Analytic dependence of a periodic analog of a fundamental solution upon the periodicity paramaters Lanza de Cristoforis, M.; Musolino, P.

Aberystwyth University

Analytic dependence of a periodic analog of a fundamental solution upon the periodicity paramaters

Lanza de Cristoforis, M.; Musolino, P.

Published in:

Annali di Matematica Pura ed Applicata DOI:

10.1007/s10231-017-0715-7 Publication date:

2017

Citation for published version (APA):

Lanza de Cristoforis, M., & Musolino, P. (2017). Analytic dependence of a periodic analog of a fundamental solution upon the periodicity paramaters. Annali di Matematica Pura ed Applicata, 1-28.

https://doi.org/10.1007/s10231-017-0715-7

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Analytic dependence of a periodic analog of a fundamental solution upon the periodicity parameters ^∗

M. Lanza de Cristoforis & P. Musolino

Abstract: We prove an analyticity result in Sobolev-Bessel potential spaces for the periodic analog of the fundamental solution of a general elliptic partial differential operator upon the parameters which determine the periodicity cell. Then we show concrete applications to the Helmholtz and the Laplace operators. In particular, we show that the periodic analogs of the fundamental solution of the Helmholtz and of the Laplace operator are jointly analytic in the spatial variable and in the parameters which determine the size of the periodicity cell. The analysis of the present paper is motivated by the application of the potential theoretic method to periodic anisotropic boundary value problems in which the ‘degree of anisotropy’ is a parameter of the problem.

Keywords: Periodic fundamental solution; Elliptic differential equation; Real analytic dependence; Helmholtz equation; Laplace equation

MSC 2010: 47H30 42B99 31B10 45A05 35J25

1 Introduction

In this paper, we analyze analyticity properties of an analog of the periodic fundamental solution of an elliptic operator with constant coefficients jointly in the spatial variable and in the parameters which determine the size of the periodicity cell. We first introduce some notation. We fix once for all

n ∈ N \ {0, 1} . Then we take

(q

11

, . . . , q

nn

) ∈]0, +∞[

ⁿ

, and we introduce a periodicity cell

Q ≡ Π

ⁿ_j=1

]0, q

jj

[ . Then we denote by q the diagonal matrix

q ≡









q

11

0 . . . 0 0 q

22

. . . 0 . . . . . . . . . . . .

0 0 . . . q

_nn









and by m

_n

(Q) the n-dimensional measure of the fundamental cell Q. Clearly, qZ

ⁿ

≡ {qz : z ∈ Z

ⁿ

} is the set of vertices of a periodic subdivision of R

ⁿ

corresponding to the fundamental cell Q, and accordingly, one can speak about q-periodic functions or distributions in R

ⁿ

. Next we introduce a family of differential operators.

Let N

₂

denote the number of multi-indexes α ∈ N

ⁿ

with |α| ≤ 2. For each c ≡ (c

_α

)

_|α|≤2

∈ C

^N2

, we set c

⁽²⁾

≡ (c

⁽²⁾_lj

)

l,j=1,...,n

c

⁽¹⁾

≡ (c

j

)

j=1,...,n

∗

The authors acknowledge the support of “Progetto di Ateneo: Singular perturbation problems for differential operators – CPDA120171/12” - University of Padova and the support of “INdAM GNAMPA Project 2015 - Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione”. M. Lanza de Cristoforis acknowledges the support of the grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK.

P. Musolino acknowledges the support of an ‘assegno di ricerca INdAM’. P. Musolino has received funding from the European

Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 663830 and

from the Welsh Government and Higher Education Funding Council for Wales through the Sˆ er Cymru National Research Network

for Low Carbon, Energy and Environment.

with c

⁽²⁾_lj

≡ 2

⁻¹

c

e_l+ej

for j 6= l, c

⁽²⁾_jj

≡ c

ej+ej

, and c

j

≡ c

ej

, where {e

j

: j = 1, . . . , n} is the canonical basis of R

ⁿ

. We note that the matrix c

⁽²⁾

is symmetric. If c ∈ C

^N2

, then we set

P [c, x] ≡ X

|α|≤2

c

α

x

^α

∀x ∈ R

ⁿ

.

We also set

E ≡



c ≡ (c

_α

)

_|α|≤2

∈ C

^N2

: inf

ξ∈Rⁿ,|ξ|=1

Re

 X

|α|=2

c

_α

ξ

^α



> 0

 .

Clearly, E coincides with the set of coefficients c ≡ (c

α

)

_|α|≤2

such that the differential operator P [c, D] ≡ X

|α|≤2

c

_α

D

^α

is strongly elliptic and has complex coefficients. As is well known, if c ∈ E , a q-periodic distribution G is a q-periodic fundamental solution of P [c, D] provided that

P [c, D]G = X

z∈Zⁿ

δ

_qz

,

where δ

qz

denotes the Dirac measure with mass at qz, for all z ∈ Z

ⁿ

.

Unfortunately however, not all operators P [c, D] admit q-periodic fundamental solutions, not even in case P [c, D] is the Laplace operator.

Instead, if we denote by E

_2πiξ

, the function defined by

E

_2πiξ

(x) ≡ e

^2πiξ·x

∀x ∈ R

ⁿ

, for all ξ ∈ R

ⁿ

and if c ∈ E , then one can show that the set

Z(c, q) ≡ {z ∈ Z

ⁿ

: P [c, 2πiq

⁻¹

z] = 0}

is finite and that the q-periodic distribution S

c,q

≡ X

z∈Zⁿ\Z(c,q)

1 m

_n

(Q)

1

P [c, 2πiq

⁻¹

z] E

_2πiq−1z

satisfies the equality

P [c, D](S

c,q

) = X

z∈Zⁿ

δ

qz

− X

z∈Z(c,q)

1

m

n

(Q) E

2πiq⁻¹z

(cf. e.g., Ammari and Kang [1, p. 53], [16, §3]). Now let c ∈ E be fixed. We are interested into the analysis of perturbation problems for the kernel S

c,q

and into the dependence of S

c,q

upon q and the spatial variable x, and we note that by perturbing q, the set Z(c, q) is not stable. To circumvent such a difficulty, we fix a finite subset Z of Z

ⁿ

, and we consider those c and q such that Z(c, q) ⊆ Z. Then we note that

S

c,q,Z

≡ X

z∈Zⁿ\Z

1 m

n

(Q)

1

P [c, 2πiq

⁻¹

z] E

_2πiq⁻¹_z

(1)

satisfies the equality

P [c, D](S

_c,q,Z

) = X

z∈Zⁿ

δ

_qz

− X

z∈Z

1

m

_n

(Q) E

_2πiq−1z

. (2)

Equality (2) can be considered as an effective substitute of equality (??), and we say that S

c,q,Z

is a Z-analog of a q-periodic fundamental solution of P [c, D].

Clearly, the distribution S

c,q,Z

differs from S

c,q

by an entire analytic function. Moreover, by interior elliptic regularity theory, both S

c,q,Z

and S

c,q

are analytic in the open set R

ⁿ

\ qZ

ⁿ

.

Let S

c

be a locally integrable real valued function in R

ⁿ

such that

P [c, D]S

c

= δ

0

in R

ⁿ

,

in the sense of distributions. Then S

c

is a fundamental solution for P [c, D] and the function S

c,q,Z

− S

c

can be extended to an analytic function in (R

ⁿ

\ qZ

ⁿ

) ∪ {0}.

We denote such an extension of S

c,q,Z

− S

c

by the symbol R

c,q,Z,S_c

, and we say that R

c,q,Z,S_c

is the regular part of S

c,q,Z

(with respect to S

c

). Obviously, R

c,q,Z,S_c

is not a q-periodic function.

In this paper we are interested into various questions on the analyticity of S

c,q,Z

and R

c,q,Z,S_c

in the variable (q, x). Here the difficulty is that the series in (1) is known to converge only in the sense of distributions. We mention that Lin and Wang [20], Mityushev and Adler [24], and Mamode [22] have proved the validity of a constructive formula for a q-periodic analog of the fundamental solution for the Laplace operator in case n = 2 via elliptic functions which would imply the analyticity of S

_c,q,Z

and R

_c,q,Z,Sc

in the variable (q, x).

However, we are not aware of such formulas for n ≥ 3 or for elliptic differential operators other than the Laplace operator.

We denote by D

ⁿ

(R) the space of n×n diagonal matrices with real entries and by D

⁺n

(R) the set of elements of D

ⁿ

(R) with diagonal entries in ]0, +∞[.

We note that if we fix s ∈ R such that s − 2 < −(n/2), then S

^c,q,Z

belongs to the Sobolev-Bessel potential space of I-periodic functions H

_I^s

(R

ⁿ

), and we prove that the map from the set of q in D

⁺n

(R) such that Z(c, q) ⊆ Z to H

I^s

(R

ⁿ

) which takes q to S

c,q,Z

◦ q is real analytic (see Theorem 3.1). Here I denotes the n × n identity matrix.

Then as an application we consider the Helmholtz operator ∆ + κ

²

for some κ ∈ C, and we denote by c(κ), the element of E such that P [c(κ), D] = ∆ + κ

²

(cf. (14)–(16)). In this paper, we consider only the case in which κ = 0 and Z(c(0), q) ⊆ Z = {0}, and the case in which κ 6= 0 and Z(c(κ), q) ⊆ Z = ∅, a case in which

−κ

²

is not an eigenvalue for ∆ in the space of q-periodic distributions in R

ⁿ

. Then we prove that if Ω is a bounded open subset of R

ⁿ

\ Z

ⁿ

, m ∈ N, α ∈]0, 1[, then the map which takes q to the restriction to clΩ of the function

S

_c(κ),q,Z

◦ q(x) ≡ S

_c(κ),q,Z

(qx) ∀x ∈ R

ⁿ

\ Z

ⁿ

, is real analytic from suitable subsets of D

⁺n

(R) to C

^m,α

(clΩ) (see Theorem 5.2).

Then we prove an analyticity result for the regular part of S

_c(κ),q,Z

in the Roumieu space C

_ω,ρ⁰

(clΩ) of real analytic functions in clΩ in case clΩ ⊆ (R

ⁿ

\ Z

ⁿ

) ∪ {0} (cf. (4) and see Theorem 5.3).

As a consequence of our results, we prove that the function R

_{c(κ),q,Z,Sc(κ)}

(qx) is analytic in the variable (q, x) (cf. Theorem 5.5), and that S

_c(κ),q,Z

(qx) is analytic in the variable (q, x) (cf. Theorem 5.7). In particular, we can deduce that the sum of the series

− X

z∈Zⁿ\{0}

1

m

_n

(Q)4π

²

|q

⁻¹

z|

²

e

^{2πi(q−1z)·x}

, (3) which converges in the sense of distributions to an analog of the q-periodic fundamental solution of the Laplace operator defines an analytic function of (q, x) ∈ D

⁺n

(R) × R

ⁿ

such that q

⁻¹

x / ∈ Z

ⁿ

, i.e., jointly in the variables q and x (see Example 5.8 at the end of the paper). For a corresponding example for the Helmholtz operator, see Example 5.9 at the end of the paper.

A central tool in periodic potential theory is represented by analogs of the periodic fundamental solution.

As an example, Ammari, Kang, and Touibi [3] have exploited an integral equation method to solve a periodic linear transmission problem for the Laplace equation and to derive effective properties of composite materials.

Such an approach has been successfully exploited also for the study of the effective parameters of elastic composites in Ammari, Kang, and Lim [2] (see also Ammari and Kang [1]).

An approach based on potential theory has been useful also for the analysis of nonlinear periodic problems.

For example, in [18] a quasi linear heat transmission problem has been investigated by means of integral equations, whereas in [10] such an analysis has been performed for a nonlinear traction problem.

We also mention that Arens, Sandfort, Schmitt, and Lechleite [4], Berman and Greengard [5], Tornberg and Greengard [26] have investigated the problem of actually computing the sum of series as that of (3).

The analysis of the present paper is motivated by the application of the potential theoretic method to boundary value problems corresponding to anisotropic periodic problems in which the sizes q

11

,. . . ,q

nn

of the periodic cell are subject to perturbation. Indeed, if one wants to apply periodic potential theory to study the dependence of the solution of a periodic boundary value problem upon the parameters q

₁₁

,. . . ,q

_nn

which determine the anisotropy of the problem, then one faces the problem to study the corresponding dependence for the fundamental solution on which the potentials are based. In particular, an analyticity result upon the parameters q

₁₁

,. . . ,q

_nn

allows to to justify representation formulas for the solutions or for functionals related to the solutions in terms of power series in q

11

,. . . ,q

nn

and therefore also polynomial asymptotic expansions of any desired degree with precise estimates on the remainder.

This paper continues the work of the authors and collaborators on the study of the behavior of the funda-

mental solution of an elliptic partial differential operator upon perturbation of the coefficients. For example,

in [7], Dalla Riva has constructed a family of fundamental solutions for elliptic partial differential operators with real constant coefficients, where the elements of such a family are expressed by means of real analytic functions of the coefficients of the operators and of the spatial variable. Then a corresponding result for elliptic partial differential operators with quaternion constant coefficients has been shown in [9].

The paper is organized as follows. In section 2 we introduce some preliminaries, in particular on periodic distributions. In section 3, we prove the analyticity result on S

c,q,Z

◦ q upon q ∈ D

⁺n

(R) in Sobolev-Bessel spaces. In section 4 we compute all the differentials of a map related to S

_c,q,Z

◦ q, which we need in the sequel.

In section 5 we consider the Helmholtz operator. We first prove the above mentioned analyticity result for S

_c(κ),q,Z

◦ q

_|clΩ

, with values in C

^m,α

(clΩ).

Then we prove the analyticity of R

_{c(κ),q,Z,Sc(κ)}

◦ q

_|clΩ

upon q in Roumieu spaces and the joint analyticity of the function R

_c(κ),q,Z,Sc

(qx) upon (q, x). Finally, we prove the analyticity S

_c(κ),q,Z

◦ q

_|clΩ

upon q in Roumieu spaces and the joint analyticity of the function S

_c(κ),q,Z

(qx) upon (q, x).

2 Preliminaries and notation

We denote the norm on a normed space X by k · k

X

. Let X and Y be normed spaces. We endow the space X × Y with the norm defined by k(x, y)k

X ×Y

≡ kxk

X

+ kyk

Y

for all (x, y) ∈ X × Y, while we use the Euclidean norm for R

ⁿ

. The symbol N denotes the set of natural numbers including 0. L

^(j)

(X , Y) denotes the space of j-linear and continuous operators from X

^j

to Y for all j ∈ N. Let E ⊆ R

ⁿ

. Then clE denotes the closure of E and ∂E denotes the boundary of E. For all R > 0, x ∈ R

ⁿ

, x

_j

denotes the j-th coordinate of x, |x|

denotes the Euclidean modulus of x in R

ⁿ

, and B

n

(x, R) denotes the ball {y ∈ R

ⁿ

: |x − y| < R}. A dot “·”

denotes the inner product in R

ⁿ

, or the matrix product between matrices. Let Ω be an open subset of R

ⁿ

. The space of m times continuously differentiable complex-valued functions on Ω is denoted by C

^m

(Ω, R), or more simply by C

^m

(Ω). Let r ∈ N \ {0}. Let f ∈ (C

^m

(Ω))

^r

. The s-th component of f is denoted f

_s

, and Df denotes the Jacobian matrix 

_∂f

s

∂xl



s=1,...,r, l=1,...,n

. Let η ≡ (η

1

, . . . , η

n

) ∈ N

ⁿ

, |η| ≡ η

1

+ · · · + η

n

. Then D

^η

f denotes

∂^|η|f

∂x^η1₁ ...∂x^ηn_n

. We denote by D(R

ⁿ

) the space of functions of class C

^∞

(R

ⁿ

) with compact support. The subspace of C

^m

(Ω) of those functions f whose derivatives D

^η

f of order |η| ≤ m can be extended with continuity to clΩ is denoted C

^m

(clΩ). The subspace of C

^m

(clΩ) whose functions have m-th order derivatives that are H¨ older continuous with exponent α ∈]0, 1] is denoted C

^m,α

(clΩ) (cf. e.g., Gilbarg and Trudinger [13]). Let E ⊆ R

^r

. Then C

^m,α

(clΩ, E) denotes {f ∈ (C

^m,α

(clΩ))

^r

: f (clΩ) ⊆ E}.

We say that a bounded open subset Ω of R

ⁿ

is of class C

^m

or of class C

^m,α

, if clΩ is a manifold with boundary imbedded in R

ⁿ

of class C

^m

or C

^m,α

, respectively (cf. e.g., Gilbarg and Trudinger [13, §6.2]). For standard properties of functions in Schauder spaces both on clΩ and on ∂Ω, we refer the reader to Gilbarg and Trudinger [13] (see also [15, §2, Lem. 3.1, 4.26, Thm. 4.28], [19, §2]).

We denote by dσ the area element of a manifold M imbedded into R

ⁿ

. We retain the standard notation for the Lebesgue space L

^p

(M ) of p-summable functions. We note that throughout the paper ‘analytic’ means always ‘real analytic’. For the definition and properties of analytic operators, we refer to Deimling [11, §15].

Next, we turn to introduce the Roumieu classes. For all bounded open subsets Ω of R

ⁿ

and ρ > 0, we set C

_ω,ρ⁰

(clΩ) ≡



u ∈ C

^∞

(clΩ) : sup

β∈Nⁿ

ρ

^|β|

|β|! kD

^β

uk

_C0(clΩ)

< +∞



, (4)

and

kuk

C⁰_ω,ρ(clΩ)

≡ sup

β∈Nⁿ

ρ

^|β|

|β|! kD

^β

uk

C⁰(clΩ)

∀u ∈ C

_ω,ρ⁰

(clΩ) , where |β| ≡ β

₁

+· · ·+β

_n

for all β ≡ (β

₁

, . . . , β

_n

) ∈ N

ⁿ

. As is well known, the Roumieu class 

C

_ω,ρ⁰

(clΩ), k · k

_C0 ω,ρ(clΩ)

 is a Banach space.

We denote by S(R

ⁿ

) the Schwartz space of rapidly decreasing functions, and by S

⁰

(R

ⁿ

) the space of tempered distributions in R

ⁿ

, and by S

_I⁰

(R

ⁿ

) the subspace of S

⁰

(R

ⁿ

) of the I-periodic elements of S

⁰

(R

ⁿ

), i.e., of the tempered distributions which are periodic with respect to the fundamental cell ]0, 1[

ⁿ

. If f is a complex valued integrable function in R

ⁿ

, then we define the Fourier transform of f as follows

f (y) ≡ (2π) ˆ

^−n/2

ˆ

Rⁿ

e

^−iy·x

f (x) dx ∀y ∈ R

ⁿ

,

and we still use the symbol ‘ ˆ ’ to denote the corresponding Fourier transform in the space of tempered

distributions. Next we introduce the following characterization of S

_I⁰

(R

ⁿ

) of Triebel [27] (see also Schmeisser

and Triebel [25, 3.2.3]).

Proposition 2.1. If {a

z

}

_z∈Zⁿ

is a family of complex numbers such that there exists m ∈ N such that sup

z∈Zⁿ

|a

z

|

(1 + |z|

²

)

^m/2

< +∞ , (5)

then the Fourier series

X

z∈Zⁿ

a

z

e

^i2πz·x

(6)

converges in S

⁰

(R

ⁿ

) endowed with the weak

^∗

-topology to an element of S

_I⁰

(R

ⁿ

).

Conversely, if u ∈ S

_I⁰

(R

ⁿ

), then there exists a unique family {a

z

(u)}

_z∈Zⁿ

in C which satisfies condition (5) for some m ∈ N and such that u equals the sum of the Fourier series in (6) with a

^z

replaced by a

z

(u).

Moreover,

a

k

(u) = (2π)

^−n/2

< u, [ϕ(· − 2πk)]

^∧

> ∀k ∈ Z

ⁿ

, (7) for all ϕ ∈ D(R

ⁿ

) which have support contained in the ball B

n

(0, 2π) and such that ϕ(0) = 1.

We also note that the map from S

_I⁰

(R

ⁿ

) endowed with the weak

^∗

-tolology to C which takes u to a

k

(u) is linear and continuous for all k ∈ Z

ⁿ

.

If s ∈ R, then we denote by H

^s

(R

ⁿ

) the Sobolev-Bessel space of tempered distributions u such that (1 + |y|

²

)

^s/2

u(y) belongs to L ˆ

²

(R

ⁿ

), and we set

kuk

_Hs(Rⁿ)

≡ k(1 + |y|

²

)

^s/2

ˆ u(y)k

_L2(Rⁿ)

∀u ∈ H

^s

(R

ⁿ

) . It is well known that (H

^s

(R

ⁿ

), k · k

H^s(Rⁿ)

) is a Banach space. Then we set

H

_loc^s

(R

ⁿ

) ≡ {u ∈ S

⁰

(R

ⁿ

) : uϕ ∈ H

^s

(R

ⁿ

) ∀ϕ ∈ D(R

ⁿ

)} , and

kuk

_H^s

loc(Rⁿ),ϕ

≡ kuϕk

_Hs(Rⁿ)

∀u ∈ H

_loc^s

(R

ⁿ

) ,

for all ϕ ∈ D(R

ⁿ

). Then it is well known that H

_loc^s

(R

ⁿ

) endowed with the family of seminorms Φ ≡ {k · k

_Hs

loc(Rⁿ),ϕ

: ϕ ∈ D(R

ⁿ

)} is a Fr´ echet space. Next we introduce the space H

_I^s

(R

ⁿ

) ≡ H

_loc^s

(R

ⁿ

) ∩ S

_I⁰

(R

ⁿ

) .

Clearly, H

_I^s

(R

ⁿ

) is a closed subspace of the Fr´ echet space H

_loc^s

(R

ⁿ

) and is accordingly a Fr´ echet space. Next we fix an arbitrary η ∈ D(R

ⁿ

) such that there exists an open neighborhood U of [0, 1]

ⁿ

such that

η(x) = 1 ∀x ∈ U . (8)

Since the tempered distributions of H

_I^s

(R

ⁿ

) are I-periodic, one can easily verify that k · k

H_loc^s (Rⁿ),η

is actually a norm on H

_I^s

(R

ⁿ

). Since k · k

H_loc^s (Rⁿ),η

belongs to Φ and H

_I^s

(R

ⁿ

) is a Fr´ echet space, we already know that (H

_I^s

(R

ⁿ

), k · k

H^s_loc(Rⁿ),η

) is complete. Then the Open Mapping Theorem in Fr´ echet spaces implies that the continuous identity map from (H

_I^s

(R

ⁿ

), Φ) to (H

_I^s

(R

ⁿ

), k · k

_Hs

loc(Rⁿ),η

) is actually a homeomorphism and that accordingly k · k

_Hs

loc(Rⁿ),η

generates the topology of the Fr´ echet space (H

_I^s

(R

ⁿ

), Φ), no matter how we choose η ∈ D(R

ⁿ

) as in (8).

Now let Ω be an open subset of R

ⁿ

, then we denote by H

^s

(Ω) the set of restrictions to Ω of the tempered distributions of H

^s

(R

ⁿ

), and we set

kuk

_Hs(Ω)

≡ inf{kvk

_Hs(Rⁿ)

: v ∈ H

^s

(R

ⁿ

), v

_|Ω

= u} ∀u ∈ H

^s

(Ω) .

It is well known that (H

^s

(Ω), k · k

_Hs(Ω)

) is a Banach space. We note that such a definition of H

^s

(Ω) coincides with other ‘intrinsic’ definitions of H

^s

(Ω) only in case Ω satisfies some regularity assumption.

If Ω is a bounded open subset of R

ⁿ

, and if we choose η ∈ D(R

ⁿ

) as in (8) such that η equals one in a neighborhood of clΩ, then the definition of the norm in H

^s

(Ω) implies that the restriction map is linear and continuous from H

_I^s

(R

ⁿ

) to H

^s

(Ω) and that

ku

_|Ω

k

_Hs(Ω)

≤ kuk

_H^s

loc(Rⁿ),η

∀u ∈ H

_I^s

(R

ⁿ

) . (9)

If c ∈ E , q ∈ D

⁺n

(R), then we set

L

c,q

[u] ≡ X

|α|≤2

(q

⁻¹

)

^α

c

α

D

^α

u ∀u ∈ S

⁰

(R

ⁿ

) ,

where (q

⁻¹

)

^α

≡ q

₁₁^−α1

. . . q

^−α_nnⁿ

for all α ∈ N

ⁿ

, and we have the following.

Lemma 2.2. Let c ∈ E . Let S

c

be a fundamental solution for P [c, D]. Let q ∈ D

⁺n

(R). Let Z be a finite subset of Z

ⁿ

such that Z(c, q) ⊆ Z. Then the following statements hold.

(i)

L

c,q

[(det q)S

c,q,Z

◦ q] = − X

z∈Z

E

2πiz

in R

ⁿ

\ Z

ⁿ

, and

L

c,q

[(det q)R

c,q,Z,Sc

◦ q] = − X

z∈Z

E

2πiz

in (R

ⁿ

\ Z

ⁿ

) ∪ {0} .

(ii) L

_c,q

[(det q)S

_c,q,Z

◦ q] = P

z∈Zⁿ

δ

_z

− P

z∈Z

E

_2πiz

in S

⁰

(R

ⁿ

).

(iii) a

_k

( P

z∈Zⁿ

δ

_z

− P

z∈Z

E

_2πiz

) = 0 for all k ∈ Z (cf. (7)).

Proof. The first equality in (i) follows by the chain rule. Since L

c,q

[(det q)S

c

◦ q] = 0 in R

ⁿ

\ {0}, the second equality in statement (i) holds true in R

ⁿ

\ Z

ⁿ

. Since R

c,q,Z,S_c

can be continued analytically in a neighborhood of 0, the second equality in statement (i) holds true in (R

ⁿ

\ Z

ⁿ

) ∪ {0}. We now prove statement (ii). The equality of statement (ii) holds if and only if

X

|α|≤2

(−q

⁻¹

)

^α

c

_α

ˆ

Rⁿ

(det q)S

_c,q,Z

(qξ)D

^α

ψ(ξ) dξ

= X

z∈Zⁿ

ψ(q

⁻¹

qz) − X

z∈Z

ˆ

Rⁿ

ψ(ξ)e

^i2πz·ξ

dξ ,

for all ψ ∈ S(R

ⁿ

). By setting x = qξ, we rewrite such an equality as ˆ

Rⁿ

S

c,q,Z

(x) X

|α|≤2

(−1)

^|α|

c

α

D

^α

(ψ(q

⁻¹

x)) dx

= X

z∈Zⁿ

< δ

qz

, ψ(q

⁻¹

·) > − X

z∈Z

ˆ

Rⁿ

ψ(q

⁻¹

x)e

^{i2πq−1z·x}

dx(det q)

⁻¹

,

for all ψ ∈ S(R

ⁿ

). Now such an equality is certainly satisfied. Indeed, ψ(q

⁻¹

·) belongs to S(R

ⁿ

) and P [c, D]S

_c,q,Z

= X

z∈Zⁿ

δ

_qz

− X

z∈Z

1

m

n

(Q) E

_2πiq−1z

in S

⁰

(R

ⁿ

) . Finally, to prove statement (iii), we note that the Poisson summation formula implies that

a

k

X

z∈Zⁿ

δ

z

− X

z∈Z

E

2πiz

!

= a

k

X

z∈Zⁿ

E

2πiz

− X

z∈Z

E

2πiz

!

= 0 ,

for all k ∈ Z.

Then we find convenient to introduce the following notation. Let Z be a finite subset of Z

ⁿ

. For each s ∈ R, we set

H

_I,Z^s

(R

ⁿ

) ≡ {u ∈ H

_I^s

(R

ⁿ

) : a

_k

(u) = 0 ∀k ∈ Z}

(cf. (7)). By the continuity of the functionals a

_k

(·) of (7) on S

_I⁰

(R

ⁿ

), the space H

_I,Z^s

(R

ⁿ

) is closed in H

_I^s

(R

ⁿ

), and it is accordingly a Banach space. Then we need the following result on the operator L

_c,q

in the space S

_I⁰

(R

ⁿ

) of I-periodic tempered distributions.

Proposition 2.3. Let c ∈ E . Let q ∈ D

⁺n

(R). Let Z be a finite subset of Z

ⁿ

such that Z(c, q) ⊆ Z. Then the following statements hold.

(i) Let u ∈ S

_I⁰

(R

ⁿ

). Then L

c,q

[u] = 0 if and only if u belongs to the complex vector space generated by {E

2πik

: k ∈ Z(c, q)}.

(ii) Let f ∈ S

_I⁰

(R

ⁿ

). Then there esists u ∈ S

_I⁰

(R

ⁿ

) such that f = L

c,q

[u] if and only if a

k

(f ) = 0 for all

k ∈ Z(c, q) (cf. (7)). Moreover, if a

^k

(f ) = 0 for all k ∈ Z we can choose u so that a

k

(u) = 0 for all

k ∈ Z.

(iii) Let s ∈ R. Then the operator L

^c,q

maps H

_I^s

(R

ⁿ

) onto the subspace H

_I,Z(c,q)^s−2

(R

ⁿ

) of H

_I^s−2

(R

ⁿ

), and restricts a linear homeomorphism from the space H

_I,Z^s

(R

ⁿ

) onto H

_I,Z^s−2

(R

ⁿ

).

Proof. (i) The sufficiency of the condition follows by the chain rule and by the definition of Z(c, q). To prove necessity, we note that equality L

c,q

[u] = 0 implies that

P [c, D]u(q

⁻¹

·) = 0 .

On the other hand, by Proposition 2.1 there exists a unique family {a

z

(u)}

_z∈Zⁿ

in C as in (5) such that u(q

⁻¹

·) = X

z∈Zⁿ

a

z

(u)E

_2πiq⁻¹_z

in S

⁰

(R

ⁿ

) .

Accordingly,

0 = P [c, D]u(q

⁻¹

·) = X

z∈Zⁿ

a

z

(u)P [c, 2πiq

⁻¹

z]E

_2πiq⁻¹_z

and so

a

z

(u)P [c, 2πiq

⁻¹

z] = 0 ∀z ∈ Z

ⁿ

.

Thus, a

z

(u) = 0 for all z ∈ Z

ⁿ

\ Z(c, q). Hence, the validity of statement (i) follows.

(ii) We first assume that u exists. Then Proposition 2.1 implies that there exist m ∈ N and a family of complex numbers {b

_z

}

_z∈Zⁿ

satisfying condition (5) and equality u = P

z∈Zⁿ

b

_z

e

^i2πz·x

in S

⁰

(R

ⁿ

) endowed with the weak

^∗

-topology. Then by applying L

_c,q

, we obtain

L

c,q

[u] = X

z∈Zⁿ\Z(c,q)

P [c, 2πiq

⁻¹

z]
b

z

e

^i2πz·x

and accordingly a

k

(L

c,q

[u]) = 0 for all k ∈ Z(c, q).

Next we assume that a

k

(f ) = 0 for all k ∈ Z and we show the existence of u. By Proposition 2.1 there exist m ∈ N and a family of complex numbers {a

^z

}

_z∈Zⁿ

satisfying condition (5) and equality f = P

z∈Zⁿ

a

z

e

^i2πz·x

in S

⁰

(R

ⁿ

) endowed with the weak

^∗

-topology. Since a

k

(f ) = 0 for all k ∈ Z, we have a

k

= 0 for all k ∈ Z.

Then we set

b

k

≡ 0 ∀k ∈ Z , b

z

≡ a

z

P [c, 2πiq

⁻¹

z] ∀z ∈ Z

ⁿ

\ Z .

Clearly, {b

z

}

_z∈Zⁿ

satisfies condition (5) with m replaced by m − 2, and accordingly the series P

z∈Zⁿ

b

z

e

^i2πz·x

converges in the weak

^∗

-topology and defines a I-periodic element u of S

⁰

(R

ⁿ

). Moreover, a

k

(u) = 0 for all k ∈ Z. By definition of the coefficients {b

_z

}

_z∈Zⁿ

, we have L

_c,q

[u] = f .

(iii) Since H

_I^s

(R

ⁿ

) = H

_loc^s

(R

ⁿ

) ∩ S

_I⁰

(R

ⁿ

), the second order differential operator L

_c,q

is linear and continuous from H

_I^s

(R

ⁿ

) to H

_I^s−2

(R

ⁿ

). By (ii), L

_c,q

maps H

_I^s

(R

ⁿ

) into H

^s−2

I,Z(c,q)

(R

ⁿ

). On the other hand if f ∈ H

_I^s−2

(R

ⁿ

) and a

k

(f ) = 0 for all k ∈ Z(c, q), then statement (ii) ensures the existence of u ∈ S

I⁰

(R

ⁿ

) such that f = L

c,q

[u]

and a

k

(u) = 0 for all k ∈ Z(c, q). Since f ∈ H

loc^s−2

(R

ⁿ

), then classical elliptic regularity theory ensures that u ∈ H

_loc^s

(R

ⁿ

) (cf. e.g., Folland [12, (6.33), p. 214]). Thus we conclude that u ∈ H

_loc^s

(R

ⁿ

) ∩ S

_I⁰

(R

ⁿ

) = H

_I^s

(R

ⁿ

).

Similarly, (ii) implies that L

c,q

maps H

_I,Z^s

(R

ⁿ

) onto H

_I,Z^s−2

(R

ⁿ

). If u ∈ H

_I^s

(R

ⁿ

), and L

c,q

[u] = 0, and a

k

(u) = 0 for all k ∈ Z, then point (i) ensures that u belongs to the complex vector space generated by {E

_2πik

: k ∈ Z(c, q)} and thus condition a

k

(u) = 0 for all k ∈ Z ensures that u = 0. Since both H

_I,Z^s

(R

ⁿ

) and H

_I,Z^s−2

(R

ⁿ

) are closed subspaces of Banach spaces, they are Banach spaces and the Open Mapping Theorem ensures that also the last part of statement (iii) holds true.

3 An analyticity result for (det q)S _c,q,Z ◦q in Sobolev-Bessel potential spaces

Theorem 3.1. Let c ∈ E . Let s ∈ R be such that s − 2 < −n/2. Let Z be a finite subset of Z

ⁿ

. Let W be an open subset of D

⁺n

(R) such that Z(c, q) ⊆ Z for all q ∈ W. Then the map S

I,c,Z^]

from W to H

_I,Z^s

(R

ⁿ

) defined by

S

^]_I,c,Z

(q) ≡ (det q)S

c,q,Z

◦ q ∀q ∈ W ,

is real analytic.

Proof. Let q ∈ W. We first prove that (det q)S

c,q,Z

◦ q is the only I-periodic tempered distribution in R

ⁿ

which satisfies the following system

 L

c,q

[v] = P

z∈Zⁿ

δ

z

− P

z∈Z

E

2πiz

,

a

k

(v) = 0 ∀k ∈ Z . (10)

By Lemma 2.2 (ii), (det q)S

_c,q,Z

◦ q satisfies the first equation in (10). Moreover, (1) implies that a

k

((det q)S

c,q,Z

◦ q) = 0 ∀k ∈ Z .

Hence, (det q)S

c,q,Z

◦ q satisfies system (10). Next we assume that v is an I-periodic tempered distribution in R

ⁿ

which satisfies system (10). Then

L

c,q

[v − (det q)S

c,q,Z

◦ q] = 0 ,

and Proposition 2.3 (i) ensures that v − (det q)S

_c,q,Z

◦ q belongs to the complex vector space generated by {E

_2πik

: k ∈ Z(c, q)}, and the second equation in (10) ensures that v − (det q)S

c,q,Z

◦ q = 0.

Next we show that the right hand side of the first equation in system (10) belongs to H

_I^s−2

(R

ⁿ

). Since s − 2 < −n/2, we have δ

z

∈ H

^s−2

(R

ⁿ

) for all z ∈ Z

ⁿ

. Hence, P

z∈Zⁿ

δ

z

belongs to H

_loc^s−2

(R

ⁿ

). Since P

z∈Zⁿ

δ

z

is obviously I-periodic, we also have P

z∈Zⁿ

δ

z

∈ H

_I^s−2

(R

ⁿ

). Accordingly, the right hand side of the first equation in system (10) belongs to H

_I^s−2

(R

ⁿ

).

Since (det q)S

c,q,Z

◦ q is the only tempered distribution which satisfies system (10), the membership of the right hand side of the first equation in system (10) to H

_I^s−2

(R

ⁿ

) and Proposition 2.3 imply in particular that (det q)S

c,q,Z

◦ q belongs to H

_I^s

(R

ⁿ

) and that (det q)S

c,q,Z

◦ q is the only element of H

_I^s

(R

ⁿ

) which satisfies system (10).

Next we consider the map A from W × H

_I,Z^s

(R

ⁿ

) to the space H

_I,Z^s−2

(R

ⁿ

) defined by the equality A(q, v) ≡ L

_c,q

[v] − X

z∈Zⁿ

δ

_z

+ X

z∈Z

E

_2πiz

for all (q, v) ∈ W × H

_I,Z^s

(R

ⁿ

). By our proof above, the set of zeros of A coincides with the graph of S

_I,c,Z^]

. Moreover, A is real analytic and if q ∈ W, then partial the Fr´ echet differential d

_v

A(q, S

_I,c,Z^]

[q]) of A at the point (q, S

_I,c,Z^]

[q]) with respect to the variable v coincides with the map L

c,q

from H

_I,Z^s

(R

ⁿ

) to H

_I,Z^s−2

(R

ⁿ

), which is a linear homeomorphism by Proposition 2.3 (iii). Then the Implicit Function Theorem in Banach spaces implies that S

_I,c,Z^]

is real analytic (cf. e.g., Deimling [11, Thm. 15.3]).

4 Explicit computation of the differentials of a map related to (det q)S c,q,Z ◦ q for a particular class of elliptic differential oper- ators

We now introduce the set E

₀

≡



c ≡ (c

_α

)

_|α|≤2

∈ E : c

_α

= 0 if |α| = 1

and c

α

= 0 if α = e

l

+ e

j

with j, l ∈ {1, . . . , n}, j 6= l

 .

Let c ∈ E

0

, q ∈ D

⁺n

(R). Since we will soon have to perform computations involving high order derivatives, we find convenient to set

L ˜

c,b

[v] ≡

n

X

j=1

b

jj

c

⁽²⁾_jj

∂

²

∂x

²_j

v + c

0

v ∀v ∈ H

_I^s

(R

ⁿ

) , for all b ∈ D

n

(R) and s ∈ R. Obviously,

L ˜

_c,q⁻²

= L

c,q

q ∈ D

⁺n

(R) , L ˜

c,b

= L

_c,b−1/2

∀b ∈ D

⁺n

(R) ,

where q

⁻²

denotes the diagonal matrix with diagonal entries q

_ij⁻²

, j = 1, . . . , n, and where b

^−1/2

denotes the diagonal matrix with diagonal entries b

^−1/2_jj

, j = 1, . . . , n. We also note that

L

c,q

[(det q)S

c,q,Z

◦ q] = L

_c,b−1/2

h

(det b

^−1/2

)S

_c,b−1/2,Z

◦ b

^−1/2

i

whenever b = q

⁻²

, q ∈ D

⁺n

(R), and that accordingly equality L

c,q

[(det q)S

c,q,Z

◦ q] = X

z∈Zⁿ

δ

z

− X

z∈Z

E

2πiz

∀q ∈ D

⁺n

(R) ,

is equivalent to the equality L

_c,b−1/2

h

(det b

^−1/2

)S

_c,b−1/2,Z

◦ b

^−1/2

i

= X

z∈Zⁿ

δ

_z

− X

z∈Z

E

_2πiz

∀b ∈ D

⁺n

(R) ,

an equality which we rewrite in the form L ˜

c,b

h

(det b

^−1/2

)S

_c,b−1/2,Z

◦ b

^−1/2

i

= X

z∈Zⁿ

δ

z

− X

z∈Z

E

2πiz

∀b ∈ D

⁺n

(R) . (11)

We also note that the following commutativity property holds.

Lemma 4.1. Let c, c

^]

∈ E

0

. Let b ∈ D

⁺n

(R). Let Z be a finite subset of Z

ⁿ

such that Z(c, b

^−1/2

) ⊆ Z. Let s ∈ R. Then ˜ L

c,b

restricts a linear homeomorphism from H

_I,Z^s

(R

ⁿ

) onto H

_I,Z^s−2

(R

ⁿ

) and

L ˜

c,b

◦ ( ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

_c],v

)[u] = ( ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

_c],v

) ◦ ˜ L

c,b

[u] ∀u ∈ H

_I,Z^s

(R

ⁿ

) , for all v ∈ D

ⁿ

(R).

Proof. The first part of the statement is an immediate consequence of Proposition 2.3 (iii). In order to prove the formula of the statement, we set u

1

≡ ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

_c],v

[u]. Then we have ˜ L

c,b

[u

1

] = ˜ L

_c],v

[u]. Since L ˜

c,b

◦ ˜ L

_c],v

= ˜ L

_c],v

◦ ˜ L

c,b

, we have ˜ L

c,b

◦ ˜ L

c,b

[u

1

] = ˜ L

_c],v

◦ ˜ L

c,b

[u] and thus the formula of the statement follows.

If c ∈ E

₀

, we denote by c

^∗

the element of E

₀

defined by

c

^∗_α

= c

_α

if |α| = 2, c

^∗_α

= 0 if |α| < 2.

Then we have the following, which provides a formula for the composite function S

^×_I,c,Z

of S

^]_I,c,Z

and of b

^−1/2

at the point b computed at (v

1

, . . . , v

j

) with v

1

= · · · = v

j

≡ v for all natural numbers j, which we need to write the Taylor formula for S

_I,c,Z^×

at the point b.

Proposition 4.2. Let c ∈ E

0

. Let s ∈ R be such that s − 2 < −n/2. Let Z be a finite subset of Z

ⁿ

. Let f W be an open subset of D

⁺n

(R) such that Z(c, b

^−1/2

) ⊆ Z for all b ∈ f W. Let S

_I,c,Z^×

be the map from f W to H

_I^s

(R

ⁿ

) defined by

S

_I,c,Z^×

(b) ≡ S

_I,c,Z^]

(b

^−1/2

) ∀b ∈ f W . Then the following statements hold.

(i) ˜ L

_c,b

[S

_I,c,Z^×

(b)] = P

z∈Zⁿ

δ

_z

− P

z∈Z

E

_2πiz

for all b ∈ f W.

(ii) Let j ∈ N \ {0}. The j-order differential of the map S

^×I,c,Z

at the point b satisfies the equality

L ˜

c,b



d

^j

S

_I,c,Z^×

(b)[

j times

z }| { v, . . . , v]



 = −j ˜ L

c^∗,v



d

^j−1

S

^×_I,c,Z

(b)[

(j − 1) times

z }| { v, . . . , v ]



 ,

for all b ∈ f W and v ∈ D

n

(R).

(iii) Let b ∈ f W. Then

d

^j

S

^×_I,c,Z

(b)[

j times

z }| {

v, . . . , v] = (−1)

^j

j!  ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

c^∗,v



j

[S

_I,c,Z^×

(b)] ∀v ∈ D

n

(R) .

for all j ∈ N.

(iv) Let b ∈ f W. Then L ˜

c,b



d

^j

S

_I,c,Z^×

(b)[

j times

z }| { v, . . . , v]



 = (−1)

^j

j!  ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

c^∗,v



^j

h X

z∈Zⁿ

δ

z

− X

z∈Z

E

2πiz

i

for all v ∈ D

n

(R) and j ∈ N \ {0}.

(v) Let j ∈ N \ {0}. If b ∈ f W and v ∈ D

n

(R), then the function d

^j

S

_I,c,Z^×

(b)[v, . . . , v] is real analytic in R

ⁿ

\ Z

ⁿ

.

Proof. Statement (i) is an immediate consequence of equality (11). We now prove statement (ii). Since L ˜

c,b

h

S

_I,c,Z^×

(b) i

= ˜ L

c^∗,b

h

S

_I,c,Z^×

(b) i

+ c

0

S

_I,c,Z^×

(b)

and ˜ L

c^∗,b

[v] is bilinear in the variable (b, v), and S

_I,c,Z^×

is differentiable, we can differentiate with respect to b and obtain

d

^j

n ˜ L

c,b

h

S

_I,c,Z^×

(b) io

[v, . . . , v] =

 j 0

 L ˜

c,b

h

d

^j

S

_I,c,Z^×

(b)[v, . . . , v] i +

 j 1

 L ˜

_c^∗_,v

h

d

^j−1

S

_I,c,Z^×

(b)[v, . . . , v] i

(12)

for all (b, v) ∈ f W × D

n

(R) and j ∈ N \ {0}. By (i), we have d

^j

n ˜ L

_c,b

h

S

_I,c,Z^×

(b) io

[v, . . . , v] = 0 for all (b, v) ∈ f W × D

n

(R) and j ∈ N \ {0}, and thus statement (ii) holds true.

We now prove statement (iii), and we argue by induction on j. If j = 0, then the statement is obvious. We now assume that the statement holds for j and we prove it for j + 1. By statement (ii) and by the inductive assumption, we have

L ˜

c,b

h

d

^j+1

S

_I,c,Z^×

(b)[v, . . . , v] i

= −(j + 1) ˜ L

c^∗,v

h

d

^j

S

_I,c,Z^×

(b)[v, . . . , v] i

= −(j + 1) ˜ L

c^∗,v



(−1)

^j

j!  ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

c^∗,v



j

[S

_I,c,Z^×

(b)]



= (−1)

^j+1

(j + 1)! ˜ L

c^∗,v

◦  ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

c^∗,v



^j

[S

^×_I,c,Z

(b)] ∀v ∈ D

n

(R) . (13) Since S

_I,c,Z^×

and its differentials have values in H

_I,Z^s

(R

ⁿ

), we can apply ˜ L

⁽⁻¹⁾_c,b

to both hand sides, and obtain the formula of the statement with (j + 1) (cf. Proposition 2.3 (iii)).

Next we prove statement (iv). By statements (i),(iii) and by Lemma 4.1, we have L ˜

c,b

h

d

^j

S

_I,c,Z^×

(b)[v, . . . , v] i

= ˜ L

c,b



(−1)

^j

j!  ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

c^∗,v



j

[S

_I,c,Z^×

(b)]



= (−1)

^j

j!  ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

c^∗,v



^j

◦ ˜ L

c,b

[S

_I,c,Z^×

(b)]

= (−1)

^j

j!  ˜ L

⁽⁻¹⁾_c,b

◦ ˜ L

c^∗,v



^j

h X

z∈Zⁿ

δ

z

− X

z∈Z

E

2πiz

i ,

for all v ∈ D

ⁿ

(R) and j ∈ N \ {0}, and thus statement (iv) holds true. Statement (v) is an immediate consequence of statement (iv), and of the analyticity of P

z∈Zⁿ

δ

_z

− P

z∈Z

E

_2πiz

in R

ⁿ

\ Z

ⁿ

, and of classical elliptic regularity theory.

5 Applications to the Helmholtz equation

5.1 An analyticity result in Schauder spaces

In this subsection we consider a concrete application of the results of Section 3 to the periodic analog of the fundamental solution of the Helmholtz equation. In order to do so, if κ ∈ C, we define c(κ) = (c

^α

(κ))

_|α|≤2

∈ C

^N2

by setting

c

e_j+e_j

(κ) ≡ 1 , ∀j ∈ {1, . . . , n} , (14)

and

c

e_l+e_j

(κ) ≡ 0 c

e_h

(κ) ≡ 0 , ∀h, j, l ∈ {1, . . . , n} , l 6= j , (15) and

c

₀

(κ) ≡ κ

²

. (16)

Thus, if κ ∈ C, we have c(κ) ∈ E

⁰

and

P [c(κ), D] = ∆ + κ

²

. Then, a straightforward computation shows that

Z(c(κ), q) ≡ {z ∈ Z

ⁿ

: κ

²

= 4π

²

|q

⁻¹

z|

²

} , (17) and we note that Z(c(κ), q) is not empty precisely when −κ

²

is an eigenvalue of the Laplace operator in the space of q-periodic distributions in R

ⁿ

(see Proposition 2.3 (i)). We also note that

L ˜

_c(κ),b

=

n

X

j=1

b

_jj

∂

²

∂x

²_j

+ κ

²

∀b ∈ D

n

(R) .

In order to proceed we need to introduce some Sobolev-Bessel potential spaces on a domain. Let Ω be an open subset of R

ⁿ

. If s ∈ R, we denote by ˜ H

^s

(Ω) the closure of D(Ω) in H

^s

(R

ⁿ

), and we endow ˜ H

^s

(Ω) with the norm of H

^s

(R

ⁿ

). Instead, as customary, we denote by H

₀^s

(Ω) the closure of D(Ω) in H

^s

(Ω) (see also McLean [23, p. 77]).

Then we have the following technical statement, which we prove in the Appendix.

Theorem 5.1. Let κ ∈ C. Let Ω be a bounded open Lipschitz subset of R

ⁿ

. Let W be an open subset of D

⁺n

(R) such that

n u ∈ H

₀¹

(Ω) : ˜ L

_c(κ),b

[u] = 0 o

= {0} ∀b ∈ W . (18)

Then the following statements hold.

(i) Let k ∈ N \ {0}. If Ω is of class C

^k−1,1

and b ∈ W, then ˜ L

_c(κ),b

is a linear homeomorphism from H

^k

(Ω) ∩ H

₀¹

(Ω) onto H

^k−2

(Ω).

(ii) Let k ∈ N \ {0}. If Ω is of class C

^k−1,1

, then the map Φ

_c(κ),k

from W × H

^k−2

(Ω) to H

^k

(Ω) ∩ H

₀¹

(Ω) which takes (b, f ) to the unique element v ∈ H

^k

(Ω) ∩ H

₀¹

(Ω) such that ˜ L

_c(κ),b

[v] = f is real analytic.

(iii) Let k ∈ Z, k ≤ 0. Let Ω be of class C

^1−k,1

. If (b

^]

, f

^]

) ∈ W × H

₀¹

(Ω), then there exists r ∈]0, +∞[ such that clB

Dn(R)

(b

^]

, r) ⊆ W and a real analytic map ˜ Φ

_c(κ),k,b],f^]

from B

Dn(R)

(b

^]

, r) × ˜ H

^k−2

(Ω) to ˜ H

^k

(Ω) such that

Φ ˜

_c(κ),k,b],f^]

[b, f ] = Φ

c(κ),1

[b, f ] ∀(b, f ) ∈ B

_Dn(R)

(b

^]

, r) × H

₀¹

(Ω) . (19) Then we can prove the following.

Theorem 5.2. Let m ∈ N, α ∈]0, 1[. Let κ ∈ C. Let Z be a finite subset of Z

ⁿ

. Let f W be an open subset of D

⁺n

(R) such that Z(c(κ), b

^−1/2

) ⊆ Z for all b ∈ f W. Let Ω be a bounded open subset of R

ⁿ

such that clΩ ⊆ R

ⁿ

\ Z

ⁿ

. Then the map from f W to C

^m,α

(clΩ), which takes b to the restriction R

Ω

S

_I,c(κ),Z^×

(b) of S

_I,c(κ),Z^×

(b) to Ω is real analytic.

Proof. Let s ∈ Z be such that s − 2 < −n/2. By the Sobolev Imbedding Theorem, it suffices to show that the map from f W to H

^s+k

(Ω) which takes b to R

Ω

S

_I,c(κ),Z^×

(b) is analytic for all bounded open subsets of R

ⁿ

such that clΩ ⊆ R

ⁿ

\ Z

ⁿ

and for all k ∈ N. Then the validity of the statement of the theorem for any m ∈ N and α ∈]0, 1[ would follow by a proof which is indeed independent of m and α.

Case k = 0 is an immediate consequence of Theorem 3.1. Indeed, the restriction operator from H

_I^s

(R

ⁿ

) to H

^s

(Ω) is linear and continuous and thus analytic (cf. (9)). We now assume that the statement holds for k and we prove it for k + 1. Since for every Ω as above there exists ζ ∈ D(R

ⁿ

\ Z

ⁿ

) which equals 1 on an open neighborhood of clΩ, it clearly suffices to show that if ζ ∈ D(R

ⁿ

\ Z

ⁿ

) and if b

1

∈ f W, then there exists an open neighborhood f W

₁

of b

₁

contained in f W and a bounded open subset Ω

₁

of R

ⁿ

of class C

^∞

such that

supp ζ ⊆ Ω

1

⊆ clΩ

1

⊆ R

ⁿ

\ Z

ⁿ

, (20)

and such that the map from f W

1

to H

^s+k+1

(Ω

₁

) which takes b to ζS

_I,c(κ),Z^×

(b) is analytic. Indeed, the restriction

map is linear and continuous and ζ is a multiplier for all Sobolev spaces on an open subset which contains the

support of ζ.

We take an arbitrary bounded open subset Ω

1

of R

ⁿ

of class C

^∞

such that (20) holds. Possibly replac- ing Ω

1

by a dilation of Ω

1

close to the identity, we can assume that −κ

²

is not a Dirichlet eigenvalue for P

n

j=1

(b

₁

)

_jj∂x^∂2₂

j

in Ω

₁

. Indeed the spectrum of the elliptic operator P

n

j=1

(b

₁

)

_jj∂x^∂2₂

j

with Dirichlet boundary conditions in a bounded open subset of R

ⁿ

of class C

^∞

is discrete and the eigenvalues scale by a positive factor if the open set undergoes a dilation by a positive factor. Since −κ

²

is not a Dirichlet eigenvalue for P

n

j=1

(b

₁

)

_jj∂x^∂22 j

in Ω

₁

, the operator ˜ L

_c(κ),b1

is a linear homeomorphism from H

₀¹

(Ω

₁

) onto H

⁻¹

(Ω

₁

). Since the map from f W to L(H

₀¹

(Ω

₁

), H

⁻¹

(Ω

₁

)) which takes b to ˜ L

_c(κ),b

is linear and continuous and the set of linear homeomorphisms is open, then there exists an open neighborhood f W

1

of b

1

contained in f W such that ˜ L

_c(κ),b

is a linear homeomorphism from H

₀¹

(Ω

1

) onto H

⁻¹

(Ω

1

) for all b ∈ f W

1

. In particular, condition (18) holds in Ω

1

for all b ∈ f W

1

.

Since Ω

1

is of class C

^∞

and ζS

_I,c(κ),Z^×

(b) vanishes on an open neighborhood of ∂Ω

1

is of class C

^∞

, we have ζS

_I,c(κ),Z^×

(b) = Φ

_c(κ),1

h

b, ˜ L

_c(κ),b

h

ζS

_I,c(κ),Z^×

(b) ii

, (21)

L ˜

_c(κ),b

h

ζS

_I,c(κ),Z^×

(b) i

∈ D(Ω

1

) ∀b ∈ f W

1

. Next we show that if Ω

2

is an open subset of R

ⁿ

of class C

^∞

such that

clΩ

1

⊆ Ω

2

⊆ clΩ

2

⊆ R

ⁿ

\ Z

ⁿ

, then the map from f W

1

to H

^s+k−1

(Ω

2

) which takes b to ˜ L

_c(κ),b

h

ζS

_I,c(κ),Z^×

(b) i

is real analytic. To do so, we note that

L ˜

_c(κ),b

h

ζ(y)S

_I,c(κ),Z^×

(b)(y) i

= ζ(y) ˜ L

_c(κ),b

h

S

^×_I,c(κ),Z

(b)(y) i

(22) +2

n

X

j=1

b

jj

∂ζ

∂x

j

(y) ∂

∂x

j

h

S

^×_I,c(κ),Z

(b) i (y)

+S

_I,c(κ),Z^×

(b)(y)

n

X

j=1

b

jj

∂

²

ζ

∂x

²_j

(y) ∀y ∈ Ω

2

, and that

ζ ˜ L

_c(κ),b

h

S

^×_I,c(κ),Z

(b) i

=

"

X

z∈Zⁿ

δ

_z

− X

z∈Z

E

_2πiz

#

ζ = − X

z∈Z

E

_2πiz

ζ in R

ⁿ

(see Proposition 4.2 (i)). Thus it suffices to show that each summand in the right hand side of (22) defines a real analytic map from f W

₁

to H

^s+k−1

(Ω

₂

).

Since ζ P

z∈Z

E

_2πiz

∈ C

^∞

(clΩ

₂

) ⊆ H

^s+k−1

(Ω

₂

) is independent of b, the first summand in the right hand side of (22) defines a real analytic map from f W

1

to H

^s+k−1

(Ω

2

).

By inductive assumption, we know that the map from f W

1

to H

^s+k−1

(Ω

₂

) which takes b to

_∂x^∂

j

h S

_I,c(κ),Z^×

(b) i is real analytic. Since the multiplication by

_∂x^∂ζ

j

is linear and continuous in H

^s+k−1

(Ω

2

), the map from f W

1

to H

^s+k−1

(Ω

2

) which takes b to the second summand in the right hand side of (22) is real analytic.

By inductive assumption, we know that the map from f W

1

to H

^s+k

(Ω

2

) which takes b to S

_I,c(κ),Z^×

(b) is analytic. Since the multiplication by

_∂x^∂2ζ2

j

is linear and continuous in H

^s+k

(Ω

₂

), and H

^s+k

(Ω

₂

) is continuously imbedded into H

^s+k−1

(Ω

₂

), the map from f W

₁

to H

^s+k−1

(Ω

₂

) which takes b to the third summand in the right hand side of (22) is real analytic.

Then we conclude that the map from f W

₁

to H

^s+k−1

(Ω

₂

) which takes b to the right hand side of (22) is real analytic.

We now discuss separately cases s + k ≥ 1 and case s + k ≤ 0. Let s + k ≥ 1. Since the restriction map is linear and continuous from H

^s+k−1

(Ω

2

) to H

^s+k−1

(Ω

1

), the map from f W

1

to H

^s+k−1

(Ω

1

) which takes b to the right hand side of (22) is real analytic and Theorem 5.1 (ii) and equality (21) imply that the map from W f

₁

to H

^s+k+1

(Ω

₁

) which takes b to ζS

^×_I,c(κ),Z

(b) is real analytic.

Now let s + k ≤ 0. Since the restriction map is linear and continuous from H

^s+k−1

(Ω

₂

) to ˜ H

^s+k−1

(Ω

₁

), the

map from f W

1

to ˜ H

^s+k−1

(Ω

₁

) which takes b to the right hand side of (22) is real analytic. We also note that

the right hand side of (22) belongs to D(Ω

1

) for all b ∈ f W

1

. Then Theorem 5.1 (iii) and equality (21) imply