

(2.2)

where ˜ u is an extension of u to the whole space R3. For a vector function u ∈ (Ck(Γ))3,

we note [∇Γu] the matrix whose the i-th column is the tangential gradient of the i-th

component of u and we set [DΓu] =T[∇Γu].

We define the surface divergence of any vectorial function u ∈ (Ck(Γ))3by

divΓu = div ˜ u|Γ−?[∇˜ u|Γ]n · n?,

and the surface scalar curl curlΓrurby

(2.3)

curlΓu = n · (curl ˜ u))

where ˜ u is an extension of u to the whole space R3. These definitions do not depend on

the extension.

Definition 2.1 For a vector function v ∈ (C∞(Ω))3and a scalar function v ∈ C∞(Ω)

we define the traces :

γv = v|Γ,

γDv := (n × v)|Γ(Dirichlet) and

γNκv := κ−1(n × curlv)|Γ(Neumann).

5

Page 6

We introduce the Hilbert spaces Hs(Γ) = γ

For s > 0, the traces

?

Hs+1

2(Ω)

?

, and THs(Γ) = γD

?

Hs+1

2(Ω)

?

γ : Hs+1

2(Ω) → Hs(Γ),

γD: Hs+1

2(Ω) → THs(Γ)

are then continuous. The dual of Hs(Γ) and THs(Γ) with respect to the L2(or L2) scalar

product is denoted by H−s(Γ) and TH−s(Γ), respectively.

The surface differential operators defined here above can be extended to the Sobolev

spaces: The tangential gradient and the tangential vector curl are linear and continuous

from Hs+1(Γ) to THs(Γ), the surface divergence and the surface scalar curl are linear

and continuous from THs+1(Γ) to Hs(Γ).

Definition 2.2 We define the Hilbert space

TH−1

2(divΓ,Γ) =

?

j ∈ TH−1

2(Γ),divΓj ∈ H−1

2(Γ)

?

endowed with the norm

|| · ||TH−1

2(divΓ,Γ)= || · ||TH−1

2(Γ)+ ||divΓ·||H−1

2(Γ).

Lemma 2.3 The operators γD and γN are linear and continuous from C∞(Ω,R3) to

TL2(Γ) and they can be extended to continuous linear operators from H(curl,Ω) and

H(curl,Ω) ∩ H(curlcurl,Ω), respectively, to TH−1

2(divΓ,Γ).

For u ∈ Hloc(curl,Ωc) and v ∈ Hloc(curlcurl,Ωc)) we define γc

same way and the same mapping properties hold true.

Recall that we assume that the boundary Γ is smooth and topologically trivial. For

a proof of the following result, we refer to [3, 8, 24].

Du and γc

Nv in the

Lemma 2.4 Let t ∈ R. The Laplace-Beltrami operator

∆Γ= divΓ∇Γ= −curlΓcurlΓ

(2.4)

is linear and continuous from Ht+2(Γ) to Ht(Γ).

It is an isomorphism from Ht+2(Γ)/R to the space Ht

∗(Γ) defined by

?

u ∈ Ht

∗(Γ) ⇐⇒u ∈ Ht(Γ) and

Γ

u = 0.

This result is due to the surjectivity of the operators divΓand curlΓfrom THt+1(Γ) to

Ht

∗(Γ).

We note the following equalities:

curlΓ∇Γ= 0 and divΓcurlΓ= 0

(2.5)

divΓ(n × j) = −curlΓj and curlΓ(n × j) = divΓj

(2.6)

6

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2.2 Pseudo-homogeneous kernels

In this paper we are concerned with boundary integral operators of the form :

?

where the integral is assumed to exist in the sense of a Cauchy principal value and

the kernel k is weakly singular, regular with respect to the variable y ∈ Γ and quasi-

homogeneous with respect to the variable z = x − y ∈ R3. We recall the regularity

properties of these operators on the Sobolev spaces Hs(Γ), s ∈ R available also for their

adjoints operators:

?

We use the class of weakly singular kernel introduced by Nedelec ([24] p. 176). More

details can be found in [13, 17, 19, 22, 35, 34].

Definition 2.5 The homogeneous kernel k(y,z) defined on Γ×?R3\{0}?is said of class

Definition 2.6 The kernel k ∈ C∞?Γ ×?R3\{0}??is pseudo-homogeneous of class −m

asymptotic expansion when z tends to 0:

KΓu(x) = vp.

Γ

k(y,x − y)u(y)dσ(y), x ∈ Γ

(2.7)

K∗

Γ(u)(x) = vp.

Γ

k(x,y − x)u(y)dσ(y), x ∈ Γ.

(2.8)

−m with m ≥ 0 if

sup

y∈Rdsup

|z|=1

?????

∂|α|

∂yα

∂|β|

∂zβk(y,z)

?????≤ Cα,β, for all multi-index α and β,

∂|β|

∂zβk(y,z) is homogeneous of degree − 2 with respect to the variable z

for all |β| = m and Dm

zk(y,z) is odd with respect to the variable z.

for an integer m such that m ? 0, if for all integer s the kernel k admit the following

k(y,z) = km(y,z) +

N−1

?

j=1

km+j(y,z) + km+N(y,z),

(2.9)

where for j = 0,1,...,N − 1 the function km+jis homogeneous of class −(m + j) and N

is chosen such that km+N is s times differentiables.

For the proof of the following theorem, we refer to [24].

Theorem 2.7 Let k be a pseudo-homogeneous kernel of class −m. The associated oper-

ator KΓgiven by (2.7) is linear and continuous from Hs(Γ) to Hs+m(Γ) for all s ∈ R.

We have similar results for the adjoint operators K∗

Γ.

The following theorem is established in [13].

7

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Theorem 2.8 Let k be a pseudo-homogeneous kernel of class −m. The potential operator

P defined by

?

is continuous from Hs−1

P(u)(x) =

Γ

k(y,x − y)u(y)dσ(y), x ∈ R3\Γ

(2.10)

2(Γ) to Hs+m(Ω) ∪ Hs+m

loc(Ωc) for all positive real number s.

2.3The electromagnetic boundary integral operators

We use some well known results about electromagnetic potentials. Details can be found

in [3, 4, 5, 6, 24].

Let κ be a complex number such that Im(κ) ≥ 0 and let

G(κ,|x − y|) =

eiκ|x−y|

4π|x − y|

be the fundamental solution of the Helmholtz equation

∆u + κ2u = 0.

The single layer potential ψκis given by :

(ψκu)(x) =

?

Γ

G(κ,|x − y|)u(y)dσ(y)x ∈ R3\Γ,

and its trace by

Vκu(x) =

?

Γ

G(κ,|x − y|)u(y)dσ(y)x ∈ Γ.

The fundamental solution is pseudo-homogeneous of class −1 (see [18, 24]). As conse-

quence we have the following result :

Lemma 2.9 Let s ∈ R. The operators

ψκ

: Hs−1

2(Γ) → Hs+1

loc(R3)

Vκ

: Hs−1

2(Γ) → Hs+1

2(Γ)

are continuous.

We define the electric potential ΨEκgenerated by j ∈ TH−1

2(divΓ,Γ) by

ΨEκj := κψκj + κ−1∇ψκdivΓj

This can be written as ΨEκj := κ−1curlcurlψκj because of the Helmholtz equation and

the identity curlcurl = −∆ + ∇div (cf [3]).

We define the magnetic potential ΨMκgenerated by m ∈ TH−1

2(divΓ,Γ) by

ΨMκm := curlψκm.

We denote the identity operator by I.

8

Page 9

Lemma 2.10 The potentials ΨEκet ΨMκare continuous from TH−1

For j ∈ TH−1

2(divΓ,Γ) to Hloc(curl,R3).

2(divΓ,Γ) we have

(curlcurl−κ2I)ΨEκj = 0 and (curlcurl−κ2I)ΨMκm = 0 in R3\Γ

and ΨEκj and ΨMκm satisfy the Silver-Müller condition.

We define the electric and the magnetic far field operators for j ∈ TH−1

element ˆ x of the unit sphere S2of R3by

2(divΓ,Γ) and an

Ψ∞

Eκj(ˆ x) = κ ˆ x ×

??

??

Γ

e−iκˆ x·yj(y)dσ(y)

?

?

× ˆ x,

Ψ∞

Mκej(ˆ x) = iκ ˆ x ×

Γ

e−iκˆ x·yj(y)dσ(y).

(2.11)

These operators are bounded from TH−1

We can now define the main boundary integral operators:

2(divΓ,Γ) to T(C∞(S2))3.

Cκ= −1

2{γD+ γc

D}ΨEκ= −1

2{γN+ γc

N}ΨMκ,

Mκ= −1

2{γD+ γc

D}ΨMκ= −1

2{γN+ γc

N}ΨEκ.

These are bounded operators in TH−1

?

=?−κ n × Vκj + κ−1curlΓVκdivΓj?(x)

and

Mκj(x)

2(divΓ,Γ). We have

Cκj(x) = −κ

Γ

n(x) × (G(κ,|x − y|)j(y))dσ(y) + κ−1

?

Γ

curlx

Γ(G(κ,|x − y|)divΓj(y))dσ(y)

= −

?

Γ

n(x) × curlx(G(κ,|x − y|)j(y))dσ(y)

= (Dκj − Bκj)(x),

with

Bκj(x)=

?

?

Γ

∇xG(κ,|x − y|)(j(y) · n(x))dσ(y),

Dκj(x)=

Γ

(∇xG(κ,|x − y|) · n(x))j(y)dσ(y).

The kernel of Dκis pseudo-homogeneous of class −1 and the operator Mκhas the same

regularity as Dκon TH−1

2(divΓ,Γ), that is compact.

We describe briefly the boundary integral equation method developped by the autors

[9] to solve the dielectric scattering problem.

Boundary integral equation method : This is based on the Stratton-Chu formula,

the jump relations of the electromagnetic potentials and the Calderón projector’s formula

(see [6, 24]).

9

Page 10

We need a variant of the operator Cκdefined for j ∈ TH−1

2(divΓ,Γ) by :

C*

0j = n × V0j + curlΓV0divΓj.

The operator C∗

integral representation of the exterior electric field Es:

0is bounded in TH−1

2(divΓ,Γ). We use the following ansatz on the

Es= −ΨEκej − iη ΨMκeC*

0j in R3\¯Ω

(2.12)

η is a positive real number and j ∈ TH−1

we have the integral representation of the interior field

2(divΓ,Γ). Thanks to the transmission conditions

E1= −1

ρ(ΨEκi{γc

NeEinc+ Nej}) − (ΨMκi{γc

DEinc+ Lej}) in Ω

(2.13)

where ρ =κiµe

κeµi

and

Le= Cκe−iη

?1

2I − Mκe

?

?

C*

0,

Ne=

?1

2I − Mκe

+ iη CκeC*

0.

We apply the exterior Dirichlet trace to the righthandside (2.13). The density j then

solves the following boundary integral equation:

Sj = ρ

?

−1

2I + Mκi

?

Lej + CκiNej = −ρ

?

−1

2I + Mκi

?

γDEinc+ CκiγNκeEincsur Γ.

The operator S is linear, bounded and invertible on TH−1

If we are concerned with the far field pattern E∞of the solution, it suffices to re-

place the potential operators ΨEκeand ΨMκeby the far field operators Ψ∞

respectively.

The solution E(Ω) = (Ei(Ω),Es(Ω)) and the far field pattern E∞(Ω) consists of ap-

plications defined by integrals on the boundary Γ and if the incident field is a fixed data,

these quantities depend on the scatterrer Ω only.

2(divΓ,Γ).

Eκeand Ψ∞

Mκe

3 Some remarks on shape derivatives

We want to study the dependance of any functionals F with respect to the shape of the

dielectric scatterer Ω. The Ω-dependance is highly nonlinear. The standard differential

calculus tools need the framework of topological vector spaces which are locally convex at

least [33], framework we do not dispose in the case of shape functionals. An interesting

approach consists in representing the variations of the domain Ω by elements of a function

space. We consider variations generated by transformations of the form

x ?→ x + r(x)

10

Page 11

of any points x in the space R3, where r is a vectorial function defined (at least) in the

neiborhood of Ω. This transformation deforms the domain Ω in a domain Ωrof boundary

Γr. The functions r are assumed to be a small enough elements of a Fréchet space X in

order that (I + r) is an isomorphism from Γ to

Γr= (I + r)Γ = {xr= x + r(x);x ∈ Γ}.

Since we consider smooth surfaces, in the remaining of this paper, the space X will be

the Fréchet space C∞

k∈N

seminorms (|| · ||k)k∈Nwhere Ck

differentiable functions whose the derivatives are bounded and

b(R3,R3) =

?

Ck

b(R3,R3) undowed with the set of non decreasing

b(R3,R3) with k ∈ N is the space of k-times continuously

||r||k= sup

0≤p≤k

sup

x∈R3

???r(p)(x)

?3, d∞(0,r) < ǫ

???.

For ǫ small enough we set

B∞

ǫ =

?

r ∈

?

C∞(R3)

?

,

where d∞is the metric induced by the seminorms.

We introduce the application

r ∈ B∞

ǫ ?→ FΩ(r) = F(Ωr).

We define the shape derivative of the functional F trough the deformation Ω → Ωξas

the Gâteaux derivative of the application FΩin the direction ξ ∈ X. We write:

DF[Ω;ξ] =∂

∂t|t=0FΩ(tξ).

3.1 Gâteaux differentiability: elementary results

Fréchet spaces are locally convex, metrisable and complete topological vector spaces on

which we can extend any elementary results available on Banach spaces. We recall some

of them. We refer to the Schwarz’s book [33] for more details.

Let X and Y be Fréchet spaces and let U be a subset of X.

Definition 3.1 (Gâteaux semi-derivatives) The application f

have Gâteaux semiderivative at r0∈ U in the direction ξ ∈ X if the following limit exists

and is finite

∂

∂rf[r0;ξ] = lim

t→0

t

: U → Y is said to

f(r0+ tξ) − f(r0)

=∂

∂t??t=0f(r0+ tξ).

Definition 3.2 (Gâteaux differentiability) The application f : U → Y is said to be

Gâteaux differentiable at r0∈ U if it has Gâteaux semiderivatives in all direction ξ ∈ X

and if the map

∂

∂rf[r0;ξ] ∈ Y

is linear and continuous.

ξ ∈ X ?→

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We say that f is continuously (or C1-) Gâteaux differentiable if it is Gâteaux differentiable

at all r0∈ U and the application

∂

∂rf : (r0;ξ) ∈ U × X ?→

∂

∂rf[r0;ξ] ∈ Y

is continuous.

Remark 3.3 Let us come to shape functionals. In calculus of shape variation, we usually

consider the Gâteaux derivative in r = 0 only. This is due to the result : If FΩis Gâteaux

differentiable on B∞

ǫ

then for all ξ ∈ X we have

∂

∂rFΩ[r0;ξ] = DF(Ωr0;ξ ◦ (I + r0)−1) =

∂

∂rFΩr0[0;ξ ◦ (I + r0)−1].

Definition 3.4 (higher order derivatives) Let m ∈ N. We say that f is (m+1)-times

continuously (or Cm+1-) Gâteaux differentiable if it is Cm-Gâteaux differentiable and

r ∈ U ?→

∂m

∂rmf[r;ξ1,...,ξm]

is continuously Gâteaux differentiable for all m-uple (ξ1,...,ξm) ∈ Xm. Then for all

r0∈ U the application

(ξ1,...,ξm+1) ∈ Xm+1?→

∂m+1

∂rm+1f[r0;ξ1,...,ξm+1] ∈ Y

is (m + 1)-linear, symetric and continuous. We say that f is C∞-Gâteaux differentiable

if it is Cm-Gâteaux differentiable for all m ∈ N.

We use the notation

∂m

∂rmf[r0,ξ] =∂m

∂tm??t=0f(r0+ tξ).

?

(3.1)

If it is Cm-Gâteaux differentiable we have

∂m

∂rmf[r0,ξ1,...,ξm] =

1

m!

m

?

p=1

(−1)m−p

1≤i1<···<ip≤m

∂m

∂rmf[r0;ξi1+ ... + ξip].

(3.2)

To determine higher order Gâteaux derivatives it is more easy to use this equality.

The chain and product rules and the Taylor expansion with integral remainder are still

available for Cm-Gâteaux differentiable maps ([33] p. 30). We use the following lemma

to study the Gâteaux differentiability of any applications mapping r on the inverse of an

element in a unitary topological algebra.

Lemma 3.5 Let X be a Fréchet space and Y be a unitary Fréchet algebra. Let U be an

open set of X. Assume that the application f : U → Y is Gâteaux differentiable at r0∈ U

and that f(r) is invertible in Y for all r ∈ U and that the application g : r ?→ f(r)−1

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is continuous at r0. Then g is Gâteaux differentiable at r0and its first derivative in the

direction ξ ∈ X is

∂

∂rf[r0,ξ] = −f(r0)−1◦∂

∂rf[r0,ξ] ◦ f(r0)−1.

(3.3)

Moreover if f is Cm-Gâteaux differentiable then g is too.

Proof.

Let ξ ∈ X and t > 0 small enough such that (r0+ tξ) ∈ U, on a:

g(r0+ tξ) − g(r0) =f(r0)−1◦ f(r0) ◦ f(r0+ tξ)−1− f(r0)−1◦ f(r0+ tξ) ◦ f(r0+ tξ)−1

=f(r0)−1◦ (f(r0) − f(r0+ tξ)) ◦ f(r0+ tξ)−1

=f(r0)−1◦ (f(r0) − f(r0+ tξ)) ◦ f(r0)−1

+f(r0)−1◦ (f(r0) − f(r0+ tξ)) ◦?f(r0+ tξ)−1− f(r0)−1?.

Since g is continuous in r0, we have lim

t→0

Gâteaux differentiable in r0we have

?f(r0+ tξ)−1− f(r0)−1?= 0 and since f is

lim

t→0

f(r0)−1◦ (f(r0) − f(r0+ tξ)) ◦ f(r0)−1

t

= −(f(r0))−1◦∂

∂rf[r0,ξ] ◦ (f(r0))−1.

As a consequence

lim

t→0

g(r0+ tξ) − g(r0)

t

= −(f(r0))−1◦∂

∂rf[r0,ξ] ◦ (f(r0))−1.

?

4 Gâteaux differentiability of pseudo-homogeneous kernels

Let xrdenote an element of Γrand let nrbe the outer unit normal vector to Γr. When

r = 0 we write n0= n. We note again dσ the area element on Γr.

In this section we want to study the Gâteaux differentiability of the application map-

ping r ∈ B∞

ǫ

to the integral operator KΓrdefined for a function ur∈ Hs(Γr) by:

?

and of the application mapping r ∈ B∞

ǫ to the potential operator Prdefined for a function

ur∈ Hs(Γr) by:

?

where kr∈ C∞?Γr×?R3\{0}??is a pseudo-homogeneous kernel of class −m with m ∈ N.

KΓrur(xr) = vp.

Γr

kr(yr,xr− yr)ur(yr)dσ(yr), xr∈ Γr

(4.1)

Prur(x) =

Γr

kr(yr,x − yr)ur(yr)dσ(yr), x ∈ K,

(4.2)

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We want to differentiate applications of the form r ?→ FΩ(r) where the domain of

definition of FΩ(r) varies with r. How do we do?

[25, 27, 29]), is that instead of studying the application

A first idea, quite classical (see

r ∈ B∞

ǫ ?→ FΩ(r) ∈ Ck(Γr)

we consider the application

r ∈ B∞

ǫ ?→ FΩ(r) ◦ (I + r) ∈ Ck(Γ).

An example is r ?→ nr. This point of view can be extended to Sobolev spaces Hs(Γ),

s ∈ R. From now we use the transformation τrwhich maps a function urdefined on Γr

to the function ur◦(I+r) defined on Γ. For all r ∈ B∞

inverse. We have

ǫ, this transformation τradmit an

(τrur)(x) = ur(x + r(x)) and (τ−1

r u)(xr) = u(x).

Then, instead of studying the application

r ∈ B∞

ǫ ?→ KΓr∈ Lc

?Hs(Γr),Hs+m(Γr)?

?Hs(Γ),Hs+m(Γ)?.

we consider the conjugate application

r ∈ B∞

ǫ ?→ τrKΓrτ−1

r

∈ Lc

In the framework of boundary integral equations, this approach is sufficient to obtain the

shape differentability of any solution to scalar boundary value problems [27, 29].

Using the change of variable x ?→ xr= x + r(x), we have for u ∈ Hs(Γ):

?

where Jris the jacobian (the determinant of the Jacobian matrix) of the change of variable

mapping x ∈ Γ to x + r(x) ∈ Γr. The differentiablility analysis of these operators begins

with the jacobian one. We have

τrKrτ−1

r (u)(x) =

Γ

kr(y + r(y),x + r(x) − y − r(y))u(y)Jr(y)dσ(y), x ∈ Γ

Jr= JacΓ(I + r) = ||ωr|| with ωr= com(I + Dr|Γ)n0= det(I + Dr|Γ)T(I + Dr|Γ)−1n,

and the normal vector nris given by

nr= τ−1

r

?

ωr

?ωr?

?

.

The first derivative at r = 0 of these applications are well known [12, 25]. Here we present

one method to obtain higher order derivative.

Lemma 4.1 The application J mapping r ∈ B∞

Gâteaux differentiable and its first derivative at r0is defined for ξ ∈ C∞

ǫ

to the jacobian Jr∈ C∞(Γ,R) is C∞

b(R3,R3) by:

∂J

∂r[r0,ξ] = Jr0(τr0divΓr0τ−1

r0)ξ.

14

Page 15

Proof.

do the proof for hypersurfaces Γ of Rn, n ∈ N, n ≥ 2. We use local coordinate system.

Assume that Γ is parametrised by an atlas (Oi,φi)1≤i≤pthen Γrcan be parametrised by

the atlas (Oi,(I + r) ◦ φi)1≤i≤p. For any x ∈ Γ, let us note e1(x),e2(x),...,en−1(x) the

vector basis of the tangent plane to Γ at x. The vector basis of the tangent plane to Γr

at x + r(x) are given by

We just have to prove the C∞-Gâteaux differentiability of W : r ?→ wr. We

ei(r,x) = [(I + Dr)(x)]ei(x)

for i = 1,...,n − 1.

Thus, we have ωr(x) =

n−1

?

n−1

?

i=1

????

ei(r,x)

i=1

ei(x)

????

. Since the applications r ?→ ei(r,x), for i = 1,...,n−1

are C∞-Gâteaux differentiable, the application W is too. Now want to compute the

derivatives using the formula (3.2). Let ξ ∈ C∞

r0∈ B∞

ǫ

b(Rn,Rn) and t small enough. We have at

∂mW

∂rm[r0,ξ] =∂m

∂tm???t=0

n−1

?

i=1

(I + Dr0+ tDξ)ei(x)

????

n−1

?

i=1

ei(x)

????

.

To simplify this expression one have to note that

[Dξ(x)]ei(x) = [Dξ(x)][(I + Dr0)(x)]−1[(I + Dr0)(x)]ei(x)

= [Dξ(x)][D(I + r0)−1(x + r0(x))][(I + Dr0)(x)]ei(x)

= [(τr0Dτ−1

r0)ξ(x)]ei(r0,x) = [(τr0DΓr0τ−1

r0)ξ(x)]ei(r0,x).

NB: given a (n × n) matrix A we have

n−1

?

i=1

··· × ei−1× Aei× ei+1× ··· = (Trace(A)I −TA)

n−1

?

i=1

ei.

Thus we have with A = [τr0DΓr0τ−1

∂rm[r0,ξ]

r0ξ] and B0= I, B1(A) = Trace(A)I −TA

(#)

W(r0)

∂W

∂r[r0,ξ]

=Jr0(τr0nr0),

?(τr0divΓr0τ−1

[B1(A)ξ]W(r0),

=Jr0

r0)ξ · τr0nr0− [(τr0∇Γr0τ−1

r0)ξ]τr0nr0

?

=

∂mW

∂rm[r0,ξ]=[Bm(A)ξ]W(r0)

m ?

0 for all m ≥ n.

=

i=1

(−1)i+1(m − 1)!

(m − i)![B1(Ai)Bm−i(A)ξ]W(r0) for m = 1,...,n − 1,

∂mW

≡

15

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