On a singularly perturbed periodic nonlinear Robin problem ∗ M. Dalla Riva, M. Lanza de Cristoforis, and P. Musolino

Abstract This paper shows some applications of a functional analytic approach to the analysis of a nonlinear Robin problem in a periodically perforated domain with small holes of size proportional to a positive parameter . The second and third authors have proved in a previous paper the existence of a particular family of solutions {u(, ·)}∈]0,0 [ uniquely determined (for  small) by its limiting behavior as  → 0. Also, the dependence of u(, ·) upon the parameter  can be described in terms of real analytic operators of  defined in a open neighborhood of 0 and of completely known functions of . Here, we exploit such a result for the family {u(, ·)}∈]0,0 [ in order to prove an analogous real analytic continuation result for the dependence of the corresponding energy integral E(u(, ·)) upon the parameter . Then we focus our attention on the limiting behavior of {u(, ·)}∈]0,0 [ as  → 0. To do so, we introduce some specific families of solutions which display a suitable property of convergence in the vicinity of the boundary of the holes. First we show that their limit is the solution of a certain “limiting boundary value problem” and then we prove a local uniqueness result for such converging families.

Keywords: Periodic nonlinear Robin problem, singularly perturbed domain, singularly perturbed data, Laplace operator, real analytic continuation in Banach space. 2010 Mathematics Subject Classification: 31B10, 47H30.

1

INTRODUCTION

As is well known, periodic boundary value problems have a large variety of applications, especially in problems for composite materials. Here we refer for example to Mityushev, Obnosov, Pesetskaya, and Rogosin [44] and to Mityushev, Pesetskaya, and Rogosin [45] (see also Ammari and Kang [3, Chs. 2, 8], Milton [40, Ch. 1]). In connection with doubly periodic problems for composite materials, we mention the monograph of Grigolyuk and Fil’shtinskij [24]. 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, and Papanicolaou [9, Ch. 1], Jikov, Kozlov, and Ole˘ınik [26, Ch. 1]). We note that the method of functional equations 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. Here we mention, e.g., Castro and Pesetskaya ∗ Dedicated

to Professor S.V. Rogosin on the occasion of his sixtieth birthday.

1

[11], Castro, Pesetskaya, and Rogosin [12], Drygas and Mityushev [22], Mityushev [42], Mityushev and Adler [43], and Rogosin, Dubatovskaya, and Pesetskaya [51]. For an extensive treatment, we refer to the monograph Mityushev and Rogosin [46]. In this paper, instead, we adopt an approach based on potential theory and on functional analysis in order to analyze a singularly perturbed periodic nonlinear Robin problem. We fix once for all a natural number n ∈ N \ {0, 1}, positive real numbers q11 , . . . , qnn ∈ ]0, +∞[, and a periodicity cell Q ≡ Πnj=1 ]0, qjj [. Then we denote by q the n × n diagonal matrix with entries q11 , . . . , qnn . Then we fix α ∈]0, 1[ and we consider a subset Ω of Rn satisfying the following assumption. Ω is a bounded open connected subset of Rn of class C 1,α which (1.1) contains the origin 0 and with a connected exterior Rn \ clΩ. Here cl denotes the closure. For the definition of functions and sets of the usual Schauder 0,α 1,α classes C 0,α , C 1,α , Cloc , and Cloc we refer for example to Gilbarg and Trudinger [23, §6.2] (see also [33, §2]). Next we fix p ∈ Q. Then there exists 0 ∈]0, +∞[ such that p + clΩ ⊆ Q

∀ ∈] − 0 , 0 [ .

(1.2)

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

∀ ∈ R .

Then we introduce the periodic domain [

S[Ωp, ]− ≡ Rn \

(qz + clΩp, ) ,

z∈Zn

for all  ∈] − 0 , 0 [. A function u from clS[Ωp, ]− to R is said to be q-periodic if u(x + qhh eh ) = u(x)

∀x ∈ clS[Ωp, ]− ,

for all h ∈ {1, . . . , n}. Here {e1 ,. . . , en } denotes the canonical basis of Rn . Next we introduce a function G ∈ C 0 (∂Ω × R) and a function γ from ]0, 0 [ to ]0, +∞[, and we consider the following nonlinear Robin problem  in S[Ωp, ]− ,  ∆u = 0 − u is q−periodic in clS[Ωp, ] , (1.3)  − ∂u (x) = 1 G((x − p)/, u(x)) ∀x ∈ ∂Ωp, , ∂νΩ γ() p,

for  ∈]0, 0 [, where νΩp, denotes the outward unit normal to ∂Ωp, . Due to the presence of the factor 1/γ(), the boundary condition may display a singularity in  if  tends to 0. In this paper, we consider the case in which  ∈ R. (1.4) γm ≡ lim+ →0 γ() By invoking [33, §§ 4, 5], one knows that possibly shrinking 0 , problem (1.3) has a solution u(, ·) ∈ Cq1,α (clS[Ωp, ]− ) for all  ∈]0, 0 [ (see also Theorem 2.1 below). Here Cq1,α (clS[Ωp, ]− ) denotes the Banach space of the q-periodic elements of C 1,α (clS[Ωp, ]− ), endowed with the norm defined by X X kukCq1,α (clS[Ωp, ]− ) ≡ sup |Dζ u(x)| + |Dζ u : clS[Ωp, ]− |α −

ζ∈Nn , |ζ|≤1 x∈clS[Ωp, ]

2

ζ∈Nn , |ζ|=1

for all u ∈ Cq1,α (clS[Ωp, ]− ), where |Dζ u : clS[Ωp, ]− |α denotes the α-H¨older constant of Dζ u. Moreover, the family {u(, ·)}∈]0,0 [ is uniquely determined (for  small) by its limiting behavior as  → 0 and the dependence of u(, ·) upon the parameter  can be described in terms of real analytic operators of  defined in an open neighborhood of 0 and of completely known functions of  (see also Theorems 2.3 and 2.6 here below). We are interested in analyzing the behavior of the energy integral Z E(u(, ·)) ≡ |Dx u(, x)|2 dx Q\clΩp,

as  approaches 0. By exploiting the real analytic representation result obtained for u(, ·) in [33] (see also Theorem 2.3 here below) we show in Theorem 3.1 that the behavior of E(u(, ·)) for  small and positive can be described in terms of real analytic functions of two real variables defined in a neighborhood of (0, γm ) and evaluated at (, /γ()). We observe that such result implies in particular that E(u(, ·)) for  small and positive is given by a converging power series of the variables  and (γm − /γ()). Then we turn to consider a sequence {εj }j∈N ⊆]0, 0 [ which converges to 0 and a family {uj }j∈N of solutions of problem (1.3) for  = εj . We denote by uj (p + εj ·) the rescaled function from Rn \ ∪z∈Zn (ε−1 j qz + clΩ) to R which takes t to uj (p + εj t) and we assume that there exist an open bounded neighborhood O of clΩ in Rn and a function v# ∈ C 1,α (clO \Ω) such that lim uj (p + εj ·)|clO\Ω = v# in C 1,α (clO \ Ω) . j→+∞

We show that under a suitable continuity assumption on the function G there exists unique 1,α a pair (u# , c# ) ∈ Cloc (Rn \ Ω) × R such that v# = u#|clO\Ω and such that  ∆u# = 0    − ∂u# (t) = γ G(t, u (t)) m # R ∂νΩ∂u#  dσ = 0 ,   ∂Ω ∂νΩ limt→∞ u# (t) = c#

in Rn \ clΩ , ∀t ∈ ∂Ω ,

(1.5)

(see Theorem 4.16). Here, νΩ denotes the outward unit normal to ∂Ω. Also, uj (p+εj ·)|clO0 \Ω converges to u#|clO0 \Ω for all open bounded neighborhoods O0 of clΩ in Rn . Finally, if G satisfies a suitable regularity assumption and if the limiting problem in (1.5) does not degenerate in a sense which we clarify below, then we prove in Theorem 4.27 that any other family {vj }j∈N of solutions of (1.3) for  = εj such that lim vj (p + εj ·)|clO\Ω = v#

j→+∞

in C 1,α (clO \ Ω) ,

must coincide with {uj }j∈N for j large enough. The functional analytic approach adopted in this paper has been previously exploited to analyze singular perturbation problems in the case of a bounded domain with a single hole in [20, 28, 29, 30, 31] and has later been extended to problems related to the system of equations of the linearized elasticity in [16, 17, 18, 19] and to the Stokes system in [15]. In particular, as far as problems in an infinite periodically perforated domain are concerned, we mention [21, 33, 47, 48]. 3

It is worth noting that singularly perturbed boundary value problems have been largely investigated with the methods of asymptotic analysis and homogenization theory. In this sense, we mention the work of Ammari and Kang [3, Ch. 5], Ammari, Kang, and Lee [5, Ch. 3], Bakhvalov and Panasenko [8], Cioranescu and Murat [13, 14], Jikov, Kozlov, and Ole˘ınik [26], Kozlov, Maz’ya, and Movchan [27], Laurain, Nazarov, and Sokolowski [35], Marˇcenko and Khruslov [36], Maz’ya and Movchan [37], Maz’ya, Movchan, and Nieves [38], Maz’ya, Nazarov, and Plamenewskij [39], Nazarov and Sokolowski [49], Ozawa [50], S´anchezPalencia [52], Ward and Keller [53]. In particular, in connection with periodic problems, we mention Ammari, Garapon, Kang, and Lee [2], Ammari, Kang, and Kim [4], Ammari, Kang, and Lim [6], and Ammari, Kang, and Touibi [7]. The paper is organized as follows. In Section 2, we recall some results of [33] concerning the existence and the local uniqueness of a family of solutions of problem (1.3). In Section 3, we prove a real analytic representation result for the energy integral E(u(, ·)). Finally, in Section 4 we investigate some limiting and uniqueness properties of converging families of solutions of problem (1.3).

2

EXISTENCE AND LOCAL UNIQUENESS OF A PARTICULAR FAMILY OF SOLUTIONS

In this section we recall some results of [33]. We use the following notation. If G ∈ C 0 (∂Ω × R), we denote by TG the (nonlinear nonautonomous) composition operator from C 0 (∂Ω) to itself which takes v ∈ C 0 (∂Ω) to the function TG [v] defined by TG [v](t) ≡ G(t, v(t))

∀t ∈ ∂Ω .

We note that if TG is real analytic, then one can prove that the partial derivative Dξ G of G with respect to its second variable exists (cf. e.g., [29, Prop. 6.3].) The following theorem asserts the existence of a family of solutions of (1.3) (see [33, Thms. 4.4, 4.5 and Def. 4.5]). Theorem 2.1. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let G ∈ C 0 (∂Ω × R) be such that TG is real analytic from C 0,α (∂Ω) to itself. Let 0 ∈]0, +∞[ be as in (1.2). Let γ 1,α be a map from ]0, 0 [ to ]0, +∞[ such that (1.4) holds. Let (˜ u, c˜) ∈ Cloc (Rn \ Ω) × R be a solution of the following boundary value problem  ∆u = 0 in Rn \ clΩ ,    − ∂u (t) = G(t, γ u(t) + c) ∀t ∈ ∂Ω , m R ∂νΩ∂u (2.2) dσ = 0 ,    ∂Ω ∂νΩ limx→∞ u(x) = 0 . If γm = 0, assume that

Z Dξ G(t, c˜) dσt 6= 0 . ∂Ω

If γm 6= 0, assume that Z Dξ G(t, γm u ˜(t) + c˜) ≤ 0 for all t ∈ ∂Ω and

Dξ G(t, γm u ˜(t) + c˜) dσt < 0 . ∂Ω

Here Dξ G denotes the partial derivative of G with respect to its last variable. Then there exist 0 ∈]0, 0 [, and a family {u(, ·)}∈]0,0 [ such that u(, ·) belongs to Cq1,α (clS[Ωp, ]− ) and solves problem (1.3) for all  ∈]0, 0 [. 4

Then we have the following statement which clarifies the behavior of u(, ·) for  small and positive (see [33, Thm. 5.1]). Theorem 2.3. Let the assumptions of Theorem 2.1 hold. Then there exist an open neighborhood Uγm of γm in R and a real analytic operator U from ] − 0 , 0 [ × Uγm to R such that /γ() ∈ Uγm for all  ∈]0, 0 [, and that U [0, γm ] = c˜ ,

(2.4)

and that the following statements hold. ˜ be an open subset of Rn with positive distance from p + qZn . Then there exist (i) Let Ω ∗ Ω˜ ∈]0, 0 [ such that ˜ ⊆ S[Ωp, ]− Ω ∀ ∈ [−∗Ω˜ , ∗Ω˜ ] , and Ω˜ ∈]0, ∗Ω˜ [ such that clS[Ωp,∗˜ ]− ⊆ S[Ωp, ]− for all  ∈ [−Ω˜ , Ω˜ ], and a real analytic Ω 1 1,α − operator US[Ω ˜ , Ω ˜ [×Uγm to the space Cq (clS[Ωp,∗˜ ] ) such that − from ] − Ω p,∗ ] Ω

˜ Ω

u(, t) = n−1

h h  i  i  1 US[Ω (t) + U , − , ∗ ] p, γ() γ() γ() ˜ Ω

∀t ∈ clS[Ωp,∗˜ ]− , Ω

for all  ∈]0, Ω˜ [. Moreover, 1 US[Ω − [0, γm ](t) = DSq,n (t − p) p,∗ ] ˜ Ω

Z sG(s, γm u ˜(s) + c˜) dσs Z∂Ω

+DSq,n (t − p)

νΩ (s)˜ u(s) dσs ∂Ω

∀t ∈ clS[Ωp,∗˜ ]− . Ω

˜ be an open bounded subset of Rn \ clΩ. Then there exist  ˜ ∈]0, 0 [ and a real (ii) Let Ω Ω,r 1,α ˜ such that analytic map UΩ˜1,r from ] − Ω,r (clΩ) ˜ , Ω,r ˜ [×Uγm to C ˜ ⊆ clS[Ωp, ]− p + clΩ ∀ ∈] − Ω,r ˜ , Ω,r ˜ [\{0} , h i h    i u(, p + t) = UΩ˜1,r , (t) + U , γ() γ() γ()

˜ ,  ∈]0,  ˜ [ . ∀t ∈ clΩ Ω,r

Moreover, UΩ˜1,r [0, γm ](t) = u ˜(t)

˜. ∀t ∈ clΩ

(2.5)

We can also show that the family {u(, ·)}∈]0,0 [ is locally unique in a sense which we clarify in the following Theorem 2.6 (see [33, Thm. 6.1]). Theorem 2.6. Let the assumptions of Theorem 2.1 hold. Let {εj }j∈N be a sequence of ]0, 0 [ converging to 0. Let {uj }j∈N be a sequence of functions such that the following condition holds for all j ∈ N: uj belongs to Cq1,α (clS[Ωp,εj ]− ) and is a solution of (1.3) with  = εj . Assume that lim uj (p + εj ·)|∂Ω = γm u ˜|∂Ω + c˜

j→+∞

in C 0,α (∂Ω) .

Then there exists j0 ∈ N such that uj (·) = u(εj , ·) for all j ≥ j0 . 5

3

A FUNCTIONAL ANALYTIC REPRESENTATION THEOREM FOR THE ENERGY INTEGRAL OF u(, ·)

In this section we consider the energy integral of our solutions u(, ·) and prove the following statement. Theorem 3.1. Let the assumptions of Theorem 2.1 hold. Then there exist ˜ ∈]0, 0 [ and real analytic operators F1 , F2 from ] − ˜, ˜[×Uγm to R such that E(u(, ·)) = 

n−2



 γ()

2

  h h  i   i n−2 F1 , + F2 , γ() γ() γ()

∀ ∈]0, ˜[ .

(3.2)

Moreover, Z F1 [0, γm ] =

|D˜ u|2 dx ,

F2 [0, γm ] = 0 .

Rn \clΩ

Proof. By applying the Divergence Theorem in the domain Q \ clΩp, and by exploiting the periodicity of u(, ·) and the boundary condition in (1.3), we obtain Z Z ∂u ∂u u(, y) E(u(, ·)) = u(, y) (, y) dσy − (, y) dσy ∂ν ∂ν Q Ωp, ∂Q ∂Ωp, Z 1 = u(, y)G((y − p)/, u(, y)) dσy γ() ∂Ωp, Z n−1 u(, p + s)G(s, u(, p + s)) dσs ∀ ∈]0, 0 [ . = γ() ∂Ω Here, νQ denotes the outward unit normal to ∂Q. Then we fix R ∈]0, +∞[ such that clΩ ⊆ Bn (0, R) and we take ˜ ≡ Bn (0,R)\clΩ,r as in Theorem 2.3 (ii), and we set Z F1 [, 1 ] ≡ UB1,r [, 1 ](s)G(s, 1 UB1,r [, 1 ](s) + U [, 1 ]) dσs n (0,R)\clΩ n (0,R)\clΩ ∂Ω Z F2 [, 1 ] ≡ U [, 1 ] G(s, 1 UB1,r [, 1 ](s) + U [, 1 ]) dσs n (0,R)\clΩ ∂Ω

for all (, 1 ) ∈] − ˜, ˜[×Uγm . Clearly, formula (3.2) holds. We note that Theorem 2.3 (ii) and the analyticity of TG ensure that the map which takes (, 1 ) in ] − ˜, ˜[×Uγm to the function TG [1 UB1,r [, 1 ](·) + U [, 1 ]] in C 0,α (∂Ω) is analytic. Then again by Theorem 2.3 n (0,R)\clΩ (ii), the maps F1 and F2 are analytic. Moreover, by formulas (2.4), (2.5), and by standard decay estimates of harmonic functions which tend to zero at infinity, we have Z Z Z ∂u ˜ (s) dσs = |D˜ u(y)|2 dy , F1 [0, γm ] = u ˜(s)G(s, γm u ˜(s) + c˜) dσs = − u ˜(s) ∂νΩ Rn \clΩ ∂Ω ∂Ω Z Z ∂u ˜ F2 [0, γm ] = c˜G(s, γm u ˜(s) + c˜) dσs = −˜ c (s) dσs = 0 . ∂Ω ∂Ω ∂νΩ 2

6

4

CONVERGING FAMILIES OF SOLUTIONS OF PROBLEM (1.3)

In this section we study some limiting and uniqueness properties of the converging families of solutions of (1.3) (see also Section 1). To do so, we first need to introduce some preliminaries of potential theory and we study an auxiliary boundary value problem and a related auxiliary boundary integral operator. This is done in the following Subsections 4.1 and 4.2, respectively.

4.1 Since

SOME PRELIMINARIES OF POTENTIAL THEORY n X j=1

−1 (2πiqjj zj )2 = −

n X

−1 4π 2 (qjj zj )2 6= 0

∀z ∈ Zn \ {0} ,

j=1

the Laplace operator is well known to have a {0}-analog of a q-periodic fundamental solution, i.e., a q-periodic tempered distribution Sq,n such that ∆Sq,n =

X

δqz −

z∈Zn

1 , meas(Q)

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

in the sense of distributions in Rn (cf. e.g., Ammari and Kang [3, p. 53], [32, Thms. 3.1, 3.5]). Since −(Zn \ {0}) = Zn \ {0}, the function Sq,n is even. Moreover, Sq,n is real analytic in Rn \ qZn and is locally integrable in Rn (cf. e.g., [32, Thm. 3.5]). Let Sn be the function from Rn \ {0} to R defined by  1 ∀x ∈ Rn \ {0}, if n = 2 , sn log |x| Sn (x) ≡ 1 2−n n |x| ∀x ∈ R \ {0}, if n > 2 , (2−n)sn where sn denotes the (n − 1)-dimensional measure of ∂Bn . Sn is well-known to be the fundamental solution of the Laplace operator. Then the function Sq,n − Sn has an analytic extension to (Rn \ qZn ) ∪ {0} (cf. e.g., Ammari and Kang [3, Lemma 2.39, p. 54]) and we denote such an extension by Rn (see also Berlyand and Mityushev [10]). Clearly, Rn (x − y) is real analytic when (x, y) is in a neighborhood of the diagonal of Rn × Rn . Let I be a bounded open connected subset of Rn of class C 1,α for some α ∈]0, 1[. If H is any of the functions Sq,n , Rn and clI ⊆ Q or if H equals Sn , we set Z v[∂I, H, µ](x) ≡ H(x − y)µ(y) dσy ∀x ∈ Rn , ∂I Z ∂ w∗ [∂I, H, µ](x) ≡ H(x − y)µ(y) dσy ∀x ∈ ∂I , ∂ν (x) I ∂I 7

for all µ ∈ C 0,α (∂I). As is well known, v[∂I, Sq,n , µ] and v[∂I, Sn , µ] are continuous on Rn . We set v − [∂I, Sq,n , µ] ≡ v[∂I, Sq,n , µ]|clS[I]−

v + [∂I, Sq,n , µ] ≡ v[∂I, Sq,n , µ]|clS[I]

v − [∂I, Sn , µ] ≡ v[∂I, Sn , µ]|clI−

v + [∂I, Sn , µ] ≡ v[∂I, Sn , µ]|clI (cf. e.g., [32, Thm. 3.7], [34, Thm. 3.1]).

4.2

AN AUXILIARY BOUNDARY VALUE PROBLEM AND AN AUXILIARY BOUNDARY INTEGRAL OPERATOR

In the following Lemma 4.1 we introduce the auxiliary boundary value problem (4.2). The corresponding proof can be deduced by a standard energy argument and is accordingly omitted. Lemma 4.1. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let ε ∈]0, +∞[ be such that clΩp,ε ⊆ Q. Let φ ∈ C 0,α (∂Ω). Then there exists at most one function u ∈ C 1,α (clS[Ωp,ε ]− ) such that  in S[Ωp,ε ]− ,  ∆u = 0 − u is q-periodic in clS[Ωp,ε ] , (4.2)  ∂u (x) = 1 φ((x − p)/ε) + 1 u(x) ∀x ∈ ∂Ω . p,ε ∂νΩ ε ε p,ε

In the sequel, C θ such that

0,α

(∂Ω)0 denotes the subspace of C 0,α (∂Ω) consisting of those functions Z θ dσ = 0 . ∂Ω

In Definition 4.3 here below we introduce the auxiliary integral operator Mε . Definition 4.3. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let ε ∈]0, +∞[ be such that clΩp,ε ⊆ Q. Then Mε denotes the operator from C 0,α (∂Ω)0 × R to C 0,α (∂Ω) which takes a pair (θ, c) to R 1 n−1 Mε [θ, c](t) ≡ DRn (ε(t − s))νΩ (t)θ(s) dσs 2 θ(t) + w∗ [∂Ω, Sn , θ](t) + ε ∂Ω   R n−2 − v[∂Ω, Sn , θ](t) + ε R (ε(t − s))θ(s) dσs + c ∀t ∈ ∂Ω . ∂Ω n Then we have the following Lemma 4.4 which relates problem (4.2) to the operator Mε . The validity of the statement can be deduced by the standard theorem on change of variables in integrals and by the jump properties of the periodic single layer potentials (cf. e.g., [32, Thm. 3.7]). Lemma 4.4. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let ε ∈]0, +∞[ be such that clΩp,ε ⊆ Q. Let φ ∈ C 0,α (∂Ω). Let (θ, c) ∈ C 0,α (∂Ω)0 × R. Then the equality Mε [θ, c] = φ Cq1,α (clS[Ωp,ε ]− )

holds if and only if the function u ∈ defined by Z u(x) ≡ εn−2 Sq,n (x − p − εs)θ(s) dσs + c ∀x ∈ clS[Ωp,ε ]− ∂Ω

satisfies problem (4.2). 8

Here we note that the definition of Mε does not display the term containing log ε in case R n = 2 due to the condition ∂Ω θ dσ = 0, which holds for all (θ, c) ∈ C 0,α (∂Ω)0 × R. Now we prove that Mε is invertible. Lemma 4.5. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let ε ∈]0, +∞[ be such that clΩp,ε ⊆ Q. Then the operator Mε is a linear homeomorphism from C 0,α (∂Ω)0 × R onto C 0,α (∂Ω). Proof. We first verify that Mε is a Fredholm operator of index zero. Indeed, by [33, Prop. 3.1], the map from C 0,α (∂Ω)0 × R to C 0,α (∂Ω) which takes (θ, c) to
1 θ + w∗ [∂Ω, Sn , θ] − v[∂Ω, Sn , θ] + c 2 is a linear homeomorphism. Also, by standard properties of integral operators with real analytic kernel and with no singularities, the map from C 0,α (∂Ω)0 × R to C 0,α (∂Ω) which takes (θ, c) to the function of the variable t ∈ ∂Ω given by Z Z εn−1 DRn (ε(t − s))νΩ (t)θ(s) dσs − εn−2 Rn (ε(t − s))θ(s) dσs ∀t ∈ ∂Ω , ∂Ω

∂Ω

is compact. It follows that Mε is a compact perturbation of a linear homeomorphism, and accordingly a Fredholm operator of index 0. Thus, in order to show that Mε is a linear homeomorphism it suffices to show that Mε [θ, c] = 0 only if (θ, c) = (0, 0). So, let (θ, c) ∈ C 0,α (∂Ω)0 × R and assume that Mε [θ, c] = 0. Then Lemmas 4.1 and 4.4 imply that Z εn−2 Sq,n (x − p − εs)θ(s) dσs + c = 0 ∀x ∈ clS[Ωp,ε ]− . ∂Ω

Let µ(x) ≡ ε−1 θ((x − p)/) for all x ∈ ∂Ωp,ε . Then v − [∂Ωp,ε , Sq,n , µ] + c = 0

in clS[Ωp,ε ]− .

It follows that (µ, c) = (0, 0) (cf. [33, Lem. 3.2]). Thus (θ, c) = (0, 0) and the validity of the lemma is verified. 2 In the following Lemma 4.6 we introduce the “limiting boundary integral operator” M# . For a proof we refer the reader to [33, Prop. 3.1]. Lemma 4.6. Let α ∈]0, 1[. Let Ω be as in (1.1). Let M# denote the operator from C 0,α (∂Ω)0 × R to C 0,α (∂Ω) which takes a pair (θ, c) to M# [θ, c](t) ≡


1 θ(t) + w∗ [∂Ω, Sn , θ](t) − v[∂Ω, Sn , θ](t) + c 2

∀t ∈ ∂Ω .

Then M# is a linear homeomorphism. (−1)

Finally, we show in the following Lemma 4.7 that the sequence of operators Mεj (−1) converges to M# as j → ∞. In the sequel, if X and Y are Banach spaces, L(X , Y) denotes the space of linear bounded operators from X to Y endowed with the usual operator norm.

9

Lemma 4.7. Let α ∈]0, 1[. Let Ω be as in (1.1). Let 0 ∈]0, +∞[ be as in (1.2). Let {εj }j∈N (−1) (−1) be a sequence of ]0, 0 [ converging to 0. Then limj→+∞ Mεj = M# in L(C 0,α (∂Ω) , C 0,α (∂Ω)0 × R). Proof. Let Nj be the operator from C 0,α (∂Ω)0 × R to C 0,α (∂Ω) which takes (θ, c) to Z n−1 Nj [θ, c](t) ≡ εj DRn (εj (t − s))νΩ (t)θ(s) dσs ∂Ω Z −εn−2 Rn (εj (t − s))θ(s) dσs ∀t ∈ ∂Ω , j ∈ N. j ∂Ω

Let UΩ be an open bounded neighborhood of clΩ. Let # be such that (t−s) ∈ (Rn \qZn )∪ {0} for all t, s ∈ UΩ and all  ∈] − # , # [. By the real analyticity of Rn in (Rn \ qZn ) ∪ {0} it follows that the map which takes (, t, s) to Rn ((t−s)) is real analytic from ]−# , # [×UΩ × ˜ n from ] − # , # [×UΩ × UΩ to R such UΩ to R. Hence, there exists a real analytic map R ˜ that Rn ((t − s)) − Rn (0) = Rn (, t, s) for all t, s ∈ UΩ and all  ∈] − # , # [. Then, by the membership of θ in C 0,α (∂Ω)0 , one has Z Nj [θ, c](t) = εn−1 DRn (εj (t − s))νΩ (t)θ(s) dσs j ∂Ω Z ˜ n (εj , t, s)θ(s) dσs R ∀t ∈ ∂Ω −εn−1 j ∂Ω

for all j such that εj ∈]0, # [ and for all (θ, c) ∈ C 0,α (∂Ω)0 × R. Then, by standard properties of integral operators with real analytic kernels and with no singularities, we can deduce that limj→+∞ Nj = 0 in L(C 0,α (∂Ω)0 × R , C 0,α (∂Ω)). Since Mεj = M# + Nj , it follows that limj→+∞ Mεj = M] in L(C 0,α (∂Ω)0 × R , C 0,α (∂Ω)). By the continuity of the map from the open subset of the invertible operators of L(C 0,α (∂Ω)0 × R , C 0,α (∂Ω)) to L(C 0,α (∂Ω) , C 0,α (∂Ω)0 × R) which takes an operator to its inverse, one deduces that (−1) (−1) = M# (cf. e.g., Hille and Phillips [25, Thms. 4.3.2 and 4.3.5]). The limj→+∞ Mεj validity of the lemma is now proved. 2

4.3

LIMITING BEHAVIOR OF A CONVERGING FAMILY OF SOLUTIONS OF PROBLEM (1.3)

We are now ready to investigate in this subsection the limiting behavior of a converging family of solutions of problem (1.3) (see also Section 1). To begin with, we consider in the following Proposition 4.8 the limiting behavior of a converging family of q-periodic harmonic functions. Proposition 4.8. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 ∈]0, +∞[ be as in (1.2). Let {εj }j∈N be a sequence of ]0, 0 [ converging to 0 and let {uj }j∈N be a sequence of functions such that uj ∈ Cq1,α (clS[Ωp,εj ]− )

and

∆uj = 0

in S[Ωp,εj ]− .

Assume that there exist an open bounded neighborhood O of clΩ in Rn and a function v# ∈ C 1,α (clO \ Ω) such that lim uj (p + εj ·)|clO\Ω = v#

j→+∞

10

in C 1,α (clO \ Ω) .

(4.9)

1,α Then there exists unique a pair (u# , c# ) ∈ Cloc (Rn \ Ω) × R such that

v# = u#|clO\Ω ,

∆u# = 0 in Rn \ clΩ,

and

lim u# (t) = c# .

t→∞

Moreover, in C 1,α (clO0 \ Ω)

lim uj (p + εj ·)|clO0 \Ω = u#|clO0 \Ω

j→+∞

(4.10)

for all open bounded neighborhoods O0 of clΩ in Rn . Proof. Let φj ∈ C 0,α (∂Ω) be defined by φj (t) ≡ εj

∂uj (p + εj t) − uj (p + εj t) ∂νΩp,εj

∀t ∈ ∂Ω , j ∈ N .

(4.11)

Let φ# ∈ C 0,α (∂Ω) be defined by φ# (t) ≡

∂v# (t) − v# (t) ∂νΩ

∀t ∈ ∂Ω .

Then observe that εj

∂uj (p + εj t) ∂νΩp,εj
= εj νΩp,εj (p + εj t) · (Duj )(p + εj t) = νΩ (t) · Dt uj (p + εj t)

∀t ∈ ∂Ω .

Hence, by the convergence of uj (p + εj ·)|clO\Ω to v# in C 1,α (clO \ Ω) it follows that lim εj

j→+∞

∂uj ∂v# (p + εj ·)|∂Ω = ∂νΩp,εj ∂νΩ

in C 0,α (∂Ω) .

(4.12)

Henceforth, in C 0,α (∂Ω) .

lim φj = φ#

j→+∞

(4.13)

(−1)

(−1)

Let (θj , cj ) ≡ Mεj [φj ] for all j ∈ N and (θ# , c# ) ≡ M# [φ# ]. Since the evaluation map from L(C 0,α (∂Ω) , C 0,α (∂Ω)0 × R) × C 0,α (∂Ω) to C 0,α (∂Ω)0 × R, which takes a pair (A, v) to A[v] is bilinear and continuous, the limiting relation (4.13) and Lemma 4.7 imply that (−1)

lim (θj , cj ) = lim Mε(−1) [φj ] = M# j

j→+∞

j→+∞

[φ# ] = (θ# , c# )

(4.14)

in C 0,α (∂Ω)0 × R. Also, one has Mεj [θj , cj ] = φj and uj satisfies problem (4.2) with φ replaced by φj by the definition (4.11) of φj . Thus, Lemmas 4.1 and 4.4 imply that Z n−2 uj (x) = εj Sq,n (x − p − εj s)θj (s) dσs + cj ∀x ∈ clS[Ωp,εj ]− . ∂Ω

Then one has uj (p + εj t) = v[∂Ω, Sn , θj ](t) +

εn−2 j

Z Rn (εj (t − s))θj (s) dσs + cj ∂Ω

11

(4.15)

for all t ∈ Rn \ ∪z∈Zn (ε−1 j qz + clΩ) (see also Subsection 4.1). By the continuity of the map from C 0,α (∂Ω) to C 1,α (clO \ Ω) which takes θ to v[∂Ω, Sn , θ]|clO\Ω (cf. e.g., Miranda [41, Thm. 5.1]), by standard propertiesR of integral operators with real analytic kernels and with no singularities, by the condition ∂Ω θ# dσ = 0, and by (4.14), one verifies that in C 1,α (clO \ Ω) .

lim uj (p + εj ·)|clO\Ω = v[∂Ω, Sn , θ# ]|clO\Ω + c#

j→+∞

Hence, the limiting relation in (4.9) implies that v# = v[∂Ω, Sn , θ# ]|clO\Ω + c# . Now the validity of the proposition follows by taking u# (t) ≡ v[∂Ω, Sn , θ# ](t) + c#

∀t ∈ Rn \ Ω .

Indeed, u# is harmonic in Rn \ clΩ and by the membership of θ# in C 0,α (∂Ω) one deduces R 1,α that u# belongs to Cloc (Rn \ Ω) (cf. [34, Thm. 3.1]). Since ∂Ω θ# dσ = 0, the single layer potential v[∂Ω, Sn , θ# ] has limit 0 at infinity and accordingly limt→∞ u# (t) = c# . Finally, the validity of (4.10) for all open bounded neighborhoods O0 of clΩ follows by equality (4.15), by the limiting relation in (4.14), by the continuity of the map from C 0,α (∂Ω) to C 1,α (clO0 \ Ω) which takes θ to v[∂Ω, Sn , θ]|clO0 \Ω (cf. e.g., Miranda [41, Thm. 5.1]), by standard properties ofRintegral operators with real analytic kernels and with no singularities, and by the condition ∂Ω θ# dσ = 0. 2 We are now ready to prove the main Theorem 4.16 of this section. Theorem 4.16. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let G ∈ C 0 (∂Ω × R) be such that TG is continuous from C 0,α (∂Ω) to itself. Let 0 ∈]0, +∞[ be as in (1.2). Let γ be a map from ]0, 0 [ to ]0, +∞[ such that (1.4) holds. Let {εj }j∈N be a sequence of ]0, 0 [ converging to 0 and let {uj }j∈N be a sequence of functions such that the following condition holds for all j ∈ N: uj belongs to Cq1,α (clS[Ωp,εj ]− ) and is a solution of (1.3) for  = εj . Assume that there exist an open bounded neighborhood O of clΩ in Rn and a function v# ∈ C 1,α (clO \ Ω) such that the limit lim uj (p + εj ·)|clO\Ω = v#

j→+∞

1,α holds in C 1,α (clO \ Ω). Then there exists unique a pair (u# , c# ) ∈ Cloc (Rn \ Ω) × R such that v# = u#|clO\Ω , ∆u# = 0 in Rn \ clΩ, lim u# (t) = c# , (4.17) t→∞

and such that −

∂u# (t) = γm G(t, u# (t)) ∂νΩ

Z ∀t ∈ ∂Ω

and

G(t, u# (t)) dσt = 0 .

(4.18)

∂Ω

Moreover, lim uj (p + εj ·)|clO0 \Ω = u#|clO0 \Ω

j→+∞

for all open bounded neighborhoods O0 of clΩ in Rn . 12

in C 1,α (clO0 \ Ω)

(4.19)

1,α Proof. The existence and uniqueness of the pair (u# , c# ) ∈ Cloc (Rn \ Ω) × R as in (4.17), (4.19) is a consequence of Proposition 4.8. So one has to prove the validity of (4.18). Observe that

−εj

εj ∂uj (p + εj t) = G(t, uj (p + εj t)) ∂νΩp,εj γ(εj )

∀t ∈ ∂Ω , j ∈ N .

(4.20)

Then, by the limiting relation in (4.12), which follows by (4.19), and by condition (1.4), and by the continuity of TG from C 0,α (∂Ω) to itself, and by taking the limit as j → +∞ in (4.20), one obtains the first equality in (4.18). Further, by (4.20), and by the periodicity of uj , and by the Divergence Theorem, one has Z Z γ(εj ) ∂uj dσ = 0 = − n−1 G(t, uj (p + εj t)) dσt . (4.21) εj ∂Ωp,εj ∂νΩp,εj ∂Ω Then, by the continuity of TG from C 0,α (∂Ω) to itself, by the limiting relation in (4.19), and by letting j → +∞ in (4.21), one deduces the validity of the second equality of (4.18). 2 We now exploit Theorem 4.16 to prove the validity of the following. Proposition 4.22. With the assumptions of Theorem 4.16 the following statements hold. (i) If γm 6= 0 and if we set u ˜ ≡ (u# − c# )/γm ,

(4.23)

then the pair (˜ u, c# ) is a solution of the “limiting boundary value problem” (2.2) and we have lim uj (p + εj ·)|clO0 \Ω = γm u ˜|clO0 \Ω + c#

j→+∞

in C 1,α (clO0 \ Ω)

(4.24)

for all open bounded neighborhoods O0 of clΩ in Rn . (ii) If γm = 0, then u# = c# in Rn \ Ω and the boundary value problem  u=0 in Rn \ clΩ ,  ∆˜ ∂u ˜ (t) = G(t, c# ) ∀t ∈ ∂Ω , −  ∂νΩ limt→∞ u ˜(t) = 0 ,

(4.25)

1,α has a unique solution u ˜ ∈ Cloc (Rn \ Ω). Moreover, the pair (˜ u, c# ) is a solution of the “limiting boundary value problem” (2.2) and we have

lim uj (p + εj ·)|clO0 \Ω = γm u ˜|clO0 \Ω + c# = c#

j→+∞

in C 1,α (clO0 \ Ω)

(4.26)

for all open bounded neighborhoods O0 of clΩ in Rn . Proof. If γm 6= 0 and u ˜ is as in (4.23), then, by (4.18), the pair (˜ u, c# ) is a solution of the limiting boundary value problem (2.2). Also, (4.24) follows by the limiting relation in (4.19) (see also Proposition 4.8). Thus statement (i) holds true. If instead γm = 0, then, by 1,α (4.18), u# is a solution in Cloc (Rn \ Ω) of the following problem  in Rn \ clΩ ,  ∆u# = 0 ∂u# − (t) = 0 ∀t ∈ ∂Ω ,  ∂νΩ limt→∞ u# (t) = c# , 13

R and accordingly u# = c# in Rn \ Ω. Moreover, by Theorem 4.16 we have ∂Ω G(t, c# ) dσt = 0. As a consequence, by classical potential theory, there exists a unique function u ˜ in 1,α Cloc (Rn \ Ω) such that (4.25) holds. Hence, the validity of (4.26) follows by the limiting relation in (4.19) (see also Proposition 4.8). 2

4.4

A LOCAL UNIQUENESS RESULT FOR CONVERGING FAMILIES OF SOLUTIONS OF PROBLEM (1.3)

In this subsection we prove that a converging family of solutions of (1.3) is essentially unique in a local sense which we clarify in the following Theorem 4.27. Theorem 4.27. Let α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let G ∈ C 0 (∂Ω × R) be such that TG is real analytic from C 0,α (∂Ω) to itself. Let 0 ∈]0, +∞[ be as in (1.2). Let γ be a map from ]0, 0 [ to ]0, +∞[ such that (1.4) holds. Let {εj }j∈N be a sequence of ]0, 0 [ converging to 0. Let {uj }j∈N and {vj }j∈N be sequences of functions such that the following condition holds for all j ∈ N: uj and vj belong to Cq1,α (clS[Ωp,εj ]− ) and both uj and vj are solutions of (1.3) for  = εj . Assume that there exist an open bounded neighborhood O of clΩ in Rn and a function v# ∈ C 1,α (clO \ Ω) such that lim uj (p + εj ·)|clO\Ω = lim vj (p + εj ·)|clO\Ω = v#

j→+∞

j→+∞

If γm = 0, assume that

in C 1,α (clO \ Ω) .

(4.28)

Z Dξ G(t, v# (t)) dσt 6= 0 .

(4.29)

∂Ω

If γm 6= 0, assume that Z Dξ G(t, v# (t)) ≤ 0 for all t ∈ ∂Ω

and

Dξ G(t, v# (t)) dσt < 0 .

(4.30)

∂Ω

Here Dξ G denotes the partial derivative of G with respect to its last variable. Then there exists a natural number j0 ∈ N such that ∀j ≥ j0 .

uj = vj

Proof. Observe that the family {uj }j∈N and the function v# satisfy the conditions in Theorem 4.16. Accordingly, Proposition 4.22 implies that there exist a function u ˜ ∈ 1,α Cloc (Rn \ Ω) and a real constant c# ∈ R such that the pair (˜ u, c# ) is a solution of the “limiting boundary value problem” (2.2) and such that lim uj (p + εj ·)|∂Ω = γm u ˜|∂Ω + c#

j→+∞

in C 1,α (∂Ω) .

Then by (4.28) one also has lim vj (p + εj ·)|∂Ω = γm u ˜|∂Ω + c#

j→+∞

14

in C 1,α (∂Ω) .

Now, by conditions (4.29) and (4.30) and by Proposition 4.22 one verifies that (˜ u, c# ) satisfies the assumptions of Theorem 2.1. Then Theorem 2.6 implies that there exist j00 , j000 ∈ N such that uj (·) = u(εj , ·) for all j ≥ j00 and vj (·) = u(εj , ·) for all j ≥ j000 . It follows that uj = vj for all j ≥ j0 ≡ max{j00 , j000 }. Thus the theorem is proved. 2 ACKNOWLEDGMENTS: M. Dalla Riva acknowledges the financial support of the Portuguese Foundation for Science and Technology (“FCT–Funda¸c˜ao para a Ciˆencia e a Tecnologia”) via the post-doctoral grant SFRH/BPD/64437/2009. His work was also supported by FEDER funds through COMPETE–Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT–Funda¸c˜ao para a Ciˆencia e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. P. Musolino acknowledges the financial support of the “Fondazione Ing. Aldo Gini” and of the “Accademia Nazionale dei Lincei” through a scholarship “Royal Society”.

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