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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 solution22
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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.
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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
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1The 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)
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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
2.1Traces 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).
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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)
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2.2Pseudo-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].
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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.
2.3 The 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.
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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]).
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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)
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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).
3.1Gâteaux 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
12
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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.
?
4Gâteaux 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)
13
Page 14
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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