ENERGY INTEGRAL OF THE SOLUTION OF A NON-IDEAL TRANSMISSION PROBLEM IN A SINGULARLY PERTURBED PERIODIC DOMAIN Matteo Dalla Riva and Paolo Musolino Abstract In this paper, the behavior of the energy integral of the solution of a non-ideal transmission problem is investigated. Such problem appears in the study of the effective thermal conductivity of a two-phase composite with thermal resistance at the interface. The composite is obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material, each of them of size proportional to a positive parameter . Under suitable assumptions, we show that the energy integral of the solution can be continued real analytically in the parameter  around the degenerate value  = 0, in correspondence of which the inclusions collapse to points.

Keywords: Non-ideal transmission problem, Energy integral, Periodic composite, Singularly perturbed perforated domain, Effective conductivity, Real analytic continuation 2010 Mathematics Subject Classification: 35B25, 26E05, 35J25, 31B10, 74E30, 74G10

1

INTRODUCTION

In this paper we present an application of a functional analytic approach for the analysis of a singularly perturbed non-ideal transmission problem in a periodic domain. We now introduce our boundary value problem. We fix once and for all (q11 , . . . , qnn ) ∈]0, +∞[n .

n ∈ N \ {0, 1} ,

Then we introduce the periodicity cell Q and the diagonal matrix q by setting Q ≡ Πni=1 ]0, qii [ ,

q ≡ (δh,i qii )(h,i)∈{1,...,n}2 .

Here δh,i ≡ 1 if h = i and δh,i ≡ 0 if h 6= i. We denote by |Q|n the n-dimensional measure of the fundamental cell Q and by q −1 the inverse matrix of q. Clearly, qZn ≡ {qz : z ∈ Zn } is the set of vertices of a periodic subdivision of Rn corresponding to the fundamental cell Q. Then we take α ∈]0, 1[ and we consider a subset Ω of Rn which satisfies the following assumption. Ω is a bounded open connected subset of Rn of class C 1,α such that Rn \ clΩ is connected and that 0 ∈ Ω.

(1.1)

The symbol ‘cl’ denotes the closure. For the definition of functions and sets of the usual Schauder classes 0,α 1,α C 0,α , C 1,α , Cloc , and Cloc we refer for example to Gilbarg and Trudinger [17, §6.2] (see also [12, §§2, 3]). If p ∈ Q is fixed, then there exists 0 ∈ R such that 0 ∈]0, +∞[ ,

p + clΩ ⊆ Q

∀ ∈] − 0 , 0 [ .

To shorten our notation, we set Ωp, ≡ p + Ω

1

∀ ∈ R .

(1.2)

We are now in the position to define the periodic domains [ (qz + Ωp, ) , S[Ωp, ]− ≡ Rn \ clS[Ωp, ] , S[Ωp, ] ≡ z∈Zn

for all  ∈] − 0 , 0 [. + − In order to introduce the transmission R problem, we take two positive0,αconstants λ , λ , a function f in 0,α the Schauder space C (∂Ω) such that ∂Ω f dσ = 0, a function g in C (∂Ω), a function ρ from ]0, 0 [ to ]0, +∞[, and a natural number j ∈ {1, . . . , n}. Then we consider the following transmission problem for a 1,α 1,α − − pair of functions (u+ j , uj ) ∈ Cloc (clS[Ωp, ]) × Cloc (clS[Ωp, ] ):                 

∆u+ j =0 ∆u− j =0 u+ (x + qhh eh ) = u+ j j (x) + δh,j qhh

in S[Ωp, ] , in S[Ωp, ]− , ∀x ∈ clS[Ωp, ] , ∀h ∈ {1, . . . , n} , ∀x ∈ clS[Ωp, ]− , ∀h ∈ {1, . . . , n} ,

− u− j (x + qhh eh ) = uj (x) + δh,j qhh

(1.3)

   ∂u− ∂u+    ∀x ∈ ∂Ωp, , λ− ∂νΩj (x) − λ+ ∂νΩj (x) = f ((x − p)/)  p, p,   +
 ∂u  − 1  λ+ ∂νΩj (x) + ρ() u+  j (x) − uj (x) = g((x − p)/) ∀x ∈ ∂Ωp, ,  p, R   u+ (x) dσ = 0 , ∂Ωp,

j

x

for all  ∈]0, 0 [, where νΩp, denotes the outward unit normal to ∂Ωp, . Here {e1 ,. . . , en } denotes the canonical basis of Rn . The boundary value problem in (1.3) is a generalization of transmission problems largely investigated in the theory of heat conduction in two-phase periodic composites with imperfect contact conditions (cf., e.g., Castro and Pesetskaya [6], Castro, Pesetskaya, and Rogosin [7], Dryga´s and Mityushev [16], Lipton [21], − Mityushev [24]). As observed in [12, §1], the functions u+ j and uj play the role of the temperature field in the inclusions occupying the periodic set S[Ωp, ] and in the matrix occupying S[Ωp, ]− , respectively. The thermal conductivities of the isotropic and homogeneous materials which fill the inclusions and the matrix are represented by the parameters λ+ and λ− , respectively. The number ρ() plays the role of the interfacial thermal resistivity. The third and fourth conditions in (1.3) imply that the temperature distributions u+ j and u− j have an increment equal to qjj in the direction ej and are periodic in all the other directions. The fifth condition in (1.3) describes the jump of the normal heat flux across the two-phase interface and the sixth condition describes the jump of the temperature field. Finally, the seventh condition in (1.3) is an auxiliary condition which we introduce in order to guarantee the uniqueness of the solution. Because of the factor 1/ρ(), the sixth condition in (1.3) may display a singularity as  tends to 0. We shall analyze problem (1.3) in the case where the limit lim

→0+

 exists finite in R. ρ()

If assumption (1.4) holds, then we set r∗ ≡ lim+ →0

 . ρ()

(1.4)

(1.5)

1,α 1,α If  ∈]0, 0 [, then the boundary value problem in (1.3) has a unique solution in Cloc (clS[Ωp, ])×Cloc (clS[Ωp, ]− ), + − + − which we denote by (uj [], uj []). By means of the pair (uj [], uj []), one can introduce the effective conductivity matrix λeff [] with (k, j)-entry λeff kj [] defined as follows:

λeff kj []

≡

! Z Z ∂u+ ∂u− 1 j [](x) j [](x) + − λ dx + λ dx |Q|n ∂xk ∂xk Ωp, Q\clΩp, Z 1 + f ((x − p)/)xk dσx ∀ ∈]0, 0 [ . |Q|n ∂Ωp, 2

(1.6)

Here, if x ∈ Rn and k ∈ {1, . . . , n}, then xk denotes the k-th component of x. Definition (1.6) extends the one of Benveniste and Miloh to the case of non-homogeneous boundary conditions and coincides with the classical definition when f and g are identically 0 (cf. Benveniste [3] and Benveniste and Miloh [4]). Then we introduce the energy integral of the solution. + − Definition 1.7. Let α ∈]0, R 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let λ , λ ∈]0, +∞[. Let f, g ∈ C 0,α (∂Ω) and ∂Ω f dσ = 0. Let ρ be a function from ]0, 0 [ to ]0, +∞[. Let j ∈ {1, . . . , n}. We set Z 2 Ej+ [] ≡ |Dx u+ j [](x)| dx , Ωp,

Ej− [] ≡

Z

2 |Dx u− j [](x)| dx ,

Q\clΩp,

for all  ∈]0, 0 [, where (1.3).

− (u+ j [], uj [])

1,α 1,α is the unique solution in Cloc (clS[Ωp, ]) × Cloc (clS[Ωp, ]− ) of problem

− We are interested in studying the asymptotic behavior of the energy integrals of (u+ j [], uj []) defined as in Definition 1.7. Accordingly, if j ∈ {1, . . . , n}, we pose the following questions.

What can be said on the function  7→ Ej+ [] for  close to 0 and positive? What can be said on the function  7→

Ej− []

for  close to 0 and positive?

(1.8) (1.9)

Questions of this type have long been investigated with the methods of Asymptotic Analysis, which aims at obtaining asymptotic expansions as the parameter  tends to 0. It is difficult to provide a complete list of the contributions. Here, we mention, e.g., Ammari, Kang, and Kim [1], Ammari, Kang, and Touibi [2], Bonnaillie-No¨el, Dambrine, Tordeux, and Vial [5], Dauge, Tordeux, and Vial [14], Il’in [18], Maz’ya, Movchan, and Nieves [22], Maz’ya, Nazarov, and Plamenewskij [23], Nazarov and Sokolowski [27]. Furthermore, boundary value problems in multiply connected periodic domains have been analyzed with the method of functional equations (cf., e.g., Castro and Pesetskaya [6], Castro, Pesetskaya, and Rogosin [7], Dryga´s and Mityushev [16], Mityushev [24], Mityushev and Adler [25], Rogosin, Dubatovskaya, and Pesetskaya [28]). Here, instead, we answer the questions in (1.8), (1.9) by showing that Ej+ [] = n Fj+ [, /ρ()] , Ej− [] = |Q|n + n Fj− [, /ρ()] , for  > 0 small, where Fj+ , Fj− are real analytic functions defined in a neighborhood of the pair (0, r∗ ). For the definition and properties of real analytic functions and maps, we refer, e.g., to Deimling [15, p. 150]. We observe that our approach has its advantages. Indeed, if for example we know that /ρ() equals for  > 0 a real analytic function defined in a whole neighborhood of  = 0, then we deduce that Ej+ [] and Ej− [] can be expanded into power series of  for  small. Such an approach has been carried out in the case of a single hole, e.g., in Lanza de Cristoforis [19] (see also [10]), and has later been extended to problems related to the system of equations of the linearized elasticity in [9] and to the Stokes system in [8], and to the case of problems in an infinite periodically perforated domain in [13, 20, 26]. − eff A detailed investigation of the behavior of u+ j [], uj [], and λkj [] can be found in [12, Thms. 7.1, 7.2, 8.1]. We also refer to [11], where these quantities have been studied under further particular assumptions. The paper is organized as follows. In Section 2, we present the results of [12] concerning the asymptotic + behavior of u− j [] and uj []. In Section 3, we exploit the results of Section 2 to answer questions (1.8) and (1.9) (see Theorems 3.1 and 3.3).

2

FUNCTIONAL ANALYTIC REPRESENTATION THEOREMS FOR THE SOLUTIONS OF THE TRANSMISSION PROBLEM

We briefly outline the strategy adopted in [12] in order to analyze the asymptotic behavior of the solution − (u+ j [], uj []) for  close to 0. First of all, we note that problem (1.3) is singular for  = 0. Then, when  3

is small and positive, we apply the potential theory to transform (1.3) into an equivalent system of integral equations defined on ∂Ωp, . However, since ∂Ωp, depends on  and degenerates when  = 0, we need to transform such a system into an equivalent system of integral equations defined on the fixed domain ∂Ω. We do so by an appropriate change of functional variables, which we choose in order to desingularize the problem. Then we analyze the dependence on  of the solutions of the new system by exploiting the Implicit Function Theorem for real analytic maps in Banach spaces. We observe that for  = 0 such solution is related 1,α to the following limiting transmission problem for the pair (u+ , u− ) ∈ C 1,α (clΩ) × Cloc (Rn \ Ω):  ∆u+ = 0 in Ω ,    ∆u− = 0  in Rn \ clΩ ,    − ∂u− + ∂u+ + −  (x))j ∀x ∈ ∂Ω ,   λ ∂νΩ (x) − λ ∂νΩ (x) = f (x) + R Ω  (λ − λ )(ν g dσ + ∂u+ + − + ∂Ω (2.1) λ (x) + r∗ u (x) − u (x) = g(x) − |∂Ω|n−1 − λ (νΩ (x))j ∀x ∈ ∂Ω ,  R ∂νΩ+    R∂Ω u (x) dσx = 0 ,     ∂Ω u− (x) dσx = 0 ,   limx→∞ u− (x) ∈ R . Problem (2.1) is analyzed by means of the following (see [12, Thm. 6.2]). + − Proposition 2.2. Let α ∈]0, R 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let λ , λ ∈]0, +∞[. 0,α Let f, g ∈ C (∂Ω) and ∂Ω f dσ = 0. Let ρ be a function from ]0, 0 [ to ]0, +∞[. Let j ∈ {1, . . . , n}. Let assumption (1.4) hold. Let r∗ be as in (1.5). Then there exists a unique pair (u+ , u− ) ∈ C 1,α (clΩ) × 1,α Cloc (Rn \ Ω) which solves (2.1) and we denote such a pair by (˜ u+ ˜− j ,u j ).

Let u ˜− j be as in Proposition 2.2. We find convenient to set ˜l− ≡ lim u ˜− j j (x) . x→∞

(2.3)

In the following Theorem 2.4 we investigate the behavior of u+ j [] for  small and positive (see [12, Thm. 7.1]). Theorem 2.4. Let α ∈]0,R 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let λ+ , λ− ∈]0, +∞[. Let f, g ∈ C 0,α (∂Ω) and ∂Ω f dσ = 0. Let ρ be a function from ]0, 0 [ to ]0, +∞[. Let j ∈ {1, . . . , n}. Let assumption (1.4) hold. Let r∗ be as in (1.5). Then there exist 1 ∈]0, 0 [, an open neighborhood Ur∗ of r∗ in R, and a real analytic map Uj+ from ] − 1 , 1 [×Ur∗ to C 1,α (clΩ) such that /ρ() ∈ Ur∗ for all  ∈]0, 1 [ and that h  i + u+ (t) ∀t ∈ clΩ , j [](p + t) = Uj , ρ() − for all  ∈]0, 1 [, where (u+ j [], uj []) is the unique solution of problem (1.3). Moreover, Z 1 + + Uj [0, r∗ ](t) = u ˜j (t) + tj − sj dσs ∀t ∈ clΩ , |∂Ω|n−1 ∂Ω

(2.5)

where u ˜+ j is defined as in Proposition 2.2. In the following Theorem 2.6 we investigate the behavior of u− j [] for  small and positive (see [12, Thm. 7.2]). In order to do so, we need to introduce a periodic analog of the fundamental solution of the Laplace operator, i.e., a periodic distribution Sq,n such that X 1 ∆Sq,n = δqz − , |Q| n n z∈Z

where δqz denotes the Dirac distribution with mass in qz. Such a distribution is determined up to an additive constant, and we can take X −1 1 e2πi(q z)·x , Sq,n (x) ≡ − 2 −1 2 |Q|n 4π |q z| n z∈Z \{0}

where the series converges in the sense of distributions on Rn (cf., e.g., [26, Thm. 2.1]). Then, Sq,n is real analytic in Rn \ qZn and is locally integrable in Rn (cf., e.g., [26, Theorem 2.1]). We are now ready to state the following. 4

Theorem 2.6. Let α ∈]0,R 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let λ+ , λ− ∈]0, +∞[. Let f, g ∈ C 0,α (∂Ω) and ∂Ω f dσ = 0. Let ρ be a function from ]0, 0 [ to ]0, +∞[. Let j ∈ {1, . . . , n}. Let − assumption (1.4) hold. Let r∗ be as in (1.5). Let (u+ j [], uj []) be the unique solution of problem (1.3) for all  ∈]0, 0 [. Let ˜lj− be as in (2.3). Let 1 , Ur∗ be as in Theorem 2.4. Then there exists a real analytic function Cj− from ] − 1 , 1 [×Ur∗ to R such that Z Z r∗ 1 − − ˜ g dσ + r∗ lj − sj dσs , Cj [0, r∗ ] = − |∂Ω|n−1 ∂Ω |∂Ω|n−1 ∂Ω and such that the following statements hold. ˜ be an open bounded subset of Rn such that clΩ ˜ ∩ (p + qZn ) = ∅. Let k ∈ N. Then there exist (i) Let Ω − ˜ such that clΩ ˜ ⊆ S[Ωp, ]− for Ω˜ ∈]0, 1 [ and a real analytic map Uj,Ω˜ from ] − Ω˜ , Ω˜ [×Ur∗ to C k (clΩ) all  ∈] − Ω˜ , Ω˜ [, and such that h h  i  i − ˜ + n Uj,−Ω˜ , (x) ∀x ∈ clΩ u− j [](x) = xj − pj + ρ()Cj , ρ() ρ() for all  ∈]0, Ω˜ [. Moreover, Uj,−Ω˜

[0, r∗ ](x) Z = DSq,n (x − p) ·

νΩ (s)˜ u− j (s) dσs

∂Ω

Z

∂ − − s u ˜j (s) dσs ∂Ω ∂νΩ

! ˜, ∀x ∈ clΩ

where u ˜− j is defined as in Proposition 2.2. ˜ be a bounded open subset of Rn \ clΩ. Then there exist # ∈]0, 1 [ and a real analytic map V − (ii) Let Ω ˜ ˜ Ω j,Ω # # # # 1,α − ˜ ˜ from ] −  ,  [×Ur to C (clΩ) such that p + clΩ ⊆ clS[Ωp, ] for all  ∈] −  ,  [, and such that ˜ Ω

˜ Ω

˜ Ω

∗

− u− j [](p + t) = ρ()Cj

h

h  i  i + Vj,−Ω˜ , (t) , ρ() ρ()

˜ Ω

˜ ∀t ∈ clΩ

for all  ∈]0, # ˜ [. Moreover, Ω ˜− Vj,−Ω˜ [0, r∗ ](t) = u ˜− j (t) − lj + tj

˜, ∀t ∈ clΩ

(2.7)

where u ˜− j is defined as in Proposition 2.2.

3

FUNCTIONAL ANALYTIC REPRESENTATION THEOREMS FOR THE ENERGY INTEGRALS OF THE SOLUTIONS OF THE TRANSMISSION PROBLEM

+ − In this section we consider the energy integral of our solutions u+ j [] and uj []. We first consider Ej [] and prove the following statement.

Theorem 3.1. Let α ∈]0,R 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let λ+ , λ− ∈]0, +∞[. Let f, g ∈ C 0,α (∂Ω) and ∂Ω f dσ = 0. Let ρ be a function from ]0, 0 [ to ]0, +∞[. Let j ∈ {1, . . . , n}. Let assumption (1.4) hold. Let r∗ be as in (1.5). Let 1 , Ur∗ be as in Theorem 2.4. Then there exists a real analytic function Fj+ from ] − 1 , 1 [×Ur∗ to R such that h  i Ej+ [] = n Fj+ , ∀ ∈]0, 1 [ . (3.2) ρ() Moreover, Fj+ [0, r∗ ]

Z =

2 |D˜ u+ j (t)|

Z dt + |Ω|n + 2

Ω

where

u ˜+ j

Ω

is defined as in Proposition 2.2. 5

∂u ˜+ j (t) dt , ∂tj

Proof. We first note that if  ∈]0, 1 [ and Uj+ is as in Theorem 2.4, then we have Z

Ej+ []

2 |Dx u+ j [](x)| dx

= Ωp,

Z


2 | Dx u+ j [] (p + t)| dt Ω Z   n−2 = |Dt u+ [](p + t) |2 dt j Ω Z h  i 2 n (t)| dt . = |Dt Uj+ , ρ() Ω = n

As a consequence, we set Fj+ [, 0 ]

Z ≡

|Dt Uj+ [, 0 ](t)|2 dt ,

Ω

for all (, 0 ) ∈] − 1 , 1 [×Ur∗ . Clearly, formula (3.2) holds. Furthermore, Theorem 2.4 and standard calculus in Schauder spaces ensure that Fj+ is a real analytic function from ] − 1 , 1 [×Ur∗ to R. Moreover, by formula (2.5), we have Z Fj+ [0, r∗ ] = |Dt Uj+ [0, r∗ ](t)|2 dt Ω Z Z   1 + = |Dt u ˜j (t) + tj − sj dσs |2 dt |∂Ω|n−1 ∂Ω ZΩ 2 = |D˜ u+ j (t) + ej | dt Ω

Z

2 |D˜ u+ j (t)| dt + |Ω|n + 2

= Ω

Z Ω

∂u ˜+ j (t) dt . ∂tj 2

Thus the proof is complete. We now turn to consider Ej− [] and we prove the following.

Theorem 3.3. Let α ∈]0,R 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let λ+ , λ− ∈]0, +∞[. Let f, g ∈ C 0,α (∂Ω) and ∂Ω f dσ = 0. Let ρ be a function from ]0, 0 [ to ]0, +∞[. Let j ∈ {1, . . . , n}. Let assumption (1.4) hold. Let r∗ be as in (1.5). Let 1 , Ur∗ be as in Theorem 2.4. Then there exist 2 ∈]0, 1 [ and a real analytic function Fj− from ] − 2 , 2 [×Ur∗ to R such that h  i Ej− [] = |Q|n + n Fj− , ρ()

∀ ∈]0, 2 [ .

(3.4)

Moreover, Fj− [0, r∗ ]

Z = Rn \clΩ

2 |D˜ u− j (t)|

Z

u ˜− j (t)(νΩ (t))j dσt − |Ω|n ,

dt − 2 ∂Ω

where u ˜− j is defined as in Proposition 2.2. 1,α Proof. For each  ∈]0, 0 [, we introduce the function vj− [] ∈ Cloc (clS[Ωp, ]− ) defined by setting

vj− [](x) ≡ u− j [](x) − xj

∀x ∈ clS[Ωp, ]− .

We note that vj− [] is periodic, i.e., vj− [](x + qz) = vj− [](x)

∀x ∈ clS[Ωp, ]− , ∀z ∈ Zn .

6

(3.5)

Then, if  ∈]0, 0 [, we have Z − 2 Ej [] = |Dx u− j [](x)| dx Q\clΩp,

Z

|Dx vj− [](x)|2 dx +

=

Z

Q\clΩp,

Z

Z

Q\clΩp,

|Dx vj− [](x)|2

=

|Dx xj |2 dx + 2 Z

n

dx + |Q|n −  |Ω|n + 2

Q\clΩp,

Dx vj− [](x) · Dx xj dx

Q\clΩp, ∂vj− []

∂xj

Q\clΩp,

(x) dx .

By the Divergence Theorem and by the periodicity of vj− [], we have Z

Z

|Dx vj− [](x)|2 dx

vj− [](x)

=

Q\clΩp,

∂Q

= −

Z

n−1

vj− [](p

Z∂Ω

= −n−2

∂vj− [] (x) dσx − ∂νQ + t)

Z

vj− [](x)

∂Ωp,

Dx vj− []




∂vj− [] (x) dσx ∂νΩp,

(p + t) · νΩ (t) dσt



(3.6)



vj− [](p + t)Dt vj− [](p + t) · νΩ (t) dσt ,

∂Ω

and Z Q\clΩp,

∂vj− [] (x) dx ∂xj

Z

vj− [](x)(νQ (x))j

=

Z

vj− [](x)(νΩp, (x))j dσx

dσx −

∂Q

∂Ωp,

= −n−1

Z

vj− [](p + t)(νΩ (t))j dσt ,

(3.7)

∂Ω

for all  ∈]0, 0 [. We observe that the Divergence Theorem and the periodicity of vj− [] imply that Z

Z

∆vj− [](x) dx

0=

=

Q\clΩp,

∂Q

∂vj− [] (x) dσx − ∂νQ

Z ∂Ωp,

∂vj− [] (x) dσx ∂νΩp,

∂vj− []

Z =−

(x) dσx ∂νΩp, Z   = −n−2 Dt vj− [](p + t) · νΩ (t) dσt ,

(3.8)

∂Ωp,

∂Ω

for all  ∈]0, 0 [. Moreover, we also have Z (νΩ (t))j dσt = 0 .

(3.9)

∂Ω

˜ ≡ {x ∈ Rn : |x| < R} \ clΩ. We Then we fix R ∈]0, +∞[ such that clΩ ⊆ {x ∈ Rn : |x| < R} and we set Ω # take 2 ≡ Ω˜ as in Theorem 2.6 (ii) and we note that h h  i  i vj− [](p + t) = ρ()Cj− , + Vj,−Ω˜ , (t) − pj − tj ρ() ρ()

˜, ∀t ∈ clΩ

for all  ∈]0, 2 [. Thus if we set − Fj,1 [, 0 ] − Fj,2 [, 0 ]

Z ≡− ≡ −2



Vj,−Ω˜ [, 0 ](t) − tj

∂Ω Z  ∂Ω



 Dt Vj,−Ω˜ [, 0 ](t) − ej · νΩ (t) dσt ,

 Vj,−Ω˜ [, 0 ](t) − tj (νΩ (t))j dσt ,

7

(3.10)

for all (, 0 ) ∈] − 2 , 2 [×Ur∗ , we deduce by equalities (3.6)− (3.10) that Z h  i − , |Dx vj− [](x)|2 dx = n Fj,1 , ρ() Q\clΩp, Z h ∂vj− []  i − 2 , (x) dx = n Fj,2 , ρ() Q\clΩp, ∂xj − for all  ∈]0, 2 [. By Theorem 2.6 (ii) and by standard calculus in Schauder spaces, the functions Fj,1 and − Fj,2 are real analytic. Moreover, by formula (2.7), we have − Fj,1 [0, r∗ ]

Z =− Z =





 D˜ u− (t) · νΩ (t) dσt j

∂Ω

Rn \clΩ − Fj,2 [0, r∗ ]

˜− u ˜− j (t) − lj

Z = −2 Z∂Ω = −2

2 |D˜ u− j (t)| dt ,



 ˜− u ˜− j (t) − lj (νΩ (t))j dσt

u ˜− j (t)(νΩ (t))j dσt .

∂Ω

Therefore, if we set − − Fj− [, 0 ] ≡ Fj,1 [, 0 ] + Fj,2 [, 0 ] − |Ω|n

for all (, 0 ) ∈] − 2 , 2 [×Ur∗ , then we deduce that Fj− is real analytic and that equalities (3.4), (3.5) hold. Thus the proof is complete. 2 Remark 3.11. We observe that if /ρ() has a real analytic continuation around  = 0, then the terms in the right hand sides of equalities (3.2) and (3.4) define real analytic functions of the variable  in a whole neighborhood of 0. As a consequence, Ej+ [] and Ej− [] can be continued real analytically for  ≤ 0 and thus can be represented for  small and positive in terms of power series of  which converge absolutely on a whole neighborhood of 0.

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