A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients Massimo Lanza de Cristoforis & Paolo Musolino

Abstract: We show that periodic layer potentials associated to parameter dependent analytic families of fundamental solutions of second order differential operators with constant coefficients, depend real-analytically upon the density (or moment) function and on a suitable parametrization of the supporting hypersurface and on the parameter. Keywords: Periodic layer potentials, second order differential operators with constant coefficients, domain perturbation, special nonlinear operators. 2000 Mathematics Subject Classification: 31B10, 47H30.

1

Introduction.

As is well known, the potential theoretic method has revealed to be a powerful tool to analyze boundary value problems for elliptic differential operators and in particular to analyze periodic versions of such problems (cf. e.g., Ammari and Kang [2, Chs. 2, 8], Shcherbina [25].) Periodic boundary value problems have a large variety of applications, especially in problems for composite materials (cf. e.g., Ammari and Kang [2, Chs. 2, 8], Milton [19, Ch. 1], Mityushev, Pesetskaya, and Rogosin [20].) In particular, such problems are relevant in the computation of effective properties of composite materials, which in turn can be justified by the Homogenization Theory (cf. e.g., Allaire [1, Ch. 1], Bensoussan, Lions, Papanicolaou [3, Ch. 1], Jikov, Kozlov, Ole˘ınik [10, Ch. 1].) In order to analyze perturbation problems with the potential theoretic method, we need to understand how layer potentials depend upon the moment and the support. In Coifman and Meyer [5, §4], and Potthast [21], [22], [23], and [6], [16],[17], [18], those authors have considered layer potentials associated to the Laplace equation and to the Helmholtz equation and to general second order differential operators with constant coefficients. In this paper, we shall extend some of those results in the frame of periodic problems. We fix once for all a natural number n ∈ N \ {0, 1} and (q11 , . . . , qnn ) ∈]0, +∞[n and a periodicity cell Q ≡ Πnj=1 ]0, qjj [ . 1

(1.1)

Then we denote by q the diagonal matrix  q11 0  0 q22 q≡  ... ... 0 0

... ... ... ...

 0 0   ...  qnn

and by |Q| the measure of the fundamental cell Q. Clearly, qZn ≡ {qz : z ∈ Zn } is the set of vertices of a periodic subdivision of Rn corresponding to the fundamental cell Q. Next we introduce a family of differential operators. Let N2 denote the number of multi-indexes α ∈ Nn with |α| ≤ 2. For each a ≡ (aα )|α|≤2 ∈ CN2 , we set (2)

a(2) ≡ (alj )l,j=1,...,n (2)

a(1) ≡ (aj )j=1,...,n

(2)

with alj ≡ 2−1 ael +ej for j 6= l, ajj ≡ aej +ej , and aj ≡ aej , where {ej : j = 1, . . . , n} is the canonical basis of Rn . We note that the matrix a(2) is symmetric. If a ∈ CN2 , then we set X P [a, x] ≡ aα xα ∀x ∈ Rn . |α|≤2

We also set  E≡

a ≡ (aα )|α|≤2 ∈ CN2 :

inf n

Re

ξ∈R ,|ξ|=1

X

aα ξ α



 >0 .

|α|=2

Clearly, E coincides with the set of coefficients a ≡ (aα )|α|≤2 such that the differential operator X P [a, D] ≡ aα Dα |α|≤2

is strongly elliptic and has complex coefficients. As is well known, if a ∈ E, a q-periodic distribution G is a q-periodic fundamental solution of P [a, D] provided that X P [a, D]G = δqz , (1.2) z∈Zn

where δqz denotes the Dirac measure with mass at qz, for all z ∈ Zn . Unfortunately however, not all operators P [a, D] admit q-periodic fundamental solutions, not even in case P [a, D] is the Laplace operator. Instead, if we denote by E2πiq−1 z , the function defined by E2πiq−1 z (x) ≡ e2πi(q

−1

z)·x

∀x ∈ Rn ,

for all z ∈ Zn and if a ∈ E, then one can show that the set Z(a) ≡ {z ∈ Zn : P [a, 2πiq −1 z] = 0}

2

(1.3)

is finite and that the q-periodic distribution X

Ga ≡

z∈Zn \Z(a)

1 1 −1 E |Q| P [a, 2πiq −1 z] 2πiq z

(1.4)

satisfies the equality P [a, D](Ga ) =

X

δqz −

z∈Zn

X z∈Z(a)

1 −1 . E |Q| 2πiq z

(1.5)

Equality (1.5) can be considered as an effective substitute of equality (1.2), and the distribution Ga can be used to introduce layer potentials, which can be employed to analyze boundary value problems on q-periodic domains. Now we are interested into the analysis of perturbation problems for layer potentials associated to the kernel Ga and we note that by perturbing a, the set Z(a) is not stable. To circumvent such a difficulty, we note that if we fix a finite subset F of Zn , and consider an element a such that Z(a) ⊆ F , then Ga,F ≡

X z∈Zn \F

1 1 −1 E |Q| P [a, 2πiq −1 z] 2πiq z

(1.6)

satisfies the equality P [a, D](Ga,F ) =

X

δqz −

z∈Zn

X 1 −1 . E |Q| 2πiq z

(1.7)

z∈F

Clearly, the distribution Ga,F differs from Ga by an entire analytic function. Moreover, by interior elliptic regularity theory, Ga,F and Ga are analytic in Rn \ qZn . Then we fix a finite subset F of Zn , and consider an analytic family of a in E such that Z(a) ⊆ F , and an analytic family of real analytic functions which enjoy the properties of Ga,F and in particular equality (1.7). Namely, we shall consider the following assumption. Let K be a real Banach space. Let O be an open subset of K.

(1.8)

Let a(·) be a real analytic map from O to E. Let Z(a(κ)) ⊆ F for all κ ∈ O. Let S(·, ·) be a real analytic map from (Rn \ qZn ) × O to C such that S(·, κ) is q−periodic for all κ ∈ O and such that S(·, κ) ∈ L1loc (Rn ) for all κ ∈ O and such that P [a(κ), D](S(·, κ)) equals the right hand side of (1.7) for all κ ∈ O . Since we are interested into perturbation results for periodic layer potentials, we now introduce our supports for the potentials. Let m ∈ N \ {0},

α ∈]0, 1[ .

Then we take a bounded open connected subset Ω of Rn of class C m,α such that Ω− ≡ Rn \ clΩ is connected. Q Then we consider a class of diffeomorphisms AQ ∂Ω from ∂Ω into Q. If φ ∈ A∂Ω , the Jordan-Leray separation theorem ensures that Rn \ φ(∂Ω) has exactly two open connected 3

components, and we denote by I[φ] and E[φ] the bounded and unbounded open connected components of Rn \ φ(∂Ω), respectively. Since φ(∂Ω) ⊆ Q, a simple topological argument shows that Q \ clI[φ] is also connected. Then we consider the periodic domains [ (qz + I[φ]) , S[I[φ]] ≡ S[I[φ]]−

(1.9)

z∈Zn n

≡ R \ clS[I[φ]] .

As is well known, one can analyze boundary value problems in S[I[φ]] or in S[I[φ]]− by the potential theoretic method by exploiting integral equations derived from the use of periodic layer potentials defined on ∂I[φ] = φ(∂Ω), which we now turn to consider. For all continuous (−1) functions f from ∂Ω to C and φ ∈ AQ defined on ∂Ω , one can consider the function f ◦ φ φ(∂Ω), and it makes sense to consider the q-periodic simple layer potential Z vq [φ, f, κ](ξ) ≡ S (ξ − η, κ) f ◦ φ(−1) (η) dση ∀ξ ∈ Rn , φ(∂Ω)

for all κ ∈ O. Then we introduce the function Vq [φ, f, κ](x) ≡ vq [φ, f, κ] ◦ φ(x)

∀x ∈ ∂Ω .

(1.10)

m−1,α We prove that the map Vq [·, ·, ·] from (C m,α (∂Ω, Rn )∩AQ (∂Ω)×O to C m,α (∂Ω) ∂Ω )×C which takes (φ, f, κ) to the function Vq [φ, f, κ] defined in (1.10) is real analytic (see Theorem 5.10.) Then we consider the functions from ∂Ω to C defined by Z Vq,l [φ, f, κ](x) ≡ ∂ξl S(φ(x) − η, κ)f ◦ φ(−1) (η) dση , (1.11) φ(∂Ω)

Vq,∗ [φ, f, κ](x) Z ≡ Dξ S(φ(x) − η, κ)a(2) (κ)νφ (φ(x))f ◦ φ(−1) (η)dση ,

(1.12)

φ(∂Ω) m−1,α for all x ∈ ∂Ω, and for all (φ, f, κ) ∈ (C m,α (∂Ω, Rn ) ∩ AQ (∂Ω) × O, and for all ∂Ω ) × C l ∈ {1, . . . , n}, and by Z Wq [φ, f, κ](x) ≡ − Dξ S(φ(x) − η, κ)a(2) (κ)νφ (η)f ◦ φ(−1) (η)dση (1.13) φ(∂Ω) Z − S(φ(x) − η, κ)νφt (η)a(1) (κ)f ◦ φ(−1) (η)dση , φ(∂Ω) m,α for all x ∈ ∂Ω and for all (φ, f, κ) ∈ (C m,α (∂Ω, Rn ) ∩ AQ (∂Ω) × O. Here ∂ξl S(·, κ) ∂Ω ) × C and Dξ S(·, κ) denote the derivative with respect to ξl and the gradient of S(ξ, κ) with respect to the first argument, respectively, and νφ denotes the exterior unit normal field to I[φ]. The functions Vq,l , Vq,∗ , Wq are associated to the φ-pull backs on ∂Ω of the derivatives of the periodic simple layer potential and of the periodic double layer potential and are well known to intervene in the integral equations associated to periodic boundary value problems for the elliptic operator P [a(κ), D]. We prove that Vq,l and Vq,∗ are real analytic

4

m−1,α from (C m,α (∂Ω, Rn ) ∩ AQ (∂Ω) × O to C m−1,α (∂Ω) and that Wq is real analytic ∂Ω ) × C Q m,α n m,α from (C (∂Ω, R ) ∩ A∂Ω ) × C (∂Ω) × O to C m,α (∂Ω). Our work stems from that of [16] for the Cauchy integral operator, and from that of [17] and [18] for the Laplace and for the Helmholtz operator, respectively, and is the periodic counterpart of the results of [6] for second order elliptic operators. The paper is organized as follows. In section 2, we collect some basic notation. In section 3, we introduce some basics on periodic fundamental solutions and on the corresponding layer potentials. In section 4, we introduce an auxiliary boundary value problem. In section 5, we prove our main results.

2

Notation

We denote the norm on a normed space X by k · kX . Let X and Y be normed spaces. We endow the space X × Y with the norm defined by k(x, y)kX ×Y ≡ kxkX + kykY for all (x, y) ∈ X ×Y, while we use the Euclidean norm for Rn . For standard definitions of Calculus in normed spaces, we refer to Prodi and Ambrosetti [24]. The symbol N denotes the set of natural numbers including 0. The inverse function of an invertible function f is denoted f (−1) , as opposed to the reciprocal of a complex-valued function g, or the inverse of a matrix A, which are denoted g −1 and A−1 , respectively. A dot “·” denotes the inner product in Rn , or the matrix product between matrices. Let A be a matrix. Then At denotes the transpose
t matrix of A and Aij denotes the (i, j)-entry of A. If A is invertible, we set A−t ≡ A−1 . Let D ⊆ Rn . Then cl D denotes the closure of D and ∂D denotes the boundary of D. For all R > 0, x ∈ Rn , xj denotes the j-th coordinate of x, |x| denotes the Euclidean modulus of x in Rn or in C, and Bn (x, R) denotes the ball {y ∈ Rn : |x−y| < R}. Let Ω be an open subset of Rn . The space of m times continuously differentiable complex-valued functions on Ω is denoted by C m (Ω, C), or more simply by C m (Ω). D(Ω) denotes the space of functions of C ∞ (Ω) with compact support. The dual D0 (Ω) denotes the space of distributions in Ω. Let n m f ∈ (C  (Ω)) . The s-th component of f is denoted fs , and Df denotes the gradient matrix ∂fs ∂xl

s,l=1,...,n

. Let η ≡ (η1 , . . . , ηn ) ∈ Nn , |η| ≡ η1 +. . .+ηn . Then Dη f denotes m

η

∂ |η| f η n ∂x1 1 ...∂xη n

.

The subspace of C (Ω) of those functions f whose derivatives D f of order |η| ≤ m can be extended with continuity to cl Ω is denoted C m (cl Ω). The subspace of C m (cl Ω) whose functions have m-th order derivatives that are H¨older continuous with exponent α ∈]0, 1] is denoted C m,α (cl Ω) (cf. e.g., Gilbarg and Trudinger [8].) The subspace of C m (cl Ω) of those functions f such that f|cl(Ω∩Bn (0,R)) ∈ C m,α (cl(Ω ∩ Bn (0, R))) for all R ∈]0, +∞[ is denoted n m,α Cloc (cl Ω). Let D ⊆ Cn . Then C m,α (cl Ω, D) denotes {f ∈ (C m,α (cl Ω)) : f (cl Ω) ⊆ D}. n m m,α Now let Ω be a bounded open subset of R . Then C (cl Ω) and C (cl Ω) are endowed with their usual norm and are well known to be Banach spaces (cf. e.g., Troianiello [27, §1.2.1].) We say that a bounded open subset Ω of Rn is of class C m or of class C m,α , if it is a manifold with boundary imbedded in Rn of class C m or C m,α , respectively (cf. e.g., Gilbarg and Trudinger [8, §6.2].) We denote by νΩ the outward unit normal to ∂Ω. For standard properties of functions in Schauder spaces, we refer the reader to Gilbarg and Trudinger [8] and to Troianiello [27] (see also [14, §2, Lem. 3.1, 4.26, Thm. 4.28], [17, §2].) If M is a manifold imbedded in Rn of class C m,α , with m ≥ 1, α ∈]0, 1[, one can define the Schauder spaces also on M by exploiting the local parametrizations. In particular, one can consider the spaces C k,α (∂Ω) on ∂Ω for 0 ≤ k ≤ m with Ω a bounded open set of class C m,α , and the trace operator of C k,α (clΩ) to C k,α (∂Ω) is linear and continuous. Moreover,

5

for each R > 0 such that clΩ ⊆ Bn (0, R), there exists a linear and continuous extension operator of C k,α (∂Ω) to C k,α (clΩ), and of C k,α (clΩ) to C k,α (clBn (0, R)) (cf. e.g., Troianiello [27, Thm. 1.3, Lem. 1.5].) We denote by dσ the area element of a manifold imbedded in Rn . We note that throughout the paper “analytic” means “real analytic”. For the definition and properties of analytic operators, we refer to Prodi and Ambrosetti [24, p. 89]. In particular, we mention that the pointwise product in Schauder spaces is bilinear and continuous, and thus analytic, and that the map which takes a nonzero function to its reciprocal, or an invertible matrix of functions to its inverse matrix is real analytic in Schauder spaces (cf. e.g., [17, pp. 141, 142].) If y ∈ Rn and f is a function defined in Rn , we set τy f (x) ≡ f (x − y) for all x ∈ Rn . If u is a distribution in Rn , then we set < τy u, f >=< u, τ−y f >

∀f ∈ D(Rn ) .

If Ω is an open subset of Rn , k ∈ N, β ∈]0, 1], we set Cbk (clΩ) ≡ {u ∈ C k (clΩ) : Dγ u is bounded ∀γ ∈ Nn such that |γ| ≤ k} , and we endow Cbk (clΩ) with its usual norm X kukCbk (clΩ) ≡ sup |Dγ u(x)| |γ|≤k

x∈clΩ

∀u ∈ Cbk (clΩ) .

Then we set Cbk,β (clΩ) ≡ {u ∈ C k,β (clΩ) : Dγ u is bounded ∀γ ∈ Nn such that |γ| ≤ k} , and we endow Cbk,β (clΩ) with its usual norm X X kukC k,β (clΩ) ≡ sup |Dγ u(x)| + |Dγ u : clΩ|β b

|γ|≤k

x∈clΩ

∀u ∈ Cbk,β (clΩ) .

|γ|=k

If clΩ ⊆ Q, then we set S[Ω]

≡

[

(qz + Ω) = qZn + Ω ,

z∈Zn

S[Ω]−

≡ Rn \ clS[Ω] .

If k ∈ N, β ∈]0, 1], then we set  Cqk (clS[Ω]) ≡ u ∈ C k (clS[Ω]) : u is q − periodic , which we regard as a Banach subspace of Cbk (clS[Ω]), and  Cqk,β (clS[Ω]) ≡ u ∈ C k,β (clS[Ω]) : u is q − periodic , which we regard as a Banach subspace of Cbk,β (clS[Ω]), and  Cqk (clS[Ω]− ) ≡ u ∈ C k (clS[Ω]− ) : u is q − periodic , which we regard as a Banach subspace of Cbk (clS[Ω]− ), and  Cqk,β (clS[Ω]− ) ≡ u ∈ C k,β (clS[Ω]− ) : u is q − periodic , n which we regard as a Banach subspace of Cbk,β (clS[Ω]− ). We denote ) the Schwartz R by S(R−2πiξ·x space of rapidly decreasing functions. Then we denote by ϕ(ξ) ˆ ≡ Rn ϕ(x)e dx for all ξ ∈ Rn , the Fourier transform of ϕ in Rn .

6

3

Some basic properties of periodic fundamental solutions and layer potentials

Let F be a finite subset of Zn . Let a ∈ E be such that Z(a) ⊆ F . Then we say that a distribution Sa,F in Rn is an F -analog of a q-periodic fundamental solution of P [a, D] provided that Sa,F is q-periodic and that X 1 X −1 . δqz − E P [a, D](Sa,F ) = |Q| 2πiq z n z∈F

z∈Z

As is well-known, a (tempered) distribution as Sa,F exists. Indeed, the following holds. Theorem 3.1 Let F be a finite subset of Zn . Let a ∈ E be such that P [a, 2πiq −1 z] 6= 0

∀z ∈ Zn \ F .

(3.2)

Then the generalized series X

Ga,F ≡

z∈Zn \F

1 1 −1 E |Q| P [a, 2πiq −1 z] 2πiq z

defines a tempered distribution in Rn such that Ga,F is q-periodic, i.e., ∀j ∈ {1, . . . , n} ,

τej qjj Ga,F = Ga,F and such that (1.7) holds.

Proof. We first consider statement (i). By assumption, there exists η > 0 such that  X aα ξ α > η|ξ|2 ∀ξ ∈ Rn \ {0} . Re |α|=2

Hence, 1 i a0 |P [a, iξ]| = |ξ|2 2 ξ t a(2) ξ − 2 ξ t a(1) − 2 |ξ| |ξ| |ξ|     1 1 1+ max{|a(1) |, |a0 |} ≥ |ξ|2 η − |ξ| |ξ| ≥ |ξ|2 η2−1 , if |ξ| ≥ max{1, 4η −1 max{|a(1) |, |a0 |}}. Hence, there exists cF > 0 such that 1 ≤ cF |P [a, 2πiq −1 z]|

∀z ∈ Zn \ F .

Now let ϕ ∈ S(Rn ). Then we have X 1 1 | < E2πiq−1 z , ϕ > | −1 z]| |Q| |P [a, 2πiq n z∈Z \F

=

X z∈Zn \F

≤ cF

1 1 −1 |ϕ(−q ˆ z)| |Q| |P [a, 2πiq −1 z]|

X 1 |ϕ(q ˆ −1 z)| < ∞ . |Q| n

z∈Z

7

(3.3)

Indeed, by Poisson’s summation formula, we have X 1 X ϕ(q ˆ −1 z) , ϕ(qz) = |Q| n n

(3.4)

z∈Z

z∈Z

and the generalized series in the right hand side converges absolutely (cf.e.g., Folland [7, p. 254].) Hence, the right hand side of (3.3) defines an element of the algebraic dual of S(Rn ). Then, since S(Rn ) is a Fr´echet space, the Banach-Steinhaus Theorem (cf. e.g., Tr`eves [26, pp. 347, 348]) ensures that Ga,F is actually a tempered distribution in Rn . Since Ga,F is the sum of a generalized series of q-periodic distributions, we conclude that Ga,F is also q-periodic. By continuity of P [a, D] in S 0 (Rn ) and by Poisson’s summation formula, we have X 1 1 P [a, D]E2πiq−1 z P [a, D]Ga,F = −1 z] |Q| P [a, 2πiq n z∈Z \F

=

1 |Q|

X

E2πiq−1 z

z∈Zn \F

1 X 1 X E2πiq−1 z − E2πiq−1 z |Q| |Q| z∈F z∈Zn X 1 X = δqz − E2πiq−1 z . |Q| n =

z∈F

z∈Z

2 By classical elliptic regularity theory, each F -analog of a q-periodic fundamental solution is analytic in Rn \ qZn . Indeed, the following holds. Theorem 3.5 Let F be a finite subset of Zn . Let a ∈ E be such that Z(a) ⊆ F . Let Sa be a fundamental solution of P [a, D]. Let Sa,F be an F -analog of a q-periodic fundamental solution of P [a, D]. Then the following statements hold. (i) Sa,F is real analytic in Rn \ qZn . (ii) The difference Sa,F − Sa is real analytic in (Rn \ qZn ) ∪ {0}. Moreover, X 1 X E2πiq−1 z in D0 (Rn ) . P [a, D](Sa,F − Sa ) = δqz − |Q| n

(3.6)

z∈F

z∈Z \{0}

(iii) Sa,F ∈ L1loc (Rn ). P P 1 Proof. Since z∈F |Q| E2πiq−1 z is an analytic function in Rn and z∈Zn \{0} δqz vanishes in (Rn \ qZn ) ∪ {0}, and P [a, D] is strongly elliptic, classical interior regularity theory implies the validity of statements (i), (ii). Statement (iii) follows by the real analyticity of Sa,F − Sa in (Rn \ qZn ) ∪ {0} and by the periodicity of Sa,F and by the local integrability of Sa in Rn (cf. e.g., John [11, Ch. III]). 2 Next we turn to consider some basic properties of the layer potentials associated to analogs of periodic fundamental solutions. We start with the following, which concerns periodic simple layer potentials. 8

Theorem 3.7 Let α ∈]0, 1[, m ∈ N \ {0}. Let Ω be a bounded connected open subset of Rn of class C m,α such that Rn \ clΩ is connected and that clΩ ⊆ Q. Let a ∈ E. Let F be a finite subset of Zn such that Z(a) ⊆ F (cf. (1.3).) Let Sa,F be an F -analog of a q-periodic fundamental solution of P [a, D]. Then the following statements hold. (i) If µ ∈ C 0,α (∂Ω), then the function vSa,F [∂Ω, µ] from Rn to C defined by Z vSa,F [∂Ω, µ](ξ) ≡ Sa,F (ξ − η)µ(η) dση ∀ξ ∈ Rn , ∂Ω

is continuous. Moreover, vSa,F [∂Ω, µ] is q-periodic and Z −1 1 X e2πi(q z)·(ξ−η) µ(η) dση P [a, D](vSa,F [∂Ω, µ])(ξ) = − |Q| ∂Ω

(3.8)

z∈F

for all ξ ∈ Rn \ ∂S[Ω]. (ii) If µ ∈ C m−1,α (∂Ω), then the function vS+a,F [∂Ω, µ] ≡ vSa,F [∂Ω, µ]|clS[Ω] belongs to Cqm,α (clS[Ω]) and the operator which takes µ to vS+a,F [∂Ω, µ] is continuous from C m−1,α (∂Ω) to Cqm,α (clS[Ω]). (iii) If µ ∈ C m−1,α (∂Ω), then the function vS−a,F [∂Ω, µ] ≡ vSa,F [∂Ω, µ]|clS[Ω]− belongs to Cqm,α (clS[Ω]− ). The operator from C m−1,α (∂Ω) to Cqm,α (clS[Ω]− ) which takes µ to vS−a,F [∂Ω, µ] is continuous. (iv) If µ ∈ C m−1,α (∂Ω), l ∈ {1, . . . , n}, then the integral Z vSa,F ,l [∂Ω, µ](ξ) ≡ ∂ξl Sa,F (ξ − η)µ(η) dση

∀ξ ∈ Rn ,

∂Ω

converges in the sense of Lebesgue for all ξ ∈ Rn \∂S[Ω] and in the sense of a principal value for all ξ ∈ ∂S[Ω]. (v) Let l ∈ {1, . . . , n}. If µ ∈ C m−1,α (∂Ω), then vSa,F ,l [∂Ω, µ]|S[Ω] admits a continuous extension vS+a,F ,l [∂Ω, µ] to clS[Ω] and vS+a,F ,l [∂Ω, µ] ∈ Cqm−1,α (clS[Ω]), and vSa,F ,l [∂Ω, µ]|S[Ω]− admits a continuous extension vS−a,F ,l [∂Ω, µ] to clS[Ω]− and vS−a,F ,l [∂Ω, µ] ∈ Cqm−1,α (clS[Ω]− ), and ∂ ± v [∂Ω, µ](ξ) ∂ξl Sa,F (νΩ (ξ))l =∓ µ(ξ) + vSa,F ,l [∂Ω, µ](ξ) , 2νΩ (ξ)t a(2) νΩ (ξ)

vS±a,F ,l [∂Ω, µ](ξ) =

(DvS±a,F [∂Ω, µ](ξ))a(2) νΩ (ξ) 1 = ∓ µ(ξ) + 2

Z ∂Ω

for all ξ ∈ ∂Ω.

9

(DSa,F (ξ − η))a(2) νΩ (ξ)µ(η) dση

(3.9)

(vi) Let l ∈ {1, . . . , n}. The operator from C m−1,α (∂Ω) to Cqm−1,α (clS[Ω]) which takes µ to vS+a,F ,l [∂Ω, µ] is continuous. The operator from C m−1,α (∂Ω) to Cqm−1,α (clS[Ω]− ) which takes µ to vS−a,F ,l [∂Ω, µ] is continuous. Proof. We first introduce some preliminary remarks. Let Sa be a fundamental solution of P [a, D]. Then Theorem 3.5 ensures that Ra,F ≡ Sa,F − Sa is real analytic in (Rn \ qZn ) ∪ {0}. Next we consider a bounded open connected subset V of class C ∞ of Rn such that clQ ⊆ V,

clV ∩ (qz + clΩ) = ∅

∀z ∈ Zn \ {0} .

(3.10)

If ξ ∈ clV and η ∈ ∂Ω, then the difference ξ − η cannot belong to qZn \ {0}, otherwise ξ ∈ ∂Ω + (qZn \ {0}) and thus there exists z ∈ Zn \ {0} with clV ∩ (∂Ω + qz) 6= ∅, a contradiction. Hence, clV − ∂Ω ⊆ (Rn \ qZn ) ∪ {0} . n

(3.11)

n

Since Ra,F is real analytic in (R \ qZ ) ∪ {0}, Lemma 7.3 of the Appendix implies that the map Z Ra,F (ξ − η)µ(η) dση ∀ξ ∈ clV , (3.12) ∂Ω

belongs to C ∞ (clV ). Next we note that Z Ra,F (ξ − η)µ(η) dση

vSa,F [∂Ω, µ](ξ) = vSa [∂Ω, µ](ξ) +

∀ξ ∈ clV ,

(3.13)

∂Ω

where

Z vSa [∂Ω, µ](ξ) ≡

Sa (ξ − η)µ(η) dση

∀ξ ∈ Rn .

∂Ω

Next we turn to prove statements (i)–(vi). We first consider statement (i). Since Sa,F is q-periodic, vSa,F [∂Ω, µ] is easily seen to be q-periodic. By classical differentiation theorems of integrals depending on a parameter, and by definition of Sa,F as F -analog of a q-periodic fundamental solution of P [a, D], equality (3.8) follows. As is well known, vSa [∂Ω, µ] is continuous in Rn (cf. e.g., [6, Thm. 3.1].) Since the function in (3.12) is of class C ∞ (clV ), equality (3.13) implies that the periodic function vSa,F [∂Ω, µ] is continuous on clV and thus in Rn . We now consider statements (ii), (iii). By classical potential theory, we have vSa [∂Ω, µ]|clΩ ∈ C m,α (clΩ) and vSa [∂Ω, µ]|clBn (0,R)\Ω ∈ C m,α (clBn (0, R) \ Ω) for all R ∈]0, +∞[ such that clΩ ⊆ Bn (0, R) (cf. e.g., [6, Thm. 3.1 (ii), (iii)].) Since the function in (3.12) belongs to C ∞ (clV ) ⊆ C m,α (clV ), equality (3.13) implies that vS+a,F [∂Ω, µ]|clΩ ∈ C m,α (clΩ) and that vS−a,F [∂Ω, µ]|clV \Ω ∈ C m,α (clV \ Ω). Then Lemmas 7.1, 7.2 of the Appendix imply that vS+a,F [∂Ω, µ] ∈ Cqm,α (clS[Ω]) and that vS−a,F [∂Ω, µ] ∈ Cqm,α (clS[Ω]− ). The continuity of the operators in (ii), (iii) follows immediately by Lemmas 7.1, 7.2, 7.3 of the Appendix, and by

10

classical potential theory (cf. e.g., [6, Thm. 3.1 (ii), (iii)].) Statement (iv) is an immediate consequence of the following equality ∂ξl Sa,F (ξ − η) = ∂ξl Sa (ξ − η) + ∂ξl Ra,F (ξ − η)

(3.14)

for all (ξ, η) ∈ clV × ∂Ω with ξ 6= η, and of the membership of the function Ra,F (ξ − η) of (ξ, η) ∈ clV × ∂Ω in C ∞ (clV × ∂Ω), and of the periodicity of Sa,F , and of classical potential theory (cf. e.g., [6, Thm. 3.1 (iv)].) We now consider statement (v). We set Z vSa ,l [∂Ω, µ](ξ) ≡ ∂ξl Sa (ξ − η)µ(η) dση ∀ξ ∈ Rn . (3.15) ∂Ω

Then we note that by standard theorems of differentiation of integrals depending on a parameter, we have Z Z ∂ ξl Ra,F (ξ − η)µ(η) dση = ∂ξl Ra,F (ξ − η)µ(η) dση , (3.16) ∂Ω

∂Ω

for all ξ ∈ clV . By (3.14), we have Z ∂ξl Ra,F (ξ − η)µ(η) dση ,

vSa,F ,l [∂Ω, µ](ξ) = vSa ,l [∂Ω, µ](ξ) +

(3.17)

∂Ω

for all ξ ∈ clV . The integral in the right hand side defines a function of class C ∞ (clV ) ⊆ C m−1,α (clV ). By classical potential theory, vSa ,l [∂Ω, µ]|Ω admits an extension to clΩ of class C m−1,α (clΩ) and vSa ,l [∂Ω, µ]|Rn \clΩ admits a continuous extension to Rn \ Ω, which belongs to C m−1,α (clBn (0, R) \ Ω) for all R > 0 such that clΩ ⊆ Bn (0, R) (cf. e.g., [6, Thm. 3.1 (v)].) Then we conclude that vSa,F ,l [∂Ω, µ]|Ω can be extended to an element of C m−1,α (clΩ), and that vSa,F ,l [∂Ω, µ]|clV \clΩ can be extended to an element of C m−1,α (clV \ Ω). Then Lemmas 7.1, 7.2 of the Appendix imply that vS+a,F ,l [∂Ω, µ] ∈ Cqm−1,α (clS[Ω]) and that vS−a,F ,l [∂Ω, µ] ∈ Cqm−1,α (clS[Ω]− ). Then formulas (3.9) follow by the corresponding classical formula for vS±a ,l [∂Ω, µ] (cf. e.g., [6, Thm. 3.1 (v)]) and by equalities (3.16), (3.17). The continuity of the operators in (vi) follows immediately by equality (3.17), and by Lemmas 7.1, 7.2, 7.3 of the Appendix, and by classical potential theory (cf. e.g., [6, Thm. 3.1 (vi)].) 2 Next we turn to consider the double layer potential. For each α, Ω, a, F , Sa,F as in the assumptions of Theorem 3.7, and µ ∈ C 0,α (∂Ω), we set Z wSa,F [∂Ω, µ, a](ξ) ≡ − (DSa,F (ξ − η))a(2) νΩ (η)µ(η) dση ∂Ω Z t − Sa,F (ξ − η)νΩ (η)a(1) µ(η) dση ∀ξ ∈ Rn . ∂Ω

Then we have the following. Theorem 3.18 Under the same assumptions of Theorem 3.7, the following statements hold. (i) Let µ ∈ C 0,α (∂Ω). Then wSa,F [∂Ω, µ, a] is q-periodic and P [a, D](wSa,F [∂Ω, µ, a])(ξ)

(3.19) 11

=

X 1 Z −1 e2πi(q z)·(ξ−η) 2πi(q −1 z)t a(2) νΩ (η)µ(η) dση |Q| ∂Ω z∈F X 1 Z −1 t e2πi(q z)·(ξ−η) νΩ (η)a(1) µ(η) dση ∀ξ ∈ Rn \ ∂S[Ω] . + |Q| ∂Ω z∈F

(ii) If µ ∈ C m,α (∂Ω), then the restriction wSa,F [∂Ω, µ, a]|S[Ω] can be extended uniquely to an element wS+a,F [∂Ω, µ, a] of Cqm,α (clS[Ω]) and the restriction wSa,F [∂Ω, µ, a]|S[Ω]− can be extended uniquely to an element wS−a,F [∂Ω, µ, a] of Cqm,α (clS[Ω]− ) and we have 1 on ∂Ω , (3.20) wS±a,F [∂Ω, µ, a] = ± µ + wSa,F [∂Ω, µ, a] 2 (DwS+a,F [∂Ω, µ, a])a(2) νΩ − (DwS−a,F [∂Ω, µ, a])a(2) νΩ = 0 on ∂Ω . (iii) If µ ∈ C 0,α (∂Ω), then we have wSa,F [∂Ω, µ, a](ξ)

=

−

n X

(2)

alj

j,l=1

Z −

∂ ∂ξl

Z Sa,F (ξ − η)(νΩ (η))j µ(η) dση ∂Ω

t Sa,F (ξ − η)νΩ (η)a(1) µ(η) dση ,

(3.21)

∂Ω

for all ξ ∈ Rn \ ∂S[Ω]. (iv) If µ ∈ C m,α (∂Ω) and U is an open neighborhood of ∂Ω in Rn and µ ˜ ∈ C m (U ), µ ˜|∂Ω = µ, then the following equality holds ∂ wS [∂Ω, µ, a](ξ) ∂ξr a,F Z n X (2) ∂ Sa,F (ξ − η) = alj ∂ξl ∂Ω j,l=1    ∂µ ˜ ∂µ ˜ · (νΩ (η))r (η) − (νΩ (η))j (η) dση ∂ηj ∂ηr   Z + (DSa,F (ξ − η))a(1) + a0 Sa,F (ξ − η) (νΩ (η))r µ(η) dση ∂Ω X 1 Z −1 (νΩ (η))r µ(η)e2πi(q z)·(ξ−η) dση + |Q| ∂Ω z∈F Z t − ∂ξr Sa,F (ξ − η)νΩ (η)a(1) µ(η) dση ∀ξ ∈ Rn \ ∂S[Ω] .

(3.22)

∂Ω

(v) The operator from C m,α (∂Ω) to Cqm,α (clS[Ω]) which takes µ to the function wS+a,F [∂Ω, µ, a] is continuous. The operator from C m,α (∂Ω) to Cqm,α (clS[Ω]− ) which takes µ to the function wS−a,F [∂Ω, µ, a] is continuous. Proof. We first consider statement (i). First we note that ξ−η ∈ / qZn

∀(ξ, η) ∈ (Rn \ ∂S[Ω]) × ∂Ω . 12

Indeed, if (ξ, η) ∈ (Rn \ ∂S[Ω]) × ∂Ω and ξ − η ∈ qZn , then we would have ξ ∈ ∂Ω + qZn = ∂S[Ω], contrary to our assumption on ξ. By definition of F -analog of a q-periodic fundamental solution, we have P [a, Dξ ](Sa,F (ξ − η)) = −

X 1 −1 e2πi(q z)·(ξ−η) , |Q|

z∈F

X 1 −1 e2πi(q z)·(ξ−η) 2πi(q −1 z)t , P [a, Dξ ](DSa,F (ξ − η)) = − |Q| z∈F

for all (ξ, η) ∈ (Rn \ ∂S[Ω]) × ∂Ω. Then by standard differentiation theorems for integrals depending on a parameter equality (3.19) holds. We now consider statement (ii). As in the proof of Theorem 3.7, we consider a fundamental solution Sa of P [a, D] and we set Ra,F ≡ Sa,F − Sa , and we take a bounded open connected subset V of class C ∞ of Rn as in (3.10). Since Ra,F is real analytic in (Rn \ qZn ) ∪ {0}, Lemma 7.3 of the Appendix implies that the map Z wRa,F [∂Ω, µ, a](ξ) ≡ − (DRa,F (ξ − η))a(2) νΩ (η)µ(η) dση (3.23) ∂Ω Z t − Ra,F (ξ − η)νΩ (η)a(1) µ(η) dση ∀ξ ∈ clV , ∂Ω

is of class C ∞ (clV ). Next we note that wSa,F [∂Ω, µ, a] = wSa [∂Ω, µ, a] + wRa,F [∂Ω, µ, a]

in clV .

(3.24)

Since Sa,F is q-periodic, the function wSa,F [∂Ω, µ, a] is easily seen to be q-periodic as well. By classical potential theory, wSa [∂Ω, µ, a]|Ω can be uniquely extended to an element wS+a [∂Ω, µ, a] of C m,α (clΩ) and wSa [∂Ω, µ, a]|Rn \Ω can be uniquely extended to an element m,α wS−a [∂Ω, µ, a] of Cloc (clΩ− ) (cf. e.g., [6, Thm. 3.1 (viii)].) Since wRa,F [∂Ω, µ, a] ∈ C ∞ (clV ), equality (3.24) implies that wSa,F [∂Ω, µ, a]|Ω can be extended uniquely to an element of C m,α (clΩ). Then the periodicity of wSa,F [∂Ω, µ, a] and Lemma 7.2 of the Appendix imply that wSa,F [∂Ω, µ, a]|S[Ω] admits a unique extension wS+a,F [∂Ω, µ, a] in Cqm,α (clS[Ω]). Since wRa,F [∂Ω, µ, a] ∈ C ∞ (clV ), equality (3.24) implies that wSa,F [∂Ω, µ, a]|clV \clΩ can be extended to a unique element of C m,α (clV \ Ω). Then the periodicity of wSa,F [∂Ω, µ, a] and Lemma 7.1 of the Appendix imply that wSa,F [∂Ω, µ, a] admits a unique extension wS−a,F [∂Ω, µ, a] in Cqm,α (clS[Ω]− ). Since wRa,F [∂Ω, µ, a] belongs to C ∞ (clV ), the jump relations (3.20) follow immediately by the corresponding classical relations for wSa [∂Ω, µ, a] and by equality (3.24) (cf. [6, Thm. 3.1 (vii)].) Statement (iii) is an immediate consequence of the definition of wSa,F [∂Ω, µ, a] and of classical differentiation theorems for integrals depending on a parameter, and of the analyticity of Sa,F (ξ − η) for ξ ∈ clV \ ∂Ω, η ∈ ∂Ω, and of the q-periodicity of wSa,F [∂Ω, µ, a] and of the right hand side of (3.21). We now consider statement (iv). By statement (iii), we have ∂ ± w [∂Ω, µ, a](ξ) ∂ξr 13

 X  Z n ∂ (2) ∂ = − Sa,F (ξ − η)(νΩ (η))j µ(η) dση alj ∂ξr ∂ξl ∂Ω j,l=1 Z ∂ t − Sa,F (ξ − η)νΩ (η)a(1) µ(η) dση ∂ξ r ∂Ω Z X    n ∂ (2) ∂ = Sa,F (ξ − η) (νΩ (η))j µ(η) dση alj ∂ξr ∂Ω ∂ηl j,l=1 Z ∂ t − Sa,F (ξ − η)νΩ (η)a(1) µ(η) dση ∂Ω ∂ξr   Z X n ∂ (2) ∂ alj Sa,F (ξ − η) dση = (νΩ (η))j µ(η) ∂ηr ∂ξl ∂Ω j,l=1 Z ∂ t − Sa,F (ξ − η)νΩ (η)a(1) µ(η) dση ∀ξ ∈ Rn \ ∂S[Ω] . ∂ξ r ∂Ω Next we note that   Z X n ∂ (2) ∂ − (νΩ (η))r µ(η) a Sa,F (ξ − η) dση ∂ηj lj ∂ξl ∂Ω j,l=1 Z n 2 X (2) ∂ Sa,F (ξ − η) dση = (νΩ (η))r µ(η) alj ∂ξl ∂ξj ∂Ω j,l=1   Z (1) = (νΩ (η))r µ(η) −DSa,F (ξ − η)a − a0 Sa,F (ξ − η) dση ∂Ω X 1 Z −1 (νΩ (η))r µ(η)e2πi(q z)·(ξ−η) dση , − |Q| ∂Ω z∈F

n

for all ξ ∈ R \ ∂S[Ω], and thus ∂ ± w [∂Ω, µ, a](ξ) ∂ξr Sa,F    Z X n ∂ ∂ ∂ (2) alj µ(η) (νΩ (η))j = − (νΩ (η))r Sa,F (ξ − η) dση ∂ηr ∂ηj ∂ξl ∂Ω j,l=1   Z (1) + (νΩ (η))r µ(η) DSa,F (ξ − η)a + a0 Sa,F (ξ − η) dση ∂Ω Z ∂ t − Sa,F (ξ − η)νΩ (η)a(1) µ(η) dση ∂ξ r ∂Ω X 1 Z −1 (νΩ (η))r µ(η)e2πi(q z)·(ξ−η) dση , + |Q| ∂Ω z∈F

for all ξ ∈ Rn \ ∂S[Ω]. Since   Z ∂ϕ ∂ϕ µ(η) (νΩ (η))j (η) − (νΩ (η))r (η) dση ∂ηr ∂ηj ∂Ω   Z ∂µ ˜ ∂µ ˜ = ϕ(η) (νΩ (η))r (η) − (νΩ (η))j (η) dση , ∂ηj ∂ηr ∂Ω 14

for all functions ϕ of class C 1 in an open neighborhood of ∂Ω, we conclude that formula (3.22) holds. The continuity of the operators in (v) follows immediately by equality (3.24), and by Lemmas 7.1, 7.2, 7.3 of the Appendix and by classical potential theory (cf. e.g., [6, Thm. 3.1 (x)].) 2

4

An auxiliary boundary value problem

As we have seen in the introduction, we wish to analyze the dependence of the φ-pull backs in (1.10)–(1.13) of periodic simple and double layer potentials upon (φ, f, κ). To do so, we exploit a method of [17], [18], [6], which consists in introducing suitable neighborhoods of the ‘base boundary’ ∂Ω and in considering suitable extensions Φ of the diffeomorphism φ and then in considering the Φ-pull backs of (1.10)–(1.13). We start with some notation. Let Ω be a bounded open connected subset of Rn of class 1 C such that Rn \ clΩ is connected. We denote by A∂Ω and by AclΩ the sets of functions of class C 1 (∂Ω, Rn ) and of class C 1 (clΩ, Rn ) which are injective and whose differential is injective at all points x ∈ ∂Ω, and at all points x ∈ clΩ, respectively. One can verify that A∂Ω is open in C 1 (∂Ω, Rn ) and that AclΩ is open in C 1 (clΩ, Rn ) (cf. [14, Cor. 4.24, Prop. 4.29], [17, Lem. 2.5].) Moreover, if φ ∈ A∂Ω , the Jordan-Leray separation Theorem ensures that Rn \ φ(∂Ω) has exactly two open connected components, and we denote by I[φ] and E[φ] the bounded and unbounded open connected components of Rn \ φ(∂Ω), respectively. Then we have the following two Lemmas (cf. [18, §2].) Lemma 4.1 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open subset of class C m,α of Rn such that both Ω and Rn \ clΩ are connected. Let νΩ denote the outward unit normal field to ∂Ω. Let ω ∈ C m,α (∂Ω, Rn ), |ω(x)| = 1, ω(x) · νΩ (x) > 1/2 for all x ∈ ∂Ω. Then the following statements hold. (i) If φ ∈ C m,α (∂Ω, Rn ) ∩ A∂Ω , then I[φ] is a bounded open connected set of class C m,α and ∂I[φ] = φ(∂Ω) = ∂E[φ]. (ii) There exists δΩ ∈]0, +∞[ such that the sets Ωω,δ

≡

{x + tω(x) : x ∈ ∂Ω, t ∈] − δ, δ[} ,

Ω+ ω,δ Ω− ω,δ

≡

{x + tω(x) : x ∈ ∂Ω, t ∈] − δ, 0[} ,

≡

{x + tω(x) : x ∈ ∂Ω, t ∈]0, δ[} ,

are connected and of class C m,α , and ∂Ωω,δ

= {x + tω(x) : x ∈ ∂Ω, t ∈ {−δ, δ}} ,

∂Ω+ ω,δ ∂Ω− ω,δ

= {x + tω(x) : x ∈ ∂Ω, t ∈ {−δ, 0}} , = {x + tω(x) : x ∈ ∂Ω, t ∈ {0, δ}} ,

− n and Ω+ ω,δ ⊆ Ω, Ωω,δ ⊆ R \ clΩ, for all δ ∈]0, δΩ [.

(iii) Let δ ∈]0, δΩ [. If Φ ∈ AclΩω,δ , then φ ≡ Φ|∂Ω ∈ A∂Ω .

15

(iv) If δ ∈]0, δΩ [, then the set A0clΩω,δ ≡

o n Φ ∈ AclΩω,δ : Φ(Ω+ is open in ω,δ ) ⊆ I[Φ|∂Ω ]

0 AclΩω,δ and Φ(Ω− ω,δ ) ⊆ E[Φ|∂Ω ] for all Φ ∈ AclΩω,δ . − (v) If δ ∈]0, δΩ [ and Φ ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ , then both Φ(Ω+ ω,δ ) and Φ(Ωω,δ ) are m,α open sets of class C , and         + − ∂Φ Ω+ ∂Φ Ω− ω,δ = Φ ∂Ωω,δ , ω,δ = Φ ∂Ωω,δ .

Then we have the following lemma (cf. e.g., [18, Prop. 2.5, 2.6].) Lemma 4.2 Let m, α, Ω, ω, δΩ be as in Lemma 4.1. Let φ0 ∈ C m,α (∂Ω, Rn ) ∩ A∂Ω . Then the following statements hold. (i) There exist δ0 ∈]0, δΩ [ and Φ0 ∈ C m,α (clΩω,δ0 , Rn ) ∩ A0clΩω,δ such that φ0 ≡ Φ0|∂Ω . 0

(ii) Let δ0 , Φ0 be as in (i). Then there exist an open neighborhood W0 of φ0 in C m,α (∂Ω, Rn )∩ A∂Ω , and a real analytic operator E0 of C m,α (∂Ω, Rn ) to C m,α (clΩω,δ0 , Rn ) which maps W0 to C m,α (clΩω,δ0 , Rn ) ∩ A0clΩω,δ and such that E0 [φ0 ] = Φ0 and E0 [φ]|∂Ω = φ 0 for all φ ∈ W0 . Now the idea is to show that layer potentials are uniquely determined by suitable prob± lems in Φ(Ω± ω,δ ) and by Φ-pull back problems in Ωω,δ . To do so, we introduce some auxiliary spaces of differentiable functions. We denote by C −1,α (clΩ) the space of distributions in Ω which equal the divergence of an element of class C 0,α (clΩ, Cn ) endowed with the quotient norm. Then as in [18, Lem. 4.5], we introduce the following. Lemma 4.3 Let m ∈ N, α ∈]0, 1[. Let Ω be an open bounded connected subset of Rn of class C max{1,m},α . The set   Z Y m,α (Ω) ≡ w ∈ C m,α (clΩ, Cn ) : (Dϕ)w dx = 0 ∀ϕ ∈ D (Ω) Ω

is a closed linear subspace of C m,α (clΩ, Cn ) and the quotient Z m,α (Ω) ≡ C m,α (clΩ, Cn ) /Y m,α (Ω) is a Banach space. Moreover, if we denote by ΠΩ the canonical projection of C m,α (clΩ, Cn ) f from Z m,α (Ω) onto C m−1,α (clΩ) onto Z m,α (Ω), there exists a unique homeomorphism div f such that div = div ◦ ΠΩ . Then we denote by Sn the function from Rn \ {0} to R defined by  1 ∀ξ ∈ Rn \ {0}, if n = 2 , sn log |ξ| Sn (ξ) ≡ 1 2−n n |ξ| ∀ξ ∈ R \ {0}, if n > 2 , (2−n)sn

(4.4)

where sn denotes the (n − 1) dimensional measure of ∂Bn . Sn is well-known to be the fundamental solution of the Laplace operator. Then for each bounded open subset Ω of Rn of class C 1 , we set Z P[f ](x) ≡

Sn (x − y)f (y) dy

∀x ∈ clΩ ,

Ω

for all f ∈ C 0 (clΩ). Then we introduce the following definition of [6]. 16

Definition 4.5 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open connected subset of Rn of class C m,α . (i) Let A[·, ·, ·] be the map from E × (C m,α (clΩ, Rn ) ∩ AclΩ ) × C m,α (clΩ) to the space C m−1,α (clΩ, Cn ) defined by n o A[a, Φ, u] ≡ (DΦ)−1 a(2) (DΦ)−t (Du)t + (DΦ)−1 a(1) u | det DΦ| ∀(a, Φ, u) ∈ E × (C m,α (clΩ, Rn ) ∩ AclΩ ) × C m,α (clΩ) . ˜ ·, ·] be the map from E × (C m,α (clΩ, Rn ) ∩ AclΩ ) × C m,α (clΩ) to Z m−1,α (Ω) (ii) Let A[·, defined by (−1) f ˜ Φ, u] ≡ ΠΩ A[a, Φ, u] + a0 div A[a, (u| det DΦ|) , (4.6) for all (a, Φ, u) ∈ E × (C m,α (clΩ, Rn ) ∩ AclΩ ) × C m,α (clΩ). (iii) Let Rn \ clΩ be connected. Let ω, δΩ be as in Lemma 4.1. Let δ ∈]0, δΩ [. Let Φ ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ , a ∈ E. Then we set B[a, Φ](v + , v − ) ≡ (Dv + )|Φ(∂Ω) a(2) νΦ|∂Ω − (Dv − )|Φ(∂Ω) a(2) νΦ|∂Ω , m,α for all (v + , v − ) ∈ C m,α (clΦ(Ω+ (clΦ(Ω− ω,δ )) × C ω,δ )).

(iv) Let Rn \ clΩ be connected. Let ω, δΩ be as in Lemma 4.1. Let T denote the map from m,α E × (C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ ) × C m,α (clΩ+ (clΩ− ω,δ ) × C ω,δ ) to the Banach space     m,α m−1,α Z ≡ Z m−1,α Ω+ Ω− (∂Ω) × C m−1,α (∂Ω) (4.7) ω,δ × Z ω,δ × C     m,α ×C m,α (∂Ω+ (∂Ω− ω,δ ) \ ∂Ω × C ω,δ ) \ ∂Ω , which takes (a, Φ, V + , V − ) to ˜ Φ, V + ], A[a, ˜ Φ, V − ], V + − V − , T [a, Φ, V + , V − ] ≡ (A[a, +

J[a, Φ, V , V

−

(4.8)

+ − ], V|(∂Ω , V|(∂Ω ), + − ω,δ )\∂Ω ω,δ )\∂Ω

where we have set J[a, Φ, V + , V − ] ≡ DV + (DΦ)−1 a(2) n[Φ] − DV − (DΦ)−1 a(2) n[Φ] and n[Φ](x) ≡

(DΦ(x))−t νΩ (x) |(DΦ(x))−t νΩ (x)|

on ∂Ω ,

(4.9)

∀x ∈ ∂Ω .

A simple computation shows that under the assumptions of Definition 4.5 (iv), we have n[Φ](x) = νΦ|∂Ω ◦ Φ(x) for all Φ ∈ C m,α (clΩω,δ , Rn ) ∩ A0clΩω,δ .

17

∀x ∈ ∂Ω ,

(4.10)

5

Real analyticity of periodic layer potentials corresponding to families of fundamental solutions

Next we turn again to periodic problems. Let Ω be a bounded open connected subset of Rn of class C 1 such that Rn \ clΩ is connected. Then we find convenient to set AQ ∂Ω

≡

{φ ∈ A∂Ω : φ(∂Ω) ⊆ Q} ,

AQ clΩ

≡

{Φ ∈ AclΩ : Φ(clΩ) ⊆ Q} .

If m, α, Ω, ω, δ are as in Lemma 4.1 (iv), we set 0

Q 0 AclΩ ≡ AQ clΩω,δ ∩ AclΩω,δ . ω,δ

If η ∈]0, 1[, then we set  E(η) ≡ a ≡ (aα )|α|≤2 ∈ E :

inf n

Re

ξ∈R ,|ξ|=1

X

aα ξ α



 > η, max |aα | < η −1 , |α|≤2

|α|=2

S for all η ∈]0, 1[. Obviously E = η∈]0,1[ E(η) and each E(η) is open in CN2 . Then we have the following Proposition, which shows that the integral of (1.10) can be identified by a boundary value problem in Ωω,δ . Proposition 5.1 Let m, α, Ω, ω, δΩ be as in Lemma 4.1. Let δ ∈]0, δΩ [. Let F be a finite subset of Zn . Let f ∈ C m−1,α (∂Ω). Let η ∈]0, 1[. Then there exists δη ∈]0, δΩ [ such 0 Q that if δ ∈]0, δη ], and if (a, Φ) belongs to E(η) × (C m,α (clΩω,δ , Rn ) ∩ AclΩ ), Z(a) ⊆ F , ω,δ −1 |det(DΦ)| ≤ η , and if Sa,F is an F -analog of a q-periodic fundamental solution of P [a, D], m,α then the pair (V + , V − ) ∈ C m,α (clΩ+ (clΩ− ω,δ ) × C ω,δ ) satisfies equation T [a, Φ, V + , V − ] = Z X 1
−1 ˜z [Φ] + ΠΩ+ E e−2πi(q z)·Φ(y) f (y)σn [Φ](y) dσy , − |clΩ ω,δ |Q| ω,δ ∂Ω z∈F Z X 1
−1 ˜z [Φ] − − ΠΩ− E e−2πi(q z)·Φ(y) f (y)σn [Φ](y) dσy , |clΩω,δ ω,δ |Q| ∂Ω z∈F ! 0, −f, h+ , h−

,

(5.2)

with h+ (x) ≡ −

h (x) ≡

Z

∀x ∈ (∂Ω+ ω,δ ) \ ∂Ω ,

Sa,F (Φ(x) − Φ(y))f (y)σn [Φ](y) dσy Z∂Ω

Sa,F (Φ(x) − Φ(y))f (y)σn [Φ](y) dσy ∀x ∈ (∂Ω− ω,δ ) \ ∂Ω , h i −1 DP |det(DΦ(·))|e2πi(q z)·Φ(·) (x) ∀x ∈ clΩω,δ , ∂Ω

˜z [Φ](x) ≡ E

for all z ∈ F , where σn [Φ] ≡ |det(DΦ)||(DΦ)−t νΩ |, if and only if V+

= vS+a,F [Φ(∂Ω), f ◦ Φ(−1) ] ◦ Φ|clΩ+ , ω,δ

V−

=

vS−a,F [Φ(∂Ω), f ◦ Φ(−1) ] ◦ Φ|clΩ− . ω,δ

18

(5.3)

Proof. First of all, we note that if δ ∈]0, δΩ [, then P maps C m−1,α (clΩω,δ ) to C m+1,α (clΩω,δ ) (cf. e.g., Gilbarg and Trudinger [8, Lem. 4.1, 4.2, pp. 54, 55].) If Φ ∈ C m,α (clΩω,δ , Rn ), ˜z [Φ] ± ∈ C m,α (clΩ± , Cn ) ⊆ C m−1,α (clΩ± , Cn ) for all z ∈ F . By we conclude that E ω,δ ω,δ |clΩω,δ ˜z [Φ], we have definition of E ˜z [Φ](x)) = |det(DΦ(x))|e2πi(q div(E

−1

z)·Φ(x)

∀x ∈ clΩω,δ ,

and thus the Piola identity implies that e2πi(q

−1

˜z [Φ]) ◦ Φ(−1) (ξ)|det(DΦ(−1) )(ξ)| = (divE    ˜z [Φ] ◦ Φ(−1) (ξ)|det(DΦ(−1) )(ξ)| = div (DΦ)E

z)·ξ

∀ξ ∈ Φ(clΩω,δ ) ,

for all z ∈ F (cf. e.g., Vodop’yanov [29, §3].) Then we can invoke [6, Thm. 4.14 (ii)] and deduce the existence of δη ∈]0, δΩ [ such that problem (5.2) has a unique solution (V + , V − ) if 0 Q δ ∈]0, δη ], and if (a, Φ) belongs to E(η)×(C m,α (clΩω,δ , Rn )∩AclΩ ), Z(a) ⊆ F , |det(DΦ)| ≤ ω,δ η −1 , and if Sa,F is an F -analog of a q-periodic fundamental solution of P [a, D]. On the other hand, [6, Thm. 4.14 (i)] guarantees that (V + , V − ) satisfies (5.2) if and only if (v + , v − ) ≡ (V + ◦ Φ(−1) , V − ◦ Φ(−1) ) satisfies problem    P + 1 +  −1 z cz [Φ, f ] E in Φ Ω , P [a, D]v = −  2πiq z∈F ω,δ |Q|      P  − 1 −  P [a, D]v = − z∈F |Q| E2πiq−1 z cz [Φ, f ] in Φ Ωω,δ ,      v+ − v− = 0 on Φ (∂Ω) , (−1) (5.4) + − B[a, Φ](v , v ) = −f ◦ Φ on Φ (∂Ω) ,  |Φ(∂Ω)      (−1)  on Φ (∂Ω+  v + = h+ ◦ Φ|Φ((∂Ω+ )\∂Ω) ω,δ ) \ ∂Ω ,   ω,δ     (−1)  −  v − = h− ◦ Φ on Φ (∂Ω ) \ ∂Ω , − ω,δ |Φ((∂Ω )\∂Ω) ω,δ

where

Z

e−2πi(q

cz [Φ, f ] ≡

−1

z)·η

f ◦ Φ(−1) (η) dση

∀z ∈ F ,

Φ(∂Ω)

for all δ ∈]0, δη ]. On the other hand, Theorem 3.7 ensures that the pair (vS+a,F [Φ(∂Ω), f ◦ Φ(−1) ], vS−a,F [Φ(∂Ω), f ◦ Φ(−1) ]) satisfies problem (5.4). Hence, (V + , V − ) as in (5.3) is the only solution of (5.2) if δ ∈]0, δη ]. 2 Next, we introduce the following Lemma on the integral operator obtained by Φ-pulling back the simple layer potential, which is an integral operator with nonsingular kernel (see also [18, Lem. 4.8].) Lemma 5.5 Let m, α, Ω, ω, δΩ be as in Lemma 4.1. Let δ ∈]0, δΩ [. Let F be a finite subset of Zn . Let assumption (1.8) hold. Then the map δ V from   0 Q C m,α (clΩω,δ , Rn ) ∩ AclΩ × C m−1,α (∂Ω) × O ω,δ to C m,α (∂Ωω,δ ) defined by Z V [Φ, f, κ](x) ≡ δ

S(Φ(x) − Φ(y), κ)f (y)σn [Φ](y) dσy

∀x ∈ ∂Ωω,δ ,

∂Ω

  0 Q for all (Φ, f, κ) ∈ C m,α (clΩω,δ , Rn ) ∩ AclΩ × C m−1,α (∂Ω) × O is real analytic. ω,δ 19

Proof. It suffices to note that Φ(∂Ωω,δ ) − Φ(∂Ω) ⊆ Rn \ qZn for all Φ ∈ C m,α (clΩω,δ , Rn) ∩ 0

0

Q AclΩ and that the map which takes a pair (Φ, f ) in ω,δ m−1,α

Q C m,α (clΩω,δ , Rn ) ∩ AclΩ ω,δ

×

1

C (∂Ω) to f σn [Φ] in L (∂Ω) is real analytic and to invoke Lemma 7.3 (ii) of the Appendix with M = ∂Ω, M1 = ∂Ωω,δ and W = {(x, y) ∈ Rn × Rn : x − y ∈ Rn \ qZn } × O. 2 We now introduce some notation. Let m, α, Ω, ω, δΩ be as in Lemma 4.1. Let F be a finite subset of Zn . If (1.8) holds, we set Z vq+ [φ, f, κ](ξ) ≡ S (ξ − η, κ) f ◦ φ(−1) (η) dση ∀ξ ∈ clS[I[φ]] , φ(∂Ω) Z − vq [φ, f, κ](ξ) ≡ S (ξ − η, κ) f ◦ φ(−1) (η) dση ∀ξ ∈ clS[I[φ]]− , φ(∂Ω) m−1,α for all (φ, f, κ) ∈ (C m,α (∂Ω, Rn ) ∩ AQ (∂Ω) × O, and ∂Ω ) × C n o 0 Q Uη,δ ≡ Φ ∈ C m,α (clΩω,δ , Rn ) ∩ AclΩ : |det(DΦ)| < 1/η , ω,δ

O(η) ≡

{κ ∈ O : a(κ) ∈ E(η)} ,

for all η ∈]0, 1[, δ ∈]0, δΩ [. Then we have the following result. Proposition 5.6 Let m, α, Ω, ω, δΩ be as in Lemma 4.1. Let F be a finite subset of Zn . Let assumption (1.8) hold. Let η ∈]0, 1[. Let δη ∈]0, δΩ [ be as in Proposition 5.1. Let δ ∈]0, δη ]. m,α (clΩω,δ , Rn ) ∩ Let Vq± [Φ, f, κ] ≡ vq± [Φ|∂Ω , f, κ] ◦ Φ|clΩ± on clΩ± ω,δ for all (Φ, f, κ) ∈ (C ω,δ

0

Q AclΩ )×C m−1,α (∂Ω)×O. Then the maps from Uη,δ ×C m−1,α (∂Ω)×O(η) to C m,α (clΩ+ ω,δ ) ω,δ − − m,α + and to C (clΩω,δ ), which take (Φ, f, κ) to Vq [Φ, f, κ] and to Vq [Φ, f, κ] are real analytic, respectively.

Proof. We follow [6, Prop. 5.2], which in turn is a generalization of [17, Prop. 3.11 (i)], and of [18, Prop. 4.10 (i)]. First we fix δ ∈]0, δη [ and we set X

≡

C m,α (clΩω,δ , Rn ) × C m−1,α (∂Ω) × K ,

Y

≡

m,α C m,α (clΩ+ (clΩ− ω,δ ) × C ω,δ ) ,

Vη,δ

≡

Uη,δ × C m−1,α (∂Ω) × O(η) .

Then we consider the map Λ from U ≡ Vη,δ × Y to the Banach space Z of (4.7) defined by Λ[Φ, f, κ, V + , V − ] ≡ T [a(κ), Φ, V + , V − ] Z X 1
−1 ˜z [Φ] + e−2πi(q z)·Φ(y) f (y)σn [Φ](y) dσy , ΠΩ+ E − − |clΩ ω,δ |Q| ω,δ ∂Ω z∈F Z X 1
−1 ˜z [Φ] − − ΠΩ− E e−2πi(q z)·Φ(y) f (y)σn [Φ](y) dσy , |clΩω,δ ω,δ |Q| ∂Ω z∈F ! 0, −f, Vq+ [Φ, f, κ]|(∂Ω+

ω,δ )\∂Ω

, Vq− [Φ, f, κ]|(∂Ω−

ω,δ )\∂Ω

20

for all (Φ, f, κ, V + , V − ) ∈ U. By Proposition 5.1, the set of zeros of Λ in U coincides with the graph of the map (Vq+ [·, ·, ·], Vq− [·, ·, ·]). Thus we can deduce the real analyticity of the operator (Vq+ [·, ·, ·], Vq− [·, ·, ·]) by showing that we can apply the Implicit Function Theorem for real analytic operators (cf. e.g., Prodi and Ambrosetti [24, Thm. 11.6]) to the equation Λ[Φ, f, κ, V + , V − ] = 0 around the point (Φ1 , f1 , κ1 , Vq+ [Φ1 , f1 , κ1 ], Vq− [Φ1 , f1 , κ1 ]) for all (Φ1 , f1 , κ1 ) in Vη,δ . The domain U = Vη,δ × Y of Λ is clearly open in X × Y. Since the exponential is an entire real analytic function, known properties of analyticity of superposition operators generated by real analytic functions imply that the map from C m,α (clΩω,δ , Rn ) to C m−1,α (clΩω,δ ) which −1 takes Φ to e2πi(q z)·Φ(·) is real analytic (cf. B¨ohme and Tomi [4, p. 10], Henry [9, p. 29], Valent [28, Thm. 5.2, p. 44].) Since the pointwise product in C m−1,α (clΩω,δ ) is continuous, and P is linear and continuous from C m−1,α (clΩω,δ ) to C m+1,α (clΩω,δ ), and ΠΩ± is linear ω,δ

n m−1,α and continuous from C m−1,α (clΩ± (Ω± ω,δ , C ) to Z ω,δ ), we conclude that the map from
± m−1,α ˜ Uη,δ to Z (Ωω,δ ) which takes Φ to ΠΩ± Ez [Φ] and the map from Uη,δ × C m−1,α (∂Ω) ω,δ R −1 to C which takes the pair (Φ, f ) to ∂Ω e−2πi(q z)·Φ(y) f (y)σn [Φ](y) dσy are real analytic for each z ∈ F . By (1.8) and by the same argument of [6, p. 26], T [·, ·, ·, ·] is real analytic. Hence, Λ is real analytic. Thus it suffices to show that the differential

d(V + ,V − ) Λ[Φ1 , f1 , κ1 , Vq+ [Φ1 , f1 , κ1 ], Vq− [Φ1 , f1 , κ1 ]] is a homeomorphism. Now by standard rules of calculus in Banach space, such a differential coincides with T [a(κ1 ), Φ1 , ·, ·]. Since δ ∈]0, δη ], a(κ1 ) ∈ E(η), Φ1 ∈ Uη,δ , Theorem 4.14 (ii) of [6] ensures that T [a(κ1 ), Φ1 , ·, ·] is a linear homeomorphism from Y onto Z, and thus the proof is complete. 2 We now show that Proposition 5.6 implies that a corresponding result holds also for the Φ-pull backs of a periodic double layer potential (see (1.12).) Corollary 5.7 Let the assumptions of Proposition 5.6 hold. Let Wq+ [Φ, f, κ] and Wq− [Φ, f, κ] − denote the continuous extensions to clΩ+ ω,δ and to clΩω,δ of the functions Z −

Dξ S(Φ(x) − η, κ)a(2) (κ)νΦ|∂Ω (η)f ◦ Φ(−1) (η) dση Z t − S(Φ(x) − η, κ)νΦ (η)a(1) (κ)f ◦ Φ(−1) (η) dση |∂Ω

∀x ∈ Ω+ ω,δ ,

Dξ S(Φ(x) − η, κ)a(2) (κ)νΦ|∂Ω (η)f ◦ Φ(−1) (η) dση Z t − S(Φ(x) − η, κ)νΦ (η)a(1) (κ)f ◦ Φ(−1) (η) dση |∂Ω

∀x ∈ Ω− ω,δ ,

Φ(∂Ω)

Φ(∂Ω)

Z −

Φ(∂Ω)

Φ(∂Ω)

  0 Q ×C m,α (∂Ω)×O, respectively. Then the maps for all (Φ, f, κ) ∈ C m,α (clΩω,δ , Rn ) ∩ AclΩ ω,δ m,α from Uη,δ × C m,α (∂Ω) × O(η) to C m,α (clΩ+ (clΩ− ω,δ ) and to C ω,δ ) which take (Φ, f, κ) to + − Wq [Φ, f, κ] and to Wq [Φ, f, κ] are real analytic, respectively.

21

Proof. We proceed as in [17, Prop. 3.11]. We first consider Wq+ [·, ·, ·]. We observe that the       + + m−1,α m−1,α n map Γ from C m,α clΩ+ to C clΩ × C clΩ , C defined by ω,δ ω,δ ω,δ   ∀g ∈ C m,α clΩ+ ω,δ ,

Γ[g] ≡ (g, ∂x1 g, . . . , ∂xn g)

  is a linear homeomorphism from C m,α clΩ+ onto the image space Im Γ, a subspace of ω,δ     m−1,α n C m−1,α clΩ+ clΩ+ . Thus it suffices to show that the nonlinear maps ω,δ × C ω,δ , C ∂ W + [·, ·, ·] for s = 1, . . . , n are real analytic from Uη,δ × C m,α (∂Ω) × O(η) Wq+ [·, ·, ·] and ∂x s  q  to C m−1,α clΩ+ ω,δ . Now let R > supx∈Ω∪Ωω,δ |x|. By Troianiello [27, Thm. 1.3, Lem. 1.5],

there exists a linear and continuous operator F from C m,α (∂Ω) to C m,α (clBn (0, R)) such that F[f ]|∂Ω = f , for all f ∈ C m,α (∂Ω). By Theorem 3.18 (iii), (iv) and by (4.10), we have the following identities Wq+ [Φ, f, κ] = −

n X l,s,j=1

(2)

alj (κ)

∂ (V + [Φ, nj [Φ]f, κ])((DΦ)−1 )sl ∂xs q

(5.8)

−Vq+ [Φ, nt [Φ]a(1) (κ)f, κ], and
∂ Wq+ [Φ, f, κ] ∂xs n n n  X X
 ∂Φr X (2) ∂ −1 = alj (κ) Vq+ [Φ, Mrj [f, Φ], κ] (DΦ) ∂xs ∂xt tl r=1 t=1

(5.9)

l,j=1

+

n X ∂Φr r=1

+ +

∂xs

n X ∂Φr r=1 n X r=1

∂xs


D Vq+ [Φ, nr [Φ]f, κ] (DΦ)−1 · a(1) (κ) a0 (κ)Vq+ [Φ, nr [Φ]f, κ] −

∂ + V [Φ, nt [Φ]a(1) (κ)f, κ] ∂xs q

Z −1 ∂Φr X E2πiq−1 z ◦ Φ nr [Φ](y)f (y)e−2πi(q z)·Φ(y) σn [Φ](y) dσy , ∂xs |Q| ∂Ω z∈F

for all s ∈ {1, . . . , n}, where −1 −t Mrj [f, Φ] = (DΦ) · νΩ · # (" n #" n   X X ∂(F[f ])  −1 −1 · (DΦ) (νΩ )l (DΦ) ∂xl lj lr l=1 l=1 " n #" n #)   X X ∂(F[f ])  −1 −1 − (DΦ) (νΩ )l (DΦ) . ∂xl lj lr l=1

l=1

Then by the real analyticity of the pointwise product in Schauder spaces and by the real analyticity of the map which takes an invertible matrix with Schauder functions as entries to its inverse, and by the real analyticity of the linear and continuous map F[·] and of the 22

trace operator, and by the real analyticity of Vq+ [·, ·, ·], and by assumption (1.8), and by identities (5.8) and (5.9), and by the proof of Proposition 5.6, we conclude that Wq+ [·, ·, ·] and   + ∂ + m,α m−1,α W [·, ·, ·] are real analytic from U × C (∂Ω) × O(η) to C clΩ η,δ q ω,δ . Similarly, ∂xs 2

we can show that Wq− [Φ, f, κ] depends real analytically on (Φ, f, κ). We are now ready to prove our main result.

Theorem 5.10 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C m,α such that both Ω and Rn \ clΩ are connected. Let F be a finite subset of Zn . Let assumption (1.8) hold. Then the following statements hold.   m−1,α (i) The map Vq [·, ·, ·] from C m,α (∂Ω, Rn ) ∩ AQ (∂Ω)×O to the space C m,α (∂Ω) ∂Ω ×C defined by (1.10) is real analytic.   m−1,α (ii) The map Vq,l [·, ·, ·] from C m,α (∂Ω, Rn ) ∩ AQ (∂Ω) × O to the space ∂Ω × C C m−1,α (∂Ω) defined by (1.11) is real analytic for all l ∈ {1, . . . , n}.   m−1,α (iii) The map Vq,∗ [·, ·, ·] from C m,α (∂Ω, Rn ) ∩ AQ (∂Ω) × O to the space ∂Ω × C C m−1,α (∂Ω) defined by (1.12) is real analytic.   m,α (iv) The map Wq [·, ·, ·] from C m,α (∂Ω, Rn ) ∩ AQ (∂Ω)×O to the space C m,α (∂Ω) ∂Ω ×C defined by (1.13) is real analytic. Proof. We follow the argument of the proof of [6, Thm. 5.6] (see also [18, Proof of Thm. 4.11].) We first consider statements (i)–(iii). It clearly suffices to show that if (φ0 , f0 , κ0 ) belongs to

m−1,α C m,α (∂Ω, Rn ) ∩ AQ (∂Ω) × O, then the operators of ∂Ω × C

(i)–(iii) are real analytic in a neighborhood of (φ0 , f0 , κ0 ). Now let ω, δ0 , W0 , E0 be as in Lemma 4.2 for φ0 . Possibly shrinking W0 , we can assume that there exists η ∈]0, 1[ such that sup sup |det(DE0 [φ](x))| < η −1 , κ0 ∈ O(η) . φ∈W0 x∈clΩω,δ0

Possibly shrinking δ0 and W0 , we can also assume that E0 [φ](clΩω,δ0 ) ⊆ Q

∀φ ∈ W0 .

Now by definition of the operators in (i)–(iii), and by Theorem 3.7, we have Vq [φ, f, κ]

=

vq+ [φ, f, κ] ◦ φ = Vq+ [E0 [φ], f, κ] ,

Vq,l [φ, f, κ]

=

nl [E0 [φ]] f t 2n [E0 [φ]]a(2) (κ)n[E0 [φ]]

+

=

nl [E0 [φ]] f t 2n [E0 [φ]]a(2) (κ)n[E0 [φ]]

+ (DVq+ [E0 [φ], f, κ]) · (DE0 [φ])−1

Vq,∗ [φ, f, κ]

= =

∂ + (v [φ, f, κ]) ◦ φ ∂ξl q


1 f + (Dvq+ [φ, f, κ]) ◦ φ a(2) (κ)νφ ◦ φ 2 1 f + (DVq+ [E0 [φ], f, κ])(DE0 [φ])−1 · a(2) (κ) · n[E0 [φ]] , 2

23


l

,

on ∂Ω for all (φ, f, κ) ∈ W0 × C m−1,α (∂Ω) × O(η) where Vq+ is as in Proposition 5.6 for some arbitrary δ ∈]0, min{δη , δ0 }]. Hence, statements (i)–(iii) follow by Lemma 4.2 and Proposition 5.6. In order to prove statement (iv), we just note that 1 Wq [φ, f, κ] = − f + Wq+ [E0 [φ], f, κ] 2

on ∂Ω ,

for all (φ, f, κ) ∈ W0 ×C m,α (∂Ω)×O(η), and then we argue as above by exploiting Corollary 5.7 instead of Proposition 5.6. 2

6

Analytic dependence results for periodic layer potentials defined on ∂Ω

Let m, α, Ω be as in Lemma 4.1 with Ω such that clΩ ⊆ Q. Let F be a finite subset of Zn . If (1.8) holds, we set Z + vq [∂Ω, f, κ](ξ) ≡ S (ξ − η, κ) f (η) dση ∀ξ ∈ clS[Ω] , Z∂Ω vq− [∂Ω, f, κ](ξ) ≡ S (ξ − η, κ) f (η) dση ∀ξ ∈ clS[Ω]− , ∂Ω

for all (f, κ) ∈ C m−1,α (∂Ω) × O, and Z wq+ [∂Ω, f, κ](ξ) ≡ − Dξ S(ξ − η, κ)a(2) (κ)νΩ (η)f (η) dση ∂Ω Z t − S(ξ − η, κ)νΩ (η)a(1) (κ)f (η) dση ∀ξ ∈ S[Ω] , ∂Ω Z wq− [∂Ω, f, κ](ξ) ≡ − Dξ S(ξ − η, κ)a(2) (κ)νΩ (η)f (η) dση ∂Ω Z t − S(ξ − η, κ)νΩ (η)a(1) (κ)f (η) dση ∀ξ ∈ S[Ω]− , ∂Ω

for all (f, κ) ∈ C (∂Ω) × O. Then by Theorem 3.18, wq+ [∂Ω, f, κ] and wq− [∂Ω, f, κ] can be extended by continuity to clS[Ω], and to clS[Ω]− , respectively. We denote the corresponding extension by the same symbol. Then we have the following, whose proof is based on Proposition 5.6 and on Corollary 5.7. m,α

Theorem 6.1 Let n ∈ N \ {0, 1}, m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open connected subset of Rn of class C m,α such that Rn \ Ω is connected and such that clΩ ⊆ Q. Let F be a finite subset of Zn . Let assumption (1.8) hold. Then the following statements hold. (i) The map from C m−1,α (∂Ω) × O to Cqm,α (clS[Ω]) which takes (f, κ) to vq+ [∂Ω, f, κ] is real analytic. (ii) The map from C m−1,α (∂Ω) × O to Cqm,α (clS[Ω]− ) which takes (f, κ) to the function vq− [∂Ω, f, κ] is real analytic.

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(iii) The map from C m,α (∂Ω) × O to Cqm,α (clS[Ω]) which takes (f, κ) to the function wq+ [∂Ω, f, κ] is real analytic. (iv) The map from C m,α (∂Ω) × O to Cqm,α (clS[Ω]− ) which takes (f, κ) to the function wq− [∂Ω, f, κ] is real analytic. Proof. We first consider statements (i), (ii) (see also [13].) It clearly suffices to show that the operators in (i), (ii), are real analytic in C m−1,α (∂Ω) × O(η) for all η in η ∈]0, 1[. Let η ∈]0, 1[. Correspondingly, we take δη ∈]0, δΩ [ as in Proposition 5.1. Possibly shrinking δη , we can assume that clΩω,δ ⊆ Q ∀δ ∈]0, δη ] . Next we take a bounded open connected subset V of Rn of class C ∞ as in (3.10). Next we turn to prove that vq+ [∂Ω, ·, ·]|clΩ is real analytic from C m−1,α (∂Ω)×O(η) to C m,α (clΩ). Let ϕ1 , ϕ2 , ϕ3 ∈ Cc∞ (Rn ) be suchSthat ϕ1 + ϕ2 + ϕ3 = 1 on clV , supp ϕ1 ⊆ Ω, supp ϕ2 ⊆ Ωω,δ , ϕ2 = 1 on ∂Ω, supp ϕ3 ⊆ Rn \ z∈Zn (qz+clΩ) ≡ Ω3 . Then we take ϕ ∈ Cc∞ (Ω) such that 0 ≤ ϕ ≤ 1 and such that ϕ = 1 in a neighborhood of supp ϕ1 . By Sard’s Theorem there exists a ˜ 1 ≡ {x ∈ Ω : ϕ(x) > c}. Obviously, Ω ˜ 1 is an open regular value c ∈]0, 1[ for ϕ. Then we set Ω ∞ ˜ ˜ subset of Ω of class C and supp ϕ1 ⊆ Ω1 ⊆ clΩ1 ⊆ Ω. Similarly, there exists an open subset ˜ 3 of Rn of class C ∞ such that supp ϕ3 ⊆ Ω ˜ 3 ⊆ clΩ ˜ 3 ⊆ Ω3 \ clΩ. By Proposition 5.6 and by Ω m,α m−1,α continuity of the pointwise multiplication in C (clΩ+ (∂Ω)×O(η) ω,δ ), the map from C + m,α + to C (clΩω,δ ) which takes (f, κ) to ϕ2 vq [∂Ω, f, κ]|clΩ+ is real analytic. Since the map ω,δ

m,α from {u ∈ C m,α (clΩ+ (clΩ), which takes u to the extension ω,δ ) : supp u ⊆ supp ϕ2 } to C + uclΩ defined by uclΩ (x) = u(x) if x ∈ clΩω,δ , uclΩ (x) = 0 if x ∈ clΩ \ clΩ+ ω,δ is clearly linear and continuous, we conclude that the map from C m−1,α (∂Ω) × O(η) to C m,α (clΩ) which takes (f, κ) to ϕ2 vq+ [∂Ω, f, κ]|clΩ is real analytic. By applying Lemma 7.3 (i) of the ˜ 1 , r = m + 1, and by Appendix with G replaced by S(x − y, κ), M = ∂Ω, V = O(η), x ∈ clΩ m+1 m,α ˜ ˜ the continuity of the imbedding from C (clΩ1 ) into C (clΩ1 ), and by continuity of the ˜ 1 ), we conclude that the map from C m−1,α (∂Ω) × O(η) pointwise multiplication in C m,α (clΩ ˜ 1 ), which takes (f, κ) to ϕ1 vq+ [∂Ω, f, κ] ˜ is real analytic. Since the map from to C m,α (clΩ |clΩ1 m,α ˜ 1 ) : supp u ⊆ supp ϕ1 } to C m,α (clΩ), which takes u to the extension uclΩ {u ∈ C (clΩ ˜ 1 , uclΩ (x) = 0 if x ∈ clΩ \ clΩ ˜ 1 is clearly linear and defined by uclΩ (x) = u(x) if x ∈ clΩ continuous, we conclude that the map from C m−1,α (∂Ω) × O(η) to C m,α (clΩ), which takes (f, κ) to ϕ1 vq+ [∂Ω, f, κ]|clΩ is real analytic. Then statement (i) follows by the following equality vq+ [∂Ω, f, κ] = ϕ1 vq+ [∂Ω, f, κ] + ϕ2 vq+ [∂Ω, f, κ] in clΩ ,

and by Lemma 7.2 of the Appendix. Statement (ii) can be proved by applying the same argument of the proof of statement (i) to equality vq− [∂Ω, f, κ] = ϕ2 vq− [∂Ω, f, κ] + ϕ3 vq− [∂Ω, f, κ]

in clV \ Ω .

and by invoking Lemma 7.1 of the Appendix. To prove statements (iii), (iv), we argue as in the proof of statements (i), (ii) by invoking Corollary 5.7 instead of Proposition 5.6, and by applying Lemmas 7.1, 7.2, 7.3 of the Appendix to the integral operators of equality Z wq [∂Ω, f, κ](ξ) ≡ − Dξ S(ξ − η, κ)a(2) (κ)νΩ (η)f (η) dση ∂Ω Z t − S(ξ − η, κ)νΩ (η)a(1) (κ)f (η) dση , Ω

25

˜ 1 in case of w+ [∂Ω, f, κ](ξ), and for all ξ ∈ clΩ ˜ 3 in case of w− [∂Ω, f, κ](ξ). 2 for all ξ ∈ clΩ q q

7

Appendix

We first introduce the following two lemmas. Lemma 7.1 Let α ∈]0, 1[, m ∈ N \ {0}. Let Ω be a bounded open connected subset of Rn of class C m,α such that clΩ ⊆ Q and such that Rn \ clΩ is connected. Let V be an open bounded connected subset of Rn such that clQ ⊆ V,

clV ∩ (qz + clΩ) = ∅

∀z ∈ Zn \ {0} ,

(cf. (3.10).) Let W ≡ V \ clΩ. Then the following statements hold. (i) Let k ∈ N. Then the restriction operator from clS[Ω]− to clW induces a linear homeomorphism from Cqk (clS[Ω]− ) onto the subspace Cqk (clW ) n o − = v ∈ C k (clW ) : ∃u ∈ CclS[Ω] u is q − periodic, v = u on clW of C k (clW ). (ii) Let k ∈ N, β ∈]0, 1]. Then the restriction operator of clS[Ω]− to clW induces a linear homeomorphism from Cqk,β (clS[Ω]− ) onto the subspace Cqk,β (clW ) o n − = v ∈ C k,β (clW ) : ∃u ∈ CclS[Ω] u is q − periodic, v = u on clW of C k,β (clW ). Proof. Statement (i) is an immediate consequence of the q-periodicity of u and of the Open Mapping Theorem. We now consider statement (ii). By definition of norm in Cqk,β (clS[Ω]− ) and by statement (i), it clearly suffices to consider case k = 0. Obviously, if u ∈ Cq0,β (clS[Ω]− ), then its restriction u|clW belongs to Cq0,β (clW ). Conversely, if v ∈ Cq0,β (clW ), then there exists a q-periodic function u from clS[Ω]− to C such that v = u|clW . Clearly, |u(x) − u(y)| ≤ |v : clW |β |x − y|β

∀x, y ∈ clW .

Next we set d ≡ inf{|x − y| : (x, y) ∈ clQ × (Rn \ V )} . Clearly, d > 0, and we have   |u(x) − u(y)| ≤ max 2d−β sup |v|, |v : clW |β |x − y|β clW

for all x, y ∈ clS[Ω]− . Accordingly, u ∈ Cq0,β (clS[Ω]− ). Hence, the restriction operator is a bijection of Cq0,β (clS[Ω]− ) onto Cq0,β (clW ). Since the restriction operator is obviously linear 26

and continuous, then the Open Mapping Theorem implies that the restriction operator is a homeomorphism. 2 Then we have a corresponding Lemma for q-periodic functions on clS[Ω]. Lemma 7.2 Let α ∈]0, 1[, m ∈ N \ {0}. Let Ω be a bounded open connected subset of Rn of class C m,α such that clΩ ⊆ Q and such that Rn \ clΩ is connected. Then the following statements hold. (i) Let k ∈ N. Then the restriction operator is a linear homeomorphism from Cqk (clS[Ω]) onto C k (clΩ). (ii) Let k ∈ N, β ∈]0, 1]. Then the restriction operator is a linear homeomorphism from Cqk,β (clS[Ω]) onto C k,β (clΩ). Proof. Statement (i) is an immediate consequence of the q-periodicity of u and of the Open Mapping Theorem. We now consider statement (ii). It clearly suffices to consider case k = 0. Obviously, if u ∈ Cq0,β (clS[Ω]), then its restriction u|clΩ belongs to Cq0,β (clΩ). Conversely, if v ∈ C 0,β (clΩ), then there exists a q-periodic function u from clS[Ω] to C such that v = u|clΩ and |u(x) − u(y)| ≤ |v : clΩ|β |x − y|β

∀x ∈ clΩ.

Then we set d ≡ inf{|x − y| : (x, y) ∈ clΩ × (Rn \ Q)} . Clearly, d > 0, and we have   |u(x) − u(y)| ≤ max 2d−β sup |v|, |v : clΩ|β |x − y|β clΩ

for all x, y ∈ clS[Ω]. Accordingly, u ∈ Cq0,β (clS[Ω]). Hence, the restriction operator is a bijection from Cq0,β (clS[Ω]) onto C 0,β (clΩ). Since the restriction operator is obviously linear and continuous, then the Open Mapping Theorem implies that the restriction operator is a homeomorphism. 2 Finally, we introduce the following Lemma on integral operators (see [15, § 3, § 4].) Lemma 7.3 Let n, s ∈ N, 1 ≤ s < n. Let M be a compact manifold of class C 1 imbedded into Rn and of dimension s. Let K be a Banach space. Then the following statements hold. (i) Let r ∈ N. Let Ω be a bounded open subset of Rn . Let V be an open subset of K. Let W be an open subset of Rn × Rn × K containing clΩ × M × V . Let G be a real analytic map from W to C. Then the map H from V × L1 (M) to C r (clΩ) defined by Z H[z, f ](x) ≡ G(x, y, z)f (y) dσy ∀x ∈ clΩ , M 1

for all (z, f ) ∈ V × L (M) is real analytic. Moreover, if K is a compact subset of V and if B is a bounded subset of L1 (M), then H(K × B) is bounded in C r (clΩ).

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(ii) Let m ∈ N\{0}, α ∈]0, 1[. Let M1 be a compact manifold of class C max{1,m},α imbedded into Rn and of dimension s. Let W be an open subset of Rn × Rn × K. Let G be a real analytic map from W to C. Let  ˜ F ≡ (ψ, φ, z) ∈ C m,α (M1 , Rn ) × C 0 (M, Rn ) × K :  ψ(M1 ) × φ(M) × {z} ⊆ W . ˜ from F˜ × L1 (M) to C m,α (M1 ) defined by The map H Z ˜ H[ψ, φ, z, f ](t) ≡ G(ψ(t), φ(y), z)f (y) dσy

∀t ∈ M1 ,

M

for all (ψ, φ, z, f ) ∈ F˜ × L1 (M), is real analytic.

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