Stokes flow in a singularly perturbed exterior domain ∗ Matteo Dalla Riva

Abstract We consider a pair of domains Ωb and Ωs in Rn and we assume that the closure of Ωb does not intersect the closure of Ωs for  ∈]0, 0 [. Then for a fixed  ∈]0, 0 [ we consider a boundary value problem in Rn \ (Ωb ∪ Ωs ) which describes the steady state Stokes flow of an incompressible viscous fluid past a body occupying the domain Ωb and past a small impurity occupying the domain Ωs . The unknown of the problem are the velocity field u and the pressure field p, and we impose the value of the velocity field u on the boundary both of the body and of the impurity. We assume that the boundary velocity on the impurity displays an arbitrarily strong singularity when  tends to 0. The goal is to understand the behaviour of (u, p) for  small and positive. The methods developed aim at representing the limiting behaviour in terms of analytic maps and possibly singular but completely known functions of , such as −1 , log .

Keywords: boundary value problem for the Stokes system, singularly perturbed exterior domain, real analytic continuation in Banach space. 2000 Mathematics Subject Classification: 76D07, 35J57, 31B10, 45F15.

1

Introduction.

In this paper we present an application of a functional analytic approach to the analysis of a boundary value problem for the Stokes system in a singularly perturbed exterior domain. We now introduce our boundary value problem. We fix once for all n ∈ N \ {0, 1}, ∗

m ∈ N \ {0},

α ∈]0, 1[ .

Dedicated to Professor Victor Burenkov on the occasion of his 70th birthday.

1

Here N denotes the set of natural numbers including 0. Then we fix two sets Ωb and Ωs in the n-dimensional Euclidean space Rn . The letter ‘b’ stands for ‘body’ and the letter ‘s’ stands for ‘small impurity’. We assume that Ωb and Ωs satisfy the following condition Ωb and Ωs are open bounded connected subsets of Rn of class C m,α with connected exterior, the origin 0 of Rn belongs to Ωs but not to the closure clΩb of Ωb .

(1)

For the definition of the functions and sets of the usual Schauder class C m,α , we refer for example to Gilbarg and Trudinger [12, §6.2] (see also §2 of this paper.) We note that condition (1) implies that Ωb and Ωs have no holes and that there exists a real number 0 such that 0 ∈]0, 1[ and clΩb ∩ (clΩs ) = ∅ for all  ∈]0, 0 [ .

(2)

Then we denote by Ωe () the exterior domain defined by Ωe () ≡ Rn \ {clΩb ∪ (clΩs )}

∀ ∈]0, 0 [.

Here the letter ‘e’ stands for ‘exterior domain’. A simple topological argument shows that Ωe () is connected, and that Rn \clΩe () has exactly the two connected components Ωb and Ωs , and that the boundary ∂Ωe () of Ωe () has exactly the two connected components ∂Ωb and ∂Ωs , for all  ∈]0, 0 [. Now, let γ be a function from ]0, 0 [ to ]0, +∞[ such that γ0 ≡ lim γ() exists in [0, +∞[ .

(3)

→0

Let (f b , f s ) ∈ C m,α (∂Ωb , Rn ) × C m,α (∂Ωs , Rn ). Let  ∈]0, 0 [. We consider the following boundary value problem for a pair of functions (u, p) m,α m−1,α of Cloc (clΩe (), Rn ) × Cloc (clΩe (), R),  ∆u − ∇p = 0 in Ωe () ,    div u = 0 in Ωe () , (4) u = fb on ∂Ωb ,    u(x) = γ()−1 f s (x/) for x ∈ ∂Ωs . As is well known problem (4) admits a unique solution which satisfies the decay condition  sup |x|n−2 |u(x)| , |x|n−1 |Du(x)| , |x|n−1 |p(x)| < +∞ (5) |x|>R

2

with R ≡ supx∈{Ωb ∪(Ωs )} |x|. The uniqueness of the solution of (4), (5) can be deduced by the results of Chang and Finn in [3, §4], we refer to Varnhorn [44, Lemma 1.1] for a proof. For a proof of the existence in case n ≥ 3 we refer to Ladyzhenskaya [25, Chap. 3, Sec. 3], for a proof of the existence in case n = 2 we refer to Power [37, §§2,3] (see also Proposition 3.4 below.) m,α We now denote by (u , p ) the unique solution in Cloc (clΩe (), Rn ) × m−1,α e Cloc (clΩ ()) of (4), (5) with R ≡ supx∈{Ωb ∪(Ωs )} |x|. The aim of this paper is to understand the behaviour of (u , p ) when  shrinks to 0. If γ0 = 0 then the velocity field u on the boundary of Ωs displays a singular behavior as  → 0 due to the presence of the factor γ()−1 in the right hand side of the fourth equation in (4). If n−2 = 0, →0 γ()(log )δ2,n

lim

(6)

then we shall prove in Theorem 5.3 that (u , p ) converges to a pair of functions (ub , pb ) as  → 0 in a sense which will be clarified. Here δ2,n denotes the Kronecker symbol and (ub , pb ) denotes the unique solution in m,α m−1,α Cloc (Rn \ Ωb , Rn )× Cloc (Rn \ Ωb , R) of the ‘unperturbed boundary value problem’   ∆ub − ∇pb = 0 in Rn \ clΩb , div ub = 0 in Rn \ clΩb , (7)  b b u =f on ∂Ωb , which satisfies the decay condition in (5) for R ≡ supx∈Ωb |x|. We note that condition (6) is verified also in case γ0 > 0. If instead n−2 →0 γ()(log )δ2,n

exists in R \ {0} ,

lim

then (u , p ) converges to a pair of functions which may be different from (ub , pb ) (cf. Theorem 5.3.) The difference is related to the unique solution m,α m−1,α (us , ps ) in Cloc (Rn \ Ωs , Rn )× Cloc (Rn \ Ωs , R) of the ‘limiting boundary value problem’   ∆us − ∇ps = 0 in Rn \ clΩs , div us = 0 in Rn \ clΩs , (8)  s s u =f on ∂Ωs , which satisfies the decay condition in (5) for R ≡ supx∈Ωs |x|. Finally, if n−2 →0 γ()(log )δ2,n

lim

does not exist in R ,

3

then (u , p ) may display a singular behaviour as  → 0. In this case we show that by taking the limit of γ()(log )δ2,n (u − ub , p − pb ) n−2 we obtain a limiting pair of functions associated to the solution (us , ps ) of the ‘limiting problem’ (8), (5) (cf. Theorem 5.3.) However, our main interest is focused on the description of the behaviour of (u , p ) when  is near 0, and not only on the limiting behaviour. Actually, we pose the following question. Let x˜ be a point of Rn \ Ωb , x˜ 6= 0. What can be said of the (9) function which takes  to (u (˜ x), p (˜ x)) when  is small and positive? Questions of this type have long been investigated with the methods of Asymptotic Analysis, which aims at giving complete asymptotic expansions of the solutions in terms of the parameter . It is perhaps difficult to provide a complete list of the contributions. Here, we mention the books of Kozlov, Maz’ya and Movchan [22] and Maz’ya, Nazarov and Plamenewskii [34] for the analysis of general elliptic boundary value problems in singularly perturbed domains by means of the so called Compound Asymptotic Expansion Method. For application of the Matched Asymptotic Expansion Method to the analysis singularly perturbed problems in fluid mechanics we refer to the book of Van Dyke [43]. For the asymptotic analysis of a Navier-Stokes flow with low Reynold number we mention the work of Kevorkian and Cole [18], Hsiao and MacCamy [16], Hsiao [15], Fischer, Hsiao and Wendland [10]. For application of the Matched Asymptotic Expansion Method to the analysis of a Stokes flow past a porous media with low or high permeability we mention Kohr, Sekhar and Wendland [20, 21]. We note that, by the techniques of the Asymptotic Analysis, one can expect to obtain results which are expressed by means of a regular functions of  plus a reminder which is smaller than a positive known function of . Instead, the approach adopted in this paper aims to express the dependence upon  in terms of real analytic maps and in terms of possibly singular but completely known functions of , γ() such as −1 , log , γ()−1 . In particular, our main Theorem 5.1 shows that for  small and positive the pair (u (˜ x), p (˜ x)) can be expressed by means of real analytic maps of four variables defined in a neighborhood of (0, 0, 1 − δ2,n , γ0 ) and evaluated at (, (log )δ2,n , (log )−δ2,n , γ()) and by means of the singular function n−2 /[γ()(log )δ2,n ] of  ∈]0, 0 [. We note that such an approach has its own advantages. For example, one could obtain asymptotic approximations which agree with those one may find in the literature. And not only, if n ≥ 3 one would also express the dependence of γ()2−n (u (˜ x), p (˜ x)) on 4

 in terms of convergent power series of the two variables , γ(). Instead, if n = 2 one would express the dependence of γ()(log )(u (˜ x), p (˜ x)) on  in terms of convergent power series of the four variables , (log ), (log )−1 , γ(). This point of view has been adopted by Lanza de Cristoforis and his collaborators Preciso, Rossi, Musolino and the author in several problems for elliptic equations (see, e.g., [27], [28], [29], [30], [31], [32], [33], [36]) and for the elliptic system of equation of linearized elastostatic (see [5], [6], [7], [8].) In this paper we aim to extend such methods to the analysis of Dirichlet boundary value problems for the Stokes system. The paper is organized as follows. In Section 2 we introduce some notation. Section 3 is a section of preliminaries where we state some known results which have been slightly modified for our purposes. In particular, in Propositions 3.2, 3.4 we exploit the so called Completed Double Layer Integral Equation Method to provide an equivalent formulation of the boundary value problems in (4), (7), (8) with the proper decay condition (5) in terms of boundary integral equations. The Completed Double Layer Integral Equation Method was introduced by Power and Miranda in [38] for the analysis of a three-dimensional exterior Stokes flow around a solid particle and was extended to the two-dimensional case by Power in [37] (see also Kohr and Pop [19, Chapter 4] and references therein.) In Section 4, we exploit the results of Section 3 to transform the problem in (4), (5), which is defined on a -depending domain, into a problem for boundary integral equations on a set which does not depend on . In Section 5, we prove our main Theorems 5.1 and 5.3. In Theorem 5.1 we answer to the question in (9) while in Theorem 5.3 we describe the limiting behavior of (u , p ) when  → 0. Finally, in the Appendix we collect some known results of Potential Theory for the Stokes system.

2

Notation

We denote the norm on a (real) normed space X by k · kX . Let X and Y be normed spaces. We endow the product space X × Y with the norm defined by k(x, y)kX ×Y ≡ kxkX + kykY ∀(x, y) ∈ X × Y, while we use the Euclidean norm for Rn . For standard definitions of Calculus in normed spaces, we refer to Prodi and Ambrosetti [39]. The inverse function of an invertible function f is denoted f (−1) , as opposed to the reciprocal of a complex-valued function g, or the inverse of a matrix A, which are denoted g −1 and A−1 , respectively. A dot ‘·’ denotes the inner product in Rn , or the matrix product between matrices with real entries. If x ∈ Rn , then xj denotes the j-th coordinate of x and |x| denotes the Euclidean modulus of x in Rn . If x ∈ Rn 5

and R > 0, then Bn (x, R) ≡ {y ∈ Rn : |x − y| < R}. We denote by sn the (n − 1) dimensional measure of ∂Bn (0, 1). Let Ω be an open subset of Rn . Let k ∈ N. The space of k times continuously differentiable realvalued functions
on Ω is denoted by C k (Ω, R), or more simply by C k (Ω). n Let f ∈ C k (Ω) . The s-th component   of f is denoted fs , and Df (or ∇f ) denotes the gradient matrix

∂fs ∂xl

. Let η ≡ (η1 , . . . , ηn ) ∈ Nn ,

s,l=1,...,n |η| f denotes ∂xη∂1 ...∂x ηn . n 1

The subspace of C k (Ω) |η| ≡ η1 + · · · + ηn . Then Dη f of those functions f such that f and its derivatives Dη f of order |η| ≤ k can be extended with continuity to clΩ is denoted C k (clΩ). The subspace of C k (clΩ) whose functions have k-th order derivatives that are H¨older continuous with exponent β ∈]0, 1] is denoted C k,β (clΩ) (cf. e.g. Gilbarg and Trudinger [12].) The subspace of C k (clΩ) of those functions f such that k,β Cloc f|cl(Ω∩Bn (0,R)) ∈ C k,β (cl(Ω∩Bn (0, R))) for all R ∈]0, +∞[ is denoted (clΩ).
n n k,β k,β Let D ⊆ R . Then C (clΩ, D) denotes f ∈ C (clΩ) : f (clΩ) ⊆ D . A similar notation holds if D is replaced by the space Mn (R) of the real n × n matrices. then C k (clΩ) endowed with the norm P If Ω is bounded, η kf kC k (clΩ) ≡ |η|≤k supclΩ |D f | is a Banach space. If f ∈ C 0,β (clΩ), then o n |f (x)−f (y)| : x, y ∈ clΩ, x 6= y . its H¨older constant |f : Ω|β is defined as sup |x−y|β If Ω is bounded, then the P space C m,β (clΩ), equipped with its usual norm kf kC k,β (clΩ) = kf kC k (clΩ) + |η|=k |Dη f : Ω|β , is well-known to be a Banach space. We say that a bounded open subset of Rn is of class C k or of class C k,β , if it is a manifold with boundary imbedded in Rn of class C k or C k,β , respectively (cf. e.g. Gilbarg and Trudinger [12, §6.2].) For standard properties of the functions of class C k,β both on a domain of Rn or on a manifold imbedded in Rn we refer to Gilbarg and Trudinger [12] (see also Lanza [26, §2, Lem. 3.1, 4.26, Thm. 4.28], Lanza and Rossi [33, §2].) We retain the standard notation of Lp spaces and of corresponding norms. We note that throughout the paper ‘analytic’ means ‘real analytic’. For the definition and properties of analytic maps, we refer to Prodi and Ambrosetti [39, p. 89]. For the definition and properties of normally solvable operators in dual systems, we refer to Kress [23, Chapters 4 and 5].

6

3

Preliminaries

i,j i We denote by SV,n ≡ (SV,n )i,j=1,...,n , SP,n ≡ (SP,n )i=1,...,n the functions from n n R \ {0} to Mn (R) and to R , respectively, defined by 
1 i,j δ2,n log |x| + (1 − δ2,n )|x|2−n δi,j SV,n (x) ≡ 2sn (δ2,n + (2 − n))  xi xj − (δ2,n + (2 − n)) n , |x| 1 xi i (x) ≡ − SP,n sn |x|n

for all i, j ∈ {1, . . . , n} and all x ∈ Rn \ {0}. Here δi,j is defined by δi,j ≡ 0 if i 6= j and δi,j ≡ 1 if i = j, for all i, j ∈ N. As is well known the pair (SV,n , SP,n ) is a fundamental solution for the differential operator in (4) in the sense that  ∆SV,n − ∇SP,n = δ0 I, div SV,n = 0, in the sense of distributions in Rn . Here δ0 denotes the Dirac delta function with mass at 0 and I denotes the n × n identity matrix. We denote by i,j S˜V,n ≡ (S˜V,n )i,j=1,...,n the function from Rn \ {0} to Mn (R) defined by   x x 1 1 i j i,j δi,j n−2 + (n − 2) n ∀x ∈ Rn \ {0}. S˜V,n (x) ≡ 2sn (δ2,n + (2 − n)) |x| |x| So that S˜V,n = SV,n for n ≥ 3 and S˜V,n = (4π)−1 I for n = 2. We also find convenient to set j i,j S˜V,n ≡ (S˜V,n )i=1,...,n ,

j i,j SV,n ≡ (SV,n )i=1,...,n ,

which we think of as column vectors for all j ∈ {1, . . . , n}. For each scalar ρ ∈ R and each matrix A ∈ Mn (R) we set T (ρ, A) ≡ −ρI + (A + At ). Let Ω be an open bounded subset of Rn of class C 1,α . We denote by νΩ the outward unit normal to ∂Ω and we denote by Ω− the set Ω− ≡ Rn \ clΩ . Let µ ∈ C 1,α (∂Ω, Rn ). Then wV [µ] denotes the function from Rn to Rn defined by wV [µ](x) Z ≡−

t

µ (y)T

j (SP,n (x

−

j y), Dx SV,n (x

∂Ω

 − y))νΩ (y) dσy j=1,...,n

7

for all x ∈ Rn , and wP [µ] denotes the function from Rn \ ∂Ω to Rn defined by Z  t wP [µ](x) ≡ −2 div µ (y)SP,n (x − y) νΩ (y) dσy ∀x ∈ Rn \ ∂Ω . ∂Ω

As is well known the restrictions wV [µ]|Ω and wP [µ]|Ω admit a unique continuous extension to clΩ, which we denote by wV+ [µ] and wP+ [µ], respectively, and wV [µ]|Ω− , wP [µ]|Ω− admit a unique continuous extension to clΩ− , which we denote by wV− [µ] and wP− [µ], respectively (cf. e.g. Theorem 6.1 in the Appendix.) Let Skewn (R) denote the subspace of Mn (R) consisting of the skewsymmetric matrices. Let Zn denote the set of the n × n-real matrices Znj1 ,j2 with j1 , j2 ∈ {1, . . . , n}, j1 < j2 , defined by Znj1 ,j2 ≡ (δi,j1 δk,j2 − δi,j2 δk,j1 )i,k=1,...,n

∀j1 , j2 ∈ {1, . . . , n}, j1 < j2 .

Then Zn is an orthogonal basis of Skewn (R) with respect to the usual scalar product. We find convenient to introduce a way to index the elements of Zn . To do so, we denote by J ≡ (J1 , J2 ) the unique injective map from {1, . . . , n(n − 1)/2} to {1, . . . , n} × {1, . . . , n} such that J1 (j) < J2 (j) for all j ∈ {1, . . . , n(n − 1)/2} and such that J1 (j1 ) ≤ J1 (j2 ), and J2 (j1 ) < J2 (j2 ) if J1 (j1 ) = J1 (j2 ), for all j1 , j2 ∈ {1, . . . , n(n − 1)/2} with j1 < j2 . Then we set Znj ≡ ZnJ1 (j),J2 (j)

∀j ∈ {1, . . . , n(n − 1)/2} . n(n−1)/2

One easily verifies that Zn = {Zn1 , . . . , Zn technical Lemma 3.1.

(10)

}. Then we have the following

Lemma 3.1. Let Ω be an open bounded subset of Rn of class C 1,α which contains the origin 0 of Rn . Then     Z Znj x i t (11) (Zn x) T 0, Dx n νΩ (x) dσx = −2sn δi,j , |x| ∂Ω   Z   Znj x T 0, Dx n νΩ (x) dσx = 0 (12) |x| ∂Ω for all i, j ∈ {1, . . . , n(n − 1)/2}. Proof. Let  > 0 be such that clBn (0, ) ⊂ Ω. By the Divergence Theorem and by the equalities Zni + (Zni )t = 0, Znj + (Znj )t = 0, we deduce that the

8

integral in the left hand side of (11) equals    Z j
1 t Zn x i t i tr Zn + (Zn ) · (Dx + Dx ) n dx (13) 2 Ω\Bn (0,) |x|     Z Znj x i t + (Zn x) T 0, Dx n νBn (0,) (x) dσx |x| ∂Bn (0,)     Z Znj x i t (Zn x) T 0, Dx n νBn (0,) (x) dσx = |x| ∂Bn (0,)  j  Z j t Zn + (Zn ) (x Znj x) + (x Znj x)t x i t (Zn x) = dσx −n |x|n |x|n+2 |x| ∂Bn (0,) Z   = −n (Zni x)t (x Znj x) + (x Znj x)t x dσx , ∂Bn (0,1)

where we understand that (x Zni

x)h,k = xh

n X

(Zni )k,l xl

∀h, k ∈ {1, . . . , n}.

l=1

Here we interpret x as a column vector. We now observe that (x Znj x)t x = (Znj x)|x|2 ,

(Zni x)t (x Znj x)x = (xt Zni x)(xt Znj x) = 0.

So that, the last integral in (13) equals Z Z i t j (Zn x) (Zn x) dσx = −nδi,j −n ∂Bn (0,1)

x2J1 (j) + x2J2 (j) dσx

∂Bn (0,1)

(see also (10).) Hence the equality in (11) follows. We now prove equation (12). By arguing as in (13), we verify that the integral in (12) equals Z   −1 −n (x Znj x) + (x Znj x)t x dσx . ∂Bn (0,1)

Since the argument of the integral is odd such integral has value 0. Now let Ω be an open bounded subset of Rn of class C m,α . We denote by Ln [·, ·, ·, ·] the operator from Ω × R2 × C m,α (∂Ω, Rn ) to C m,α (∂Ω, Rn ) defined by Z 1 ˜ Ln [ˆ y , ρ1 , ρ2 , µ](x) ≡ − µ(x) + wV [µ](x) + ρ1 SV,n (x − yˆ) µ dσ 2 ∂Ω n(n−1)/2 X Z j · (x − yˆ) Z n +ρ2 µt (y)Znj · (y − yˆ) dσy ∀x ∈ ∂Ω, n |x − y ˆ | ∂Ω j=1 9

for all yˆ ∈ Ω, (ρ1 , ρ2 ) ∈ R2 , µ ∈ C m,α (∂Ω, Rn ). Let Ω1 and Ω2 be open bounded subsets of Rn of class C m,α such that clΩ1 ∩ clΩ2 = ∅. We denote ˜ n ≡ (L ˜ n,1 , L ˜ n,2 ) the operator from Ω1 × Ω2 × R4 × C m,α (∂Ω1 , Rn ) × by L m,α n C (∂Ω2 , R ) to C m,α (∂Ω1 , Rn ) × C m,α (∂Ω2 , Rn ) defined by 1 ˜ n,i [ˆ L y(1) , yˆ(2) , ρ, µ1 , µ2 ] ≡ − µi (x) + wV [µ1 ](x) + wV [µ2 ](x) 2 Z +ρ1 S˜V,n (x − yˆ(1) ) µ1 dσ ∂Ω1 Z µ2 dσ +ρ2 (SV,n (x − yˆ(1) ) − SV,n (x − yˆ(2) )) ∂Ω2 n(n−1)/2

+ρ3

X j=1 n(n−1)/2

+ρ4

X j=1

Znj · (x − yˆ(1) ) |x − yˆ(1) |n

Z

Znj · (x − yˆ(2) ) |x − yˆ(2) |n

Z

µt1 (y)Znj · (y − yˆ(1) ) dσy

∂Ω1

µt2 (y)Znj · (y − yˆ(2) ) dσy

∀x ∈ ∂Ωi

∂Ω2

for all i ∈ {1, 2}, yˆ(1) ∈ Ω1 , yˆ(2) ∈ Ω2 , ρ ≡ (ρ1 , ρ2 , ρ3 , ρ4 ) ∈ R4 , (µ1 , µ2 ) ∈ C m,α (∂Ω1 , Rn )×C m,α (∂Ω2 , Rn ). Then we have the following Propositions 3.2, 3.4 whose statements are just slight modifications of the results obtained by Power in [37, §§2,3], and by Power and Miranda in [38]. For the sake of completeness, we include a proof. To do so, we need the following notation. We denote by R the set of functions from Rn to Rn defined by R ≡ {ρ ∈ C 1 (Rn , Rn ) : ρ(x) = Ax + b ∀x ∈ Rn , with A ∈ Skewn (R) and b ∈ Rn } .

(14)

Moreover, if Ω is an open subset of Rn we denote by RΩ the set of restrictions to Ω of the functions of R, and we denote by RΩ,loc the set of functions from Ω to Rn which equal an element of R on each connected component of Ω, and we denote by (RΩ,loc )|∂Ω the set of functions on ∂Ω which are trace on ∂Ω of functions of RΩ,loc . Proposition 3.2. Let Ω be an open bounded connected subset of Rn of class C m,α with Rn \ clΩ connected. Let (ˆ y , ρ1 , ρ2 ) ∈ Ω × (R \ {0})2 . Then the operator Ln [ˆ y , ρ1 , ρ2 , ·] is a homeomorphism from C m,α (∂Ω, Rn ) onto itself. Moreover, if f ∈ C m,α (∂Ω, Rn ), (ˆ y , ρ1 , ρ2 , µ) ∈ Ω × R2 × C m,α (∂Ω) and

10

Ln [ˆ y , ρ1 , ρ2 , µ] = f , then the pair of functions (u, p) defined by Z − ˜ u(x) ≡ wV [µ](x) + ρ1 SV,n (x − yˆ) µ dσ

(15)

∂Ω n(n−1)/2

+ρ2

X j=1

p(x) ≡

wP− [µ](x)

Znj · (x − yˆ) |x − yˆ|n

+ (1 −

Z

µt (y)Znj · (y − yˆ) dσy ,

∂Ω

t (x δ2,n )ρ1 SP,n

Z − yˆ)

µ dσ

∀x ∈ Rn \ Ω,

∂Ω m−1,α m,α (Rn \ Ω) which satisfies (Rn \ Ω, Rn ) × Cloc is the unique element of Cloc the decay condition in (5) for R ≡ supx∈Ω |x| and such that  ∆u − ∇p = 0 , div u = 0 in Rn \ clΩ , (16) u=f on ∂Ω.

Proof. As a first step we show that Ln [ˆ y , ρ1 , ρ2 , ·] is a Fredholm operator of index 0 for all fixed (ˆ y , ρ1 , ρ2 ) ∈ Ω × (R \ {0})2 . We note that the map of C m,α (∂Ω, Rn ) into itself which takes µ to the function ρ1 S˜V,n (x − yˆ)

n(n−1)/2

Z µ dσ + ρ2 ∂Ω

X j=1

Znj · (x − yˆ) |x − yˆ|n

Z

µt (y)Znj · (y − yˆ) dσy

∂Ω

of the variable x ∈ ∂Ω, is compact as a linear map with finite dimensional image. Hence, Ln [ˆ y , ρ1 , ρ2 , ·] is a compact perturbation of the operator 1 m,α − 2 I + WV of C (∂Ω, Rn ) to itself (cf. definition (49) of the Appendix.) Since a compact perturbation of a Fredholm operator of index 0 is still a Fredholm operator of index 0, and since − 21 I +WV is a Fredholm operator of index 0 from C m,α (∂Ω, Rn ) to itself, we deduce that Ln [ˆ y , ρ1 , ρ2 , ·] is a Fredholm operator of index 0 (cf. e.g. Theorem 6.6 of the Appendix.) We now show that Ln [ˆ y , ρ1 , ρ2 , ·] is an isomorphism from C m,α (∂Ω, Rn ) to itself by proving that the null space of Ln [ˆ y , ρ1 , ρ2 , ·] is trivial. So, let µ ∈ C m,α (∂Ω, Rn ) be such that Ln [ˆ y , ρ1 , ρ2 , µ] = 0. We show that µ = 0. By classical jump properties of the double layer Stokes potentials (cf. e.g. Kohr and Pop [19, Theorems 3.4.10, 3.4.14]), we deduce that the pair of functions (u, p) defined by (15) is a solution of the boundary value problem in (16) with f = 0 and satisfies the decay condition in (5) for R ≡ supx∈Ω |x|. Thus (u, p) = (0, 0).

11

In particular, n(n−1)/2

wV− [µ](x)

+ ρ2

X j=1

Znj · (x − yˆ) |x − yˆ|n

Z

µt (y)Znj · (y − yˆ) dσy

(17)

∂Ω

Z

= −ρ1 S˜V,n (x − yˆ) µ dσ ∂Ω Z − t µ dσ wP [µ](x) = −(1 − δ2,n )ρ1 SP,n (x − yˆ)

∀x ∈ Rn \ Ω , ∀x ∈ Rn \ Ω . (18)

∂Ω

We note that the function on the left hand side of equation (17) is o(|x|2−n ) for x in a neighborhood of ∞ and that R 2 t Z Z µ dσ ∂Ω µ dσ S˜V,n (x − yˆ) µ dσ ≥ |x − yˆ|n−2 2sn (δ2,n + (n − 2)) ∂Ω ∂Ω for all x ∈ Rn \ Ω. Then, by multiplying both the left and right hand side of
t R equation (17) by |x − yˆ|n−2 ∂Ω µ dσ , and by taking the limit as |x| → ∞, we deduce that Z µ dσ = 0. (19) ∂Ω

Then equations (17) and (18) imply that we have n(n−1)/2

wV− [µ](x)

+ ρ2

X j=1

Znj · (x − yˆ) |x − yˆ|n

Z

µt (y)Znj · (y − yˆ) dσy = 0,

(20)

∂Ω

wP− [µ](x) = 0,

(21)

for all x ∈ Rn \ Ω. Proposition 6.2 of the Appendix, and the Divergence Theorem, and the obvious equality Zni + (Zni )t = 0, imply that Z  t T (0, DwV− [µ](x))νΩ (x) Zni · (x − yˆ) dσx (22) ∂Ω Z  t = T (wP+ [µ](x), DwV+ [µ](x))νΩ (x) Zni · (x − yˆ) dσx ∂Ω Z  1 = tr (DwV+ [µ](x) + Dt wV+ [µ](x))(Zni + (Zni )t ) dx = 0 2 Ω for all i ∈ {1, . . . , n(n − 1)/2}. Moreover, by Lemma 3.1,    t Znj · (x − yˆ) T 0, Dx νΩ (x) Zni · (x − yˆ) dσx = −2sn δi,j |x − yˆ|n ∂Ω

Z

12

(23)

for all i ∈ {1, . . . , n(n − 1)/2}. Then, by equations (20), (22) and (23), we obtain Z ∀j ∈ {1, . . . , n(n − 1)/2} , (24) µt (y)Znj · (y − yˆ) dσy = 0 ∂Ω

and thus wV− [µ] = 0 by equation (20). Then, by the known jump properties of the Stokes double layer potentials we have − 21 µ + WV [µ] = 0. Then, by Theorem 6.4 in the Appendix, µ ∈ (RΩ,loc )|∂Ω . By equations (19) and (24), we also have that µ is orthogonal to (RΩ,loc )|∂Ω . Hence µ = 0. The last statement of the Proposition follows immediately by classical jump properties of Stokes double layer potentials and by the uniqueness of the solution of the boundary value problem in (16) under the condition in (5) with R ≡ supx∈Ω |x|. The following Remark 3.3 is an immediate consequence of Proposition 3.2. We observe that the same remark also follows by the results of Chang and Finn (cf. [3, Theorems 1 and 2].) Remark 3.3. Let the notation of Proposition 3.2 hold. Let n = 2. Then the limit lim|x|→+∞ u(x) exists in R2 . Proposition 3.4. Let Ω1 and Ω2 be open bounded connected subsets of Rn of class C m,α with clΩ1 ∩clΩ2 = ∅. Let Rn \clΩ1 and Rn \clΩ2 be connected. Let ˜ y(1) , yˆ(2) , ρ, ·, ·] is a (ˆ y(1) , yˆ(2) , ρ) ∈ Ω1 × Ω2 × (R \ {0})4 . Then the operator L[ˆ m,α n m,α n homeomorphism of C (∂Ω1 , R ) × C (∂Ω2 , R ) onto itself. Moreover, if ˜ y(1) , yˆ(2) , ρ, µ1 , µ2 ] = (µ1 , µ2 ) belongs to C m,α (∂Ω1 , Rn )×C m,α (∂Ω2 , Rn ) and L[ˆ (f1 , f2 ), then the pair of functions (u, p) defined by Z − − ˜ u(x) ≡ wV [µ1 ](x) + wV [µ2 ](x) + ρ1 SV,n (x − yˆ(1) ) µ1 dσ (25) ∂Ω1 Z +ρ2 (SV,n (x − yˆ(1) ) − SV,n (x − yˆ(2) )) µ2 dσ ∂Ω2 n(n−1)/2

+ρ3

X j=1 n(n−1)/2

+ρ4

X j=1

p(x) ≡

wP− [µ1 ](x)

+

Znj · (x − yˆ(1) ) |x − yˆ(1) |n

Z

Znj · (x − yˆ(2) ) |x − yˆ(2) |n

Z

wP− [µ2 ](x)

µt1 (y)Znj · (y − yˆ(1) ) dσy

∂Ω1

µt2 (y)Znj · (y − yˆ(2) ) dσy ,

∂Ω2

+ (1 −

t δ2,n )ρ1 SP,n (x

Z − yˆ(1) )

µ1 dσ ∂Ω1

t +ρ2 (SP,n (x

− yˆ(1) ) −

t SP,n (x

Z − yˆ(2) ))

µ2 dσ ∂Ω2

∀x ∈ Rn \ (Ω1 ∪ Ω2 ) 13

m−1,α m,α (Rn \ (Ω1 ∪ Ω2 )) (Rn \ (Ω1 ∪ Ω2 ), Rn ) × Cloc is the unique element of Cloc which satisfies the decay condition in (5) for R ≡ supx∈(Ω1 ∪Ω2 ) |x| and such that   ∆u − ∇p = 0 , div u = 0 in Rn \ cl(Ω1 ∪ Ω2 ), u = f1 on ∂Ω1 , (26)  u = f2 on ∂Ω2 .

Proof. Let i, j ∈ {1, 2}, i 6= j. Let Uj be an open bounded neighborhood of ∂Ωj such that clUj ∩ ∂Ωi = ∅. By standard properties of integral operators with real analytic kernel and with no singularity, the map from L1 (∂Ωi , Rn ) to C m+1 (clUj , Rn ) which takes µ to w[µ]|clUj is continuous. Since C m,α (∂Ωi , Rn ) is continuously imbedded into L1 (∂Ωi , Rn ), and since C m+1 (clUj , Rn ) is compactly imbedded into C m,α (clUj , Rn ), and since the restriction operator is continuous from C m,α (clUj , Rn ) to C m,α (∂Ωj , Rn ), we deduce that the operator which takes µ ∈ C m,α (∂Ωi , Rn ) to wV [µ]|∂Ωj ∈ C m,α (∂Ωj , Rn ) is compact. Then, by arguing so as in the proof of Proposition 3.2, we deduce that ˜ n [ˆ L y(1) , yˆ(2) , ρ, ·, ·] is a Fredholm operator of index 0 for all fixed (ˆ y(1) , yˆ(2) , ρ) ∈ 4 ˜ Ω1 × Ω2 × (R \ {0}) . Thus, in order to show that Ln [ˆ y(1) , yˆ(2) , ρ, ·, ·] is a m,α n m,α n homeomorphism from C (∂Ω1 , R ) × C (∂Ω2 , R ) to itself it suffices to ˜ n [ˆ prove that the null space of L y(1) , yˆ(2) , ρ, ·, ·] is trivial. So, let (µ1 , µ2 ) ∈ m,α n m,α n ˜ n [ˆ C (∂Ω1 , R ) × C (∂Ω2 , R ) be such that L y(1) , yˆ(2) , ρ, µ1 , µ2 ] = (0, 0). We show that (µ1 , µ2 ) = (0, 0). By classical jump properties of the double layer Stokes potential, we deduce that the pair of functions (u, p) defined by (25) is the solution of the boundary value problem in (26) with f1 = 0, f2 = 0 and satisfies (5) for R ≡ supx∈Ω1 ∪Ω2 |x|. Thus, the uniqueness of the solution of (26), (5) implies Rthat (u, p) = (0, 0) R(cf. e.g. Varnhorn [44, pp.16–17].) We now show that ∂Ω1 µ1 dσ = 0 and ∂Ω2 µ2 dσ = 0. By equation u = 0, we have wV− [µ1 ](x) + wV− [µ2 ](x)

(27) Z

+ρ2 (SV,n (x − yˆ(1) ) − SV,n (x − yˆ(2) ))

µ2 dσ ∂Ω2

n(n−1)/2

+ρ3

X j=1

Znj · (x − yˆ(1) ) |x − yˆ(1) |n

Z

µt1 (y)Znj · (y − yˆ(1) ) dσy

∂Ω1

n(n−1)/2

Z Znj · (x − yˆ(2) ) +ρ4 µt2 (y)Znj · (y − yˆ(2) ) dσy 2 |x − y ˆ | (2) ∂Ω 2 j=1 Z = ρ1 S˜V,n (x − yˆ(1) ) µ1 dσ ∀x ∈ Rn \ (Ω1 ∪ Ω2 ). X

∂Ω1

14

We note that the function on the left hand side of equation (27) is o(|x|2−n ) as |x| → ∞ and that R 2 Z t Z µ dσ ∂Ω1 1 n−2 µ1 dσ S˜V,n (x − yˆ(1) ) µ1 dσ ≥ |x − yˆ(1) | 2sn (δ2,n + (n − 2)) ∂Ω1 ∂Ω1 for all x ∈ Rn \ Ω1 . Thus, by multiplying both the left and right hand side R t of (27) by |x − yˆ(1) |n−2 ∂Ω1 µ1 dσ and by taking the limit as |x| → ∞, we R deduce that ∂Ω1 µ1 dσ = 0. Now, Proposition 6.2 of the Appendix and the Divergence Theorem imply that Z
T wP− [µi ](x), DwV− [µi ](x) νΩ1 dσx (28) ∂Ω1 Z
= T wP+ [µi ](x), DwV+ [µi ](x) νΩ1 dσx = 0 ∀i ∈ {1, 2}. ∂Ω1

Moreover

  Znj · (x − yˆ(i) ) T 0, Dx νΩ1 (x) dσx = 0, |x − yˆ(i) |n ∂Ω1

Z

(29)

for all i ∈ {1, 2}, j ∈ {1, . . . , n(n − 1)/2} by Lemma 3.1. By the Green Formula (cf. e.g. the equations in (48) of the Appendix) we have Z
k k − T SP,n (x − yˆ(i) ), DSV,n (x − yˆ(i) ) νΩ1 (x) dσx = wV [ek ](ˆ y(i) ) = δ1,i ek ∂Ω1

(30) for all k ∈ {1, . . . , n} and i ∈ {1, 2}, where ek ≡ (δj,k )j=1,...,n . Since (u, p) = (0, 0), we clearly have Z T (p, Du)νΩ1 dσ = 0 . (31) ∂Ω1

R We now exploit definition (25). By equality ∂Ω1 µ1 dσ = 0, and by equations R (28)–(31), we deduce that ∂Ω2 µ2 dσ = 0. Thus we have n(n−1)/2

wV− [µ1 ](x)

+

wV− [µ2 ](x)

+ ρ3

X j=1

n(n−1)/2

+ρ4

X j=1

Znj · (x − yˆ(1) ) |x − yˆ(1) |n

Znj · (x − yˆ(2) ) |x − yˆ(2) |n

15

Z ∂Ω

Z

µt1 (y)Znj · (y − yˆ(1) ) dσy

∂Ω

µt2 (y)Znj · (y − yˆ(2) ) dσy = 0,

and wP− [µ1 ](x) + wP− [µ2 ](x) = 0 for all x ∈ Rn \ (Ω1 ∪ Ω2 ). We observe that we have Z  t T (0, DwV− [µ1 ](x) + DwV− [µ2 ](x))νΩ (x) Zni · (x − yˆ(i) ) dσx = 0, ∂Ωi

 t   Znj · (x − yh ) νΩ (x) Znk · (x − yˆ(i) ) dσx = −2sn δi,h δj,k , T 0, Dx n |x − yh | ∂Ωi

Z

for all i, h ∈ {1, 2}, j, k ∈ {1, . . . , n(n − 1)/2} (cf. equation (24) and Lemma 3.1). We deduce that Z µti (y)Znj · (y − yˆ(i) ) dσy = 0 ∀j ∈ {1, . . . , n(n − 1)/2}, (32) ∂Ωi

and wV− [µi ] = 0, for all i ∈ {1, 2}. Now, if we set µ ≡ (µ1 , µ2 ), then we have − 21 µ + WV [µ] = 0R and thus µ ∈ (RΩ1 ∪Ω2 ,loc )|∂Ω by Theorem 6.4 in the Appendix. By equations ∂Ω1 µ1 dσ = 0, R and ∂Ω2 µ2 dσ = 0, and (32) we verify that µ is orthogonal to (RΩ1 ∪Ω2 ,loc )|∂Ω . Hence µ = 0. The last statement of the Proposition follows immediately by the classical properties of the Stokes second layer potential and by the uniqueness of the solution of the boundary value problem in (26) with condition (5).

4

The auxiliary boundary operator M

We now provide a formulation of problem (4), (5) in terms of integral equations. We can exploit Proposition 3.4. However, we note that the corresponding integral equations include integration on the -dependent set ∂Ωe (). In order to get rid of such a dependence, we introduce the following proposition in which we properly rescale the restriction of the unknown functions to ∂Ωs . We find convenient to introduce the following notation. We set Ym,α ≡ C m,α (∂Ωb , Rn ) × C m,α (∂Ωs , Rn ) . Then we have the following Proposition 4.1 which immediately follows by the rule of change of variables over ∂Ωs and by Propositions 3.2 and 3.4. Proposition 4.1. Let Ωb and Ωs be as in (1). Let 0 be as in (2). Let γ, γ0 be as in (3). Let y b ∈ Ωb . Let g b ∈ C m,α (∂Ωb , Rn ), g s ∈ C m,α (∂Ωs , Rn ). Let η ] ∈ C m,α (∂Ωb , Rn ) be the unique solution of the equation Ln [y b , 1, 1, η ] ] = g b .

16

We denote by M ≡ (M b , M s ) the operator from ] − 0 , 0 [× R3 × Ym,α to Ym,α defined by M b [, 1 , 2 , 3 , η b , η s ](x) Z

η s dσ ≡ Ln [y , 1, 1, η ](x) + (SV,n (x − y ) − SV,n (x)) ∂Ωs Z −1 (η s (y))t T (SP,n (x − y), DSV,n (x − y))νΩs (y) dσy b

b

∂Ωs n(n−1)/2

X

+1

j=1

Znj x |x|n

b

Z ∂Ωs

(η s (y))t Znj y dσy

∀x ∈ ∂Ωb ,

1 M s [, 1 , 2 , 3 , η b , η s ](x) ≡ − η s (x) + wV [η s ](x) 2  Z δ2,n n−2 b +  2 SV,n (x − y ) − 2 SV,n (x) − η s dσ 4π ∂Ωs Z n(n−1)/2 X Zj x n (η s (y))t Znj y dσy − g s (x) + n |x| s ∂Ω j=1 +n−2 2 wV [η b ](x) + 3 wV [η ] ](x) n(n−1)/2 X Z j · (x − y b ) Z n n−2 (η b (y))t Znj · (y − y b ) dσy + 2 b |n |x − y b ∂Ω j=1 n(n−1)/2

Znj · (x − y b ) +3 |x − y b |n j=1  Z b n−2 ˜ +SV,n (x − y )  2 X

Z ∂Ωb

(η ] (y))t Znj · (y − y b ) dσy

b

Z

η dσ + 3

∂Ωb

]



η dσ

∀x ∈ ∂Ωs ,

∂Ωb

where we intend 0 = 1 for all  ∈] − 0 , 0 [. Let  ∈]0, 0 [. Then the pair (η b , η s ) ∈ Ym,α is a solution of M [, (log )δ2,n , (log )−δ2,n , γ(), η b , η s ] = 0,

(33)

if and only if ˜ n [y b , 0, 1, −1 (log )−δ2,n , 1, −1 , µ1 , µ2 ] = (f1 , f2 ) L where the pairs (µ1 , µ2 ), (f1 , f2 ) ∈ C m,α (∂Ωb , Rn )×C m,α (∂Ωs , Rn ) are defined by µ1 ≡ γ()η ] + n−2 (log )−δ2,n η b , f1 ≡ γ()g b , µ2 (x) ≡ η s (x/), f2 (x) ≡ g s (x/) for all x ∈ ∂Ωs . In particular, the solution (η b , η s ) in Ym,α of (33) exists and is unique. 17

Proposition 4.1 reduces the analysis of problem (4), (5), which has been considered only for  ∈]0, 0 [ to the analysis of the set of zeros of M . However, equation M = 0, contrary to problem (4), (5) makes sense also for  = 0. Then we state the following Proposition 4.2 where we clarify what equation M = 0 means for  = 0. The proof of Proposition 4.2 immediately follows by the definition of M in Proposition 4.1 and by Proposition 3.2. Proposition 4.2. Let the notation and assumption of Proposition 4.1 hold. Then the pair (η b , η s ) ∈ Ym,α is a solution of M [0, 0, 1 − δ2,n , γ0 , η b , η s ] = 0,

(34)

if and only if R  b b b s  Ln [y , 1, 1, η ] = (SV,n (x − y ) − SV,n (x)) ∂Ωs η Rdσ, s s ] b Ln [0, −1, 1, η ] = g − γ0 wV [η ](0) − γ0 S˜V,n (y ) ∂Ωb η ] dσ Pn(n−1)/2 j b R  (Zn y ) ∂Ωb (η ] (y))t Znj · (y − y b ) dσ. −γ0 |y b |−n j=1 In particular, there exists a unique (η b , η s ) ∈ Ym,α which satisfies (34). In the following Proposition 4.4 we exploit the Implicit Function Theorem for real analytic maps to investigate the dependence of the solution (η b , η s ) of equation (33) upon  (cf. e.g. Deimling [9, Theorem 15.3].) To do so we need to analyze the regularity of the operator M . The definition of M involves the double layer potential wV [·] and also integral operators which display no singularity. To analyze their regularity we need the following. e Lemma 4.3. Let F be a real analytic map from Rn \{0} to Mn (R). Let Ω, Ω e be of be open bounded connected subsets of Rn . Let Ω be of class C m,α . Let Ω 1 m,α n m,α e R )×C (∂Ω, Rn )× class C . Then the map H from {(Φ, φ, f ) ∈ C (clΩ, e ∩ φ(∂Ω) = ∅} to C m,α (clΩ, e Rn ) which takes (Φ, φ, f ) to L1 (∂Ω, Rn ) : Φ(clΩ) the function H[Φ, φ, f ] defined by Z e H[Φ, φ, f ](x) ≡ F (Φ(x) − φ(y))f (y) dσy ∀x ∈ clΩ, ∂Ω

is real analytic. Proof. The proof follows by a known result in composition operators (cf. B¨ohme and Tomi [2, p.10], Henry [14, p.29], Valent [42, Theorem 5.2, p.44]), and its proof is a straightforward modification of the corresponding argument of Lanza [29, Theorem 6.2]. We observe that the map which takes (Φ, φ, f ) to e e the function Φ(x)−φ(y) of (x, y) ∈ clΩ×∂Ω in C m,α (clΩ×∂Ω, Rn \{0}), and e × ∂Ω, Rn \ {0}) to its composite the map which takes a function of C m,α (clΩ 18

e function with F in C m,α (clΩ×∂Ω, Mn (R)) are real analytic, and that the map e × ∂Ω, Mn (R)) × L1 (∂Ω, Rn ) which takes a pair of functions (g, f ) of C m,α (clΩ R m,α n e R ) is bilinear and continuous. to ∂Ω g(·, y)f (y) dσy in C (clΩ, Proposition 4.4. Let the notation of Proposition 4.1 hold. Let (˜ η b , η˜s ) be b s the unique solution in Ym,α of equation M [0, 0, 1 − δ2,n , γ0 , η˜ , η˜ ] = 0. Then there exist 0 ∈]0, 0 [, and an open neighborhood U of (0, 1 − δ2,n , γ0 ) in R3 , and an open neighborhood V of (˜ η b , η˜s ) in Ym,α , and a real analytic operator E ≡ (E b , E s ) from ] − 0 , 0 [×U to V, such that the set of zeros of M in ] − 0 , 0 [×U × V coincides with the graph of E. Proof. We note that the existence and uniqueness of the solution (˜ η b , η˜s ) ∈ Ym,α of M [0, 0, 1 − δ2,n , γ0 , η˜b , η˜s ] = 0 follows by Proposition 4.2. We now prove the statement by applying the Implicit Function Theorem to equation M [, 1 , 2 , 3 , η b , η s ] = 0 around the point (0, 0, 1 − δ2,n , γ0 , η˜b , η˜s ). By standard properties of the Stokes double layer potential (cf. e.g. Theorem 6.1 in the Appendix) and by Lemma 4.3, we deduce that the map M is real analytic. By standard Calculus in Banach space, the differential of M at the point (0, 0, 1 − δ2,n , γ0 , η˜b , η˜s ) with respect to the variable (η b , η s ) is delivered by the formula ∂(ηb ,ηs ) M b [0, 0, 1 − δ2,n , γ0 , η˜b , η˜s ](¯ η b , η¯s )(x) b

b

b

Z

≡ Ln [y , 1, 1, η¯ ](x) − (SV,n (x − y ) + SV,n (x))

η¯s dσ

∀x ∈ ∂Ωb ,

∂Ωs

∂(ηb ,ηs ) M s [0, 0, 1 − δ2,n , γ0 , η˜b , η˜s ](¯ η b , η¯s ) ≡ Ln [0, −1, 1, η¯s ], for all (¯ η b , η¯s ) ∈ Ym,α . We verify that ∂(ηb ,ηs ) M [0, 0, 1 − δ2,n , γ0 , η˜b , η˜s ] is a linear homeomorphism of Ym,α onto itself. By the Open Mapping Theorem if suffices to show that, for each (f¯b , f¯s ) ∈ Ym,α there exists a unique (¯ η b , η¯s ) ∈ Ym,α such that ∂(ηb ,ηs ) M [0, 0, 1 − δ2,n , γ0 , η˜b , η˜s ](¯ η b , η¯s ) = (f¯b , f¯s ). By Proposition 3.2 we immediately deduce the existence and uniqueness of η¯s , which in turn implies that of η¯b . Now we can invoke the Implicit Function Theorem for analytic maps and deduce the existence of the operator E as in the statement (cf. e.g. Deimling [9, Theorem 15.3].)

5

Real analytic representation for (u, p)

To simplify our notation we introduce the following definition. Let 0 be as in (2). Let γ, γ0 be as in (3). Then we denote by Ψn [·] the function from ]0, 0 [ to R4 defined by
∀ ∈]0, 0 [ . Ψn [] ≡ , (log )δ2,n , (log )−δ2,n , γ() 19

We note that lim→0 Ψn [] = (0, 0, 1 − δ2,n , γ0 ). Then we have the following Theorem 5.1 where we describe the behavior of the solution (u , p ) of (4), (5) for  small and positive in terms of real analytic functions of Ψn [] and singular but completely known functions of , γ(). Theorem 5.1. Let Ωb and Ωs be as in (1). Let 0 be as in (2). Let γ, γ0 be as in (3). Let (f b , f s ) ∈ C m,α (∂Ωb , Rn ) × C m,α (∂Ωs , Rn ). Let (ub , pb ) denote the m,α m−1,α unique solution in Cloc (Rn \ Ωb , Rn ) × Cloc (Rn \ Ωb ) of the boundary value problem in (7) which satisfies the decay condition in (5) for R ≡ supx∈Ωb |x|. ˜ be an open bounded subset of Rn \ clΩb such that 0 ∈ ˜ Then, there Let Ω / clΩ. exist Ω˜ ∈]0, 0 [, and an open neighborhood U of (0, 1 − δ2,n , γ0 ) in R3 , and real analytic maps ˜ Rn ) , U : ] − Ω˜ , Ω˜ [×U → C m,α (clΩ, ˜ P : ] − Ω˜ , Ω˜ [×U → C m−1,α (clΩ) such that ˜ ∩ (clΩs ) = ∅ clΩ

and

Ψn [] ∈] − Ω˜ , Ω˜ [×U

∀ ∈]0, Ω˜ [

(35)

and n−2 U [Ψn []] , γ()(log )δ2,n n−2 = pb|clΩ˜ + P [Ψn []] γ()(log )δ2,n

u|clΩ˜ = ub|clΩ˜ + p|clΩ˜

(36) ∀ ∈]0, Ω˜ [.

m,α m−1,α Here (u , p ) denotes the unique solution in Cloc (clΩ(), Rn )×Cloc (clΩ()) of the boundary value problem in (4) which satisfies the decay condition in (5) for R ≡ supx∈(Ωb ∪(Ωs )) |x|.

Proof. Let y b be a point of Ωb . Let U be the neighborhood of (0, 1 − δ2,n , γ0 ) introduced in Proposition 4.4. Then condition (35) is easily seen to be verified

20

for Ω˜ ∈]0, 0 [ small enough. We now define Z − b b ˜ U [e](x) ≡ wV [E [e]](x) + SV,n (x − y )

E b [e] dσ

(37)

∂Ωb

n(n−1)/2

+

X j=1

Znj · (x − y b ) |x − y b |n

Z ∂Ωb

(E b [e](y))t Znj · (y − y b ) dσy

Z

E s [e] dσ + SV,n (x − y ) − SV,n (x) ∂Ωs Z +1 (E s [e](y))t T (SP,n (x − y), DSV,n (x − y))νΩs (y) dσy b

∂Ωs n(n−1)/2

X

+1

j=1

Znj x |x|n

b




Z ∂Ωs

P [e](x) ≡ wP [E [e]](x) + (1 −

(E s [e](y))t Znj y dσy ,

t δ2,n )SP,n (x

b

Z

−y )

E b [e] dσ

(38)

∂Ωb

Z

E s [e] dσ + SP,n (x − y ) − SP,n (x) ∂Ωs Z  s t (E [e](y)) SP,n (x − y)νΩs (y) dσy −21 div b




∂Ωs

˜ and all e ≡ (, 1 , 2 , 3 ) ∈] −  ˜ ,  ˜ [×U. Then by Proposifor all x ∈ clΩ, Ω Ω tions 4.1 and 4.4, we deduce the validity of (36). The real analyticity of the maps U and P follows by the real analyticity of E b , E s (cf. Proposition 4.4), and by the standard properties of the Stokes double layer potentials (cf. e.g. Theorem 6.1 in the Appendix), and by Lemma 4.3. We note that if n ≥ 3, and if γ has a real analytic extension to a whole neighborhood of 0, then possibly shrinking Ω˜ , the maps U [Ψn []], P [Ψn []] of the variable  ∈]0, Ω˜ [ have a real analytic continuation on ] − Ω˜ , Ω˜ [. In the next Theorem 5.3 we describe the limiting behavior of (u , p ) when  → 0. To do so, we need the following Lemma 5.2 where we introduce the pair (gΩV , gΩP ). Lemma 5.2 can be verified by a standard argument based on the ellipticity properties of the Stokes system. For the sake of completeness we include here a proof. Lemma 5.2. Let Ω be an open bounded subset of Rn of class C m,α such that 0∈ / clΩ. Then there exists a unique pair of functions (gΩV , gΩP ) in the space L1loc (Rn \ Ω, Mn (R)) × L1loc (Rn \ Ω, Rn ) such that ∆gΩV − ∇gΩP = δ0 I and divgΩV = 0 in the sense of distributions in Rn \ clΩ and such that the following conditions hold. 21

V (i) If B is an open bounded subset of Rn \ clΩ with 0 ∈ / clB, then gΩ|B exm,α P tends to a function of C (clB, Mn (R)) and gΩ|B extends to a function of C m−1,α (clB, Rn ). V = 0. (ii) gΩ|∂Ω

(iii) gΩV and gΩP satisfy the decay condition in (5) for R ≡ supx∈Ω |x|. Proof. Let (uΩ , pΩ ) be the unique pair of functions in the product space m,α m−1,α Cloc (Rn \Ω, Mn (R))×Cloc (Rn \Ω, Rn ) which satisfies the decay condition in (5) for R ≡ supx∈Ω |x| and such that ∆uΩ − ∇pΩ = 0 , div uΩ = 0 in Rn \ clΩ ,

u(x) = −SV,n (x) ∀x ∈ ∂Ω.

Then the pair of functions (gΩV , gΩP ) defined by gΩV (x) ≡ uΩ (x) + SV,n (x) , gΩP (x) ≡ pΩ (x) + SP,n (x) ∀x ∈ Rn \ (Ω ∪ {0}) satisfy the conditions in the statement of the Lemma. Moreover, if (fΩV , fΩP ) is another pair of functions of L1loc (Rn \ Ω, Mn (R)) × L1loc (Rn \ Ω, Rn ) which satisfies conditions in the statement of the Lemma, then the pair (hVΩ , hPΩ ) ≡ (gΩV , gΩP ) − (fΩV , fΩP ) satisfies conditions (i), (ii) and (iii) of the Lemma and ∆hVΩ − ∇hPΩ = 0, div hVΩ = 0 in the sense of the distributions in Rn \ clΩ. Hence ∆hPΩ = div(∆hVΩ − ∇hPΩ ) = 0, which implies that hPΩ is real analytic on Rn \ clΩ. Then equation ∆hVΩ − ∇hPΩ = 0 implies that also hVΩ is real analytic on Rn \ clΩ. We deduce that (hVΩ , hPΩ ) extends to a pair of functions m,α m−1,α of Cloc (Rn \ Ω, Mn (R)) × Cloc (Rn \ Ω, Rn ). By the uniqueness of the classical solution of the Dirichlet exterior boundary value problem for the Stokes system with the decay condition in (5) it follows that (hVΩ , hPΩ ) = (0, 0). Hence (fΩV , fΩP ) = (gΩV , gΩP ). We are now ready to state the following Theorem 5.3. Let the notation of Theorem 5.1 hold. Let (gΩV b , gΩPb ) be as in Lemma 5.2 with Ω = Ωb . Let (us , ps ) denote the unique solution in m,α m−1,α Cloc (Rn \ Ωs , Rn ) × Cloc (Rn \ Ωs ) of the problem in (8) which satisfy the decay condition in (5) for R ≡ supx∈Ωs |x|. Let τ s , λs ∈ Rn be defined by Z s τ ≡ T (ps , Dus )νΩs dσ , λs ≡ lim us (x) . |x|→∞

∂Ωs

Then the limits  γ()(log )δ2,n  b u − u = −gΩV b |clΩ˜ (δ2,n λs + (1 − δ2,n )τ s ), (39) ˜ ˜ |clΩ |clΩ →0 n−2  γ()(log )δ2,n  b lim p|clΩ˜ − p|clΩ˜ = −gΩPb |clΩ˜ · (δ2,n λs + (1 − δ2,n )τ s ) n−2 →0 

lim

22

˜ Rn ) × C m−1,α (clΩ). ˜ In particular if hold in C m,α (clΩ, n−2 γ1 ≡ lim →0 γ()(log )δ2,n

exists in R ,

then the limits lim u|clΩ˜ = ub|clΩ˜ − γ1 gΩV b |clΩ˜ (δ2,n λs + (1 − δ2,n )τ s ) , →0

(40)

lim p|clΩ˜ = pb|clΩ˜ − γ1 gΩPb |clΩ˜ · (δ2,n λs + (1 − δ2,n )τ s ) →0

hold in C

m,α

˜ Rn ) × C m−1,α (clΩ). ˜ (clΩ,

Proof. By Propositions 3.2, 4.2 and 4.4 and by equations (28), (30), we deduce that Z E s [0, 0, 1 − δ2,n , γ] dσ = (δ2,n λs + (1 − δ2,n )τ s ) ∂Ωs

(see also Remark 3.3.) Then, by definitions (37), (38), and by Lemma 5.2, and by Proposition 3.2, we have U [0, 0, 1 − δ2,n , γ0 ] = −gΩV b |clΩ˜ (δ2,n λs + (1 − δ2,n )τ s ) , P [0, 0, 1 − δ2,n , γ0 ] = −gΩPb |clΩ˜ · (δ2,n λs + (1 − δ2,n )τ s ) . ˜ Rn )× Then, by the equations in (36) the limits in (39), (40) hold in C m,α (clΩ, ˜ C m−1,α (clΩ). Acknowledgment. The research of this paper has been supported by Centro de Matem´atica da Universidade do Porto, (CMUP)-Portugal, and by Funda¸c˜ao para a Ciˆencia e a Tecnologia, (FCT)-Portugal, with the research grant SFRH/BPD/64437/2009.

6

Appendix: classical potential theoretic results for the Stokes system

We collect here some known results of Potential Theory for the Stokes system (cf. e.g. Ladyzhenskaya [25, Chap. 3], Kohr and Pop [19, Chap. 3], Hsiao and Wendland [17, Chap. 2].) We shall use the following notation. Let Ω be an open bounded subset of Rn of class C 1,α . Let µ ∈ C 0,α (∂Ω, Rn ). Then we set Z vV [µ](x) ≡ SV,n (x − y)µ(y) dσy ∀ x ∈ Rn , Z∂Ω vP [µ](x) ≡ µt (y)SP,n (x − y) dσy ∀ x ∈ Rn \ ∂Ω . ∂Ω

23

e be open bounded subsets of Rn with Ω of class C m,α Theorem 6.1. Let Ω, Ω e Then the following statements hold. and clΩ ⊂ Ω. (i) If µ ∈ C 0,α (∂Ω, Rn ), then vV [µ] ∈ C 0 (Rn , Rn ) and vP [µ]|Ω admits a unique continuous extension to clΩ, which we denote by vP+ [µ], and vP [µ]|Ω− admits a unique continuous extension to clΩ− , which we denote by vP− [µ]. The maps which take µ ∈ C m−1,α (∂Ω, Rn ) to vV [µ]|clΩ ∈ e \ Ω, Rn ), and to v + [µ] ∈ C m,α (clΩ, Rn ), and to vV [µ]|clΩ\Ω ∈ C m,α (clΩ e P − m−1,α m−1,α e C (clΩ), and to v [µ] e ∈ C (clΩ \ Ω) are continuous. P

|clΩ\Ω

(ii) If µ ∈ C 1,α (∂Ω, Rn ), then the functions wV [µ]|Ω and wP [µ]|Ω admit a unique continuous extension to clΩ, which we denote by wV+ [µ] and by wP+ [µ], respectively. The functions wV [µ]|Ω− and wP [µ]|Ω− admit a unique continuous extension to clΩ− , which we denote by wV− [µ] and by wP− [µ], respectively. The maps which take µ ∈ C m,α (∂Ω, Rn ) to e \ Ω, Rn ), and wV+ [µ] ∈ C m,α (clΩ, Rn ), and to wV− [µ]|clΩ\Ω ∈ C m,α (clΩ e e \ Ω) are to wP+ [µ] ∈ C m−1,α (clΩ), and to wP− [µ]|clΩ\Ω ∈ C m−1,α (clΩ e continuous. Proof. We first prove statement (i). Let Sn and Bn be the functions from Rn \ {0} to R defined by  1 log |ξ| if n = 2 , sn Sn (ξ) ≡ 1 2−n |ξ| if n ≥ 3 , (2−n)sn and  Bn (ξ) ≡

(−1)(n−2)/2 (4sn )−1 |ξ|4−n log |ξ| [2(n − 2)(n − 4)sn ]−1 |ξ|4−n

if n ∈ {2, 4} , if n ∈ N \ {0, 1, 2, 4} ,

for all ξ ∈ Rn \ {0}. As is well known, Sn is a fundamental solution of the Laplace operator ∆ and Bn is a fundamental solution of the biharmonic operator ∆2 . If φ ∈ Lp (∂Ω) with p ∈]1, +∞[, then we denote by vSn [φ] and vBn [φ] the single layer potential corresponding to the kernels Sn and Bn , respectively, of moment φ. If µ ∈ Lp (∂Ω, Rn ) with p ∈]1, +∞[, then a straightforward computation shows that vV [µ](x) = (∆ − ∇div) (vBn [µi ](x))i=1,...,n vP [µ](x) = −div (vSn [µi ](x))i=1,...,n

∀ x ∈ Rn , (41) n ∀ x ∈ R \ ∂Ω. (42)

Now, statement (i) follows by the classical regularity results for the single layer potential of an elliptic operator (see Miranda [35, Thm. 5.1].) 24

We now show the validity of statement (ii). Let Mij,x denote the tangential differential operator defined by Mij,x ≡ νΩ,i (x)

∂ ∂ − νΩ,j (x) ∂xj ∂xi

for all x ∈ ∂Ω and i, j ∈ {1, . . . , n}. Then we denote by Mx the matrix (Mi,j,x )i,j=1,...,n . By an elementary computation, we can verify that i i T (SP,n (x − y), DSV,n (x − y))νΩ (y)


j

∂ = −δij Sn (x − y) + Mij,y Sn (x − y) + 2 (My SV,n (x − y))ji ∂νΩ (y) for all x, y ∈ ∂Ω, x 6= y and i, j ∈ {1, . . . , n}. By the Divergence Theorem, we have Z Z (My Sn (x − y))µ(y) dσy = − Sn (x − y)(My µ(y)) dσy , (43) Z ∂Ω Z ∂Ω SV,n (x − y)(My µ(y)) dσy , (44) (My SV,n (x − y))t µ(y) dσy = ∂Ω

∂Ω n

for all x ∈ R (see also Kupradze et al. [24, Chap. V, §1].) Hence, we deduce that  Z  ∂ wV [µ](x) = Sn (x − y) µ(y) dσy (45) ∂νΩ (y) ∂Ω Z
+ Sn (x − y) My µ(y) dσy ∂Ω Z
SV,n (x − y) My µ(y) dσy ∀x ∈ Rn . −2 ∂Ω

Moreover, we clearly have Z µt (y)Sn (x − y) νΩ (y) dσy = 0 ∆

∀x ∈ Rn \ ∂Ω.

(46)

∂Ω

By exploiting (42), (43) and (46) we can show that Z  wP [µ](x) = 2 div (DSn (x − y) · µ(y)) νΩ (y) dσy Z ∂Ω −2∆ µt (y)Sn (x − y) νΩ (y) dσy ∂Ω Z
= −2 div My Sn (x − y) µ(y) dσy Z ∂Ω
= 2 div Sn (x − y) My µ(y) dσy ∀ x ∈ Rn \∂Ω. ∂Ω

25

(47)

Now statement (ii) follows by equations (45) and (47), and by statement (i), and by known properties of regularity of single and double layer potentials corresponding to the fundamental solution Sn of the Laplace operator (cf. e.g. Miranda [35, Thm. 5.I].) We also need the following classical result. Proposition 6.2. Let Ω be an open and bounded subset of Rn of class C 1,α . Let µ ∈ C 1,α (∂Ω, Rn ). Then T (wP+ [µ], DwV+ [µ])νΩ = T (wP− [µ], DwV− [µ])νΩ

on ∂Ω .

Proof. Let Bn be the function from Rn \ {0} to R introduced in the proof of Theorem 6.1. By a straightforward computation we verify that SV,n (ξ) = (∆ − ∇div)(Bn (ξ)I) for all ξ ∈ Rn \ {0}. Then, by equations (45) and (47) and by exploiting equalities (43) and (46) we can show that {T (wP [µ](x), DwV [µ](x))νΩ (ξ)}i Z n  X = −(2νΩ,i (ξ)Dj + νΩ,j (ξ)Di )


Sn (x − y) Mjk,y µk (y) dσy

∂Ω

j,k=1

Z +νΩ,j (ξ)Dk Z∂Ω +νΩ,j (ξ)Dk Z ∂Ω −νΩ,j (ξ)Dj


Sn (x − y) Mik,y µj (y) dσy
Sn (x − y) Mjk,y µi (y) dσy
Sn (x − y) Mik,y µk (y) dσy

∂Ω

+4νΩ,j (ξ)Di Dj

n X

Z Dh


Bn (x − y) Mhk,y µk (y) dσy



∂Ω

h=1

for all x ∈ Rn \ ∂Ω, ξ ∈ ∂Ω. Then, by the known jump properties of the single layer potentials corresponding to an elliptic partial differential operator with constant coefficients, we deduce the validity of the proposition (cf. e.g. Cialdea [4, Theorem 3].) We also note that if α, m and Ω are as in Theorem 6.1, then we can write the Green formula in the form  u(x) if x ∈ Ω , wV [u|∂Ω ](x) − vV [T (p, Du)|∂Ω νΩ ](x) = (48) 0 if x ∈ Ω− ,  p(x) if x ∈ Ω , wP [u|∂Ω ](x) − vP [T (p, Du)|∂Ω νΩ ](x) = 0 if x ∈ Ω− ,

26

for all u ∈ C 1,α (clΩ, Rn ), p ∈ C 0,α (clΩ) which solve ∆u − ∇p = 0, div u = 0 in Ω (cf. e.g. Varnhorn [44, pp.16–17].) We now introduce the following notation. Let Ω be an open bounded subset of Rn of class C m,α . We set
i i Kji (x, y) ≡ T SP,n (x − y), Dx SV,n (x − y) (δj,1 νΩ (y) + δj,2 νΩ (x)) for all x, y ∈ ∂Ω, x 6= y, i ∈ {1, . . . , n} and j ∈ {1, 2}. Then we denote by WV [µ](x) the function of x ∈ ∂Ω defined by Z  t i WV [µ](x) ≡ − µ(y) K1 (x, y) dσy ∀ x ∈ ∂Ω , (49) ∂Ω

i=1,...,n

for all µ ∈ C m,α (∂Ω, Rn ) and we denote by WV∗ [µ](x) the function of x ∈ ∂Ω defined by WV∗ [µ](x)

Z ≡

n X

K2j (x, y)µj (y) dσy

∀ x ∈ ∂Ω

(50)

∂Ω j=1

for all µ ∈ C m−1,α (∂Ω, Rn ). By the standard jump properties for wV [µ] and T (vP [µ], DvV [µ])νΩ and by Theorem 6.1 we deduce that the map from C m,α (∂Ω, Rn ) to itself which takes µ to WV [µ] is continuous and that the map from C m−1,α (∂Ω, Rn ) to itself which takes µ to WV∗ [µ] is continuous. Then we have the following classical result. Theorem 6.3. Let Ω an open bounded subset of Rn of class C 1,α . The operators − 21 I + WV from C 1,α (∂Ω, Rn ) to itself and − 21 I + WV∗ from C 0,α (∂Ω, Rn ) to itself are Fredholm of index 0 and are adjoint with respect to the standard bilinear product of L2 (∂Ω, Rn ). Proof. By a straightforward computation we verify that there exists a positive constant CΩ such that i Kj (x, y) ≤ CΩ |x − y|α+1−n , (51) i i 0 0 α 0 1−n Kj (x, y) − Kj (x , y) ≤ CΩ |x − x | inf{|x − y|, |x − y|} , i α−n D∂Ω,x K1 (x, y) ≤ CΩ |x − y| , i i 0 0 D∂Ω,x K1 (x, y) − D∂Ω,x0 K1 (x , y) ≤ CΩ |x − x| inf{|x0 − y|, |x − y|}α−1−n for all x, x0 , y ∈ ∂Ω, x 6= y, x0 6= y, i ∈ {1, . . . , n}, j ∈ {1, 2}, where D∂Ω,x denotes the tangential operator defined by D∂Ω,x ≡ Dx − νΩ (x) νΩ (x) · Dx for all x ∈ ∂Ω (cf. G¨ unter [13, Chapt. 1].) Let β ∈]0, α[. By a classical argument we deduce that WV [µ] ∈ C 1,α (∂Ω, Rn ) for all µ ∈ C 1,β (∂Ω, Rn ) 27

and WV∗ [µ] ∈ C 0,α (∂Ω, Rn ) for all µ ∈ C 0,β (∂Ω, Rn ) (cf. Schauder [40, Hilfssatz VII, XI] and [41, Hilfssatz XII], where the double layer potential associated to a fundamental solution of the Laplace operator in R3 and the corresponding adjoint operator are considered, but the proof for WV and WV∗ is similar.) Then, by the compactness of the embedding of C 1,α (∂Ω, Rn ) in C 1,β (∂Ω, Rn ) and of C 0,α (∂Ω, Rn ) in C 0,β (∂Ω, Rn ) it follows that the operators WV of C 1,α (∂Ω, Rn ) to itself and WV∗ of C 0,α (∂Ω, Rn ) to itself are compact. Since a compact perturbation of an isomorphism is a Fredholm operator of index 0, − 21 I + WV and − 21 I + WV∗ are Fredholm of index 0. Finally, by the first inequality in (51), and by the known properties of weakly singular integral operators, and by a straightforward computation we verify that − 12 I + WV and − 12 I + WV∗ are adjoint one to the other. We now turn to describe the null spaces of − 21 I + WV and of − 12 I + WV∗ in a fashion which generalizes that of Folland [11, Ch. 3] for the potentials associated to a fundamental solution of the Laplace operator. To do so, we find convenient to denote by RnΩ the set of functions from Ω to Rn which are constant, and by (RnΩ )|∂Ω the set of functions on ∂Ω which are trace on ∂Ω of functions of RnΩ . Also, if X is a vector subspace of L1 (∂Ω, Rn ) with Ω of class C 1 , we set   Z X0 ≡

f ∈X :

f dσ = 0

.

∂Ω

Moreover we denote by Ker(− 12 I + WV ) and Ker(− 21 I + WV∗ ) the null spaces of − 21 I + WV in C 1,α (∂Ω, Rn ) and of − 12 I + WV∗ in C 0,α (∂Ω, Rn ), respectively. Then we have the following Theorem 6.4. Theorem 6.4. Let Ω be an open bounded subset of Rn of class C 1,α . Then the following statements hold. (i) vV [µ]|∂Ω ∈ Ker(− 21 I + WV ) for all µ ∈ Ker(− 12 I + WV∗ ).
(ii) The map from Ker(− 21 I + WV∗ ) 0 to Ker(− 21 I + WV ) which takes µ to vV [µ]|∂Ω is injective. (iii) If n ≥ 3, then the map from Ker(− 21 I + WV∗ ) to Ker(− 21 I + WV ) which takes µ to vV [µ]|∂Ω is an isomorphism.
(iv) Ker(− 21 I + WV ) is the direct sum of vV [ Ker(− 21 I + WV∗ ) 0 ]|∂Ω and of (RnΩ )|∂Ω . Such a sum however, is not necessarily orthogonal. (v) Ker(− 21 I + WV ) = (RΩ,loc )|∂Ω (see right below (14) for a definition.)

28

Proof. Let µ ∈ Ker(− 21 I + WV∗ ). Theorem 6.1 (i) implies that vV+ [µ] ∈ C 1,α (clΩ, Rn ). Then by the Green formula in (48) applied to the function vV [µ], we have wV [vV [µ]|∂Ω ](x) − vV [T (vP+ [µ]|∂Ω , DvV+ [µ]|∂Ω )νΩ ](x) = 0 , for all x ∈ Rn \ clΩ (cf. (48).) By standard jump properties for Stokes single layer potentials, we have 1 wV− [vV [µ]|∂Ω ](x) = vV− [− µ + WV∗ [µ]](x) = vV− [0](x) = 0 , 2 for all x ∈ ∂Ω, i.e., wV− [vV [µ]|∂Ω ] vanishes on the boundary of Rn \ clΩ. Then standard jump properties of Stokes double layer potentials imply that statement (i) holds. We now prove statement (ii). Let µ ∈ C 0,α (∂Ω, Rn ) be such that − 12 µ + WV∗ [µ] = T (vP+ [µ], DvV+ [µ])νΩ = 0 on ∂Ω and vV [µ]|∂Ω = 0. Since the pair (vV− [µ], vP− [µ]) solves the homogeneous Dirichlet problem for the Stokes system in Ω− , the known layer potentials R properties of decay at infinity for the single − and condition ∂Ω µ dσ = 0 in case n = 2, imply that (vV [µ], vP− [µ]) = (0, 0). Hence T (vP− [µ], DvV− [µ])νΩ = 0 and by standard jump properties of Stokes single layer potentials, we deduce that µ = T (vP− [µ], DvV− [µ])νΩ − T (vP+ [µ], DvV+ [µ])νΩ = 0

on ∂Ω ,

which Rimplies the validity of statement (ii). Note that here we exploit condition ∂Ω µ dσ = 0 only in case n = 2. We now prove statement (iii). By Theorem 6.3, we know that the null spaces in statement (iii) are of equal finite dimension. Thus it suffices to show that the map which takes µ to vV [µ]|∂Ω induces an injection, a fact which follows by the proof of statement (ii). We now prove statement (iv). We first prove that if µ ∈ (Ker(− 21 I+WV∗ ))0 and vV [µ]|∂Ω = ρ ∈ (RnΩ )|∂Ω , then µ = 0. By standard jump relations for Stokes simple layer potentials, we have µ = T (vP− [µ]|∂Ω , DvV− [µ]|∂Ω )νΩ .

(52)

By the behaviour at infinity of vV− [µ], vP− [µ] and by the Divergence Theorem applied to the exterior of Ω, we conclude that Z
1 tr (DvV− [µ] + Dt vV− [µ]) · (DvV− [µ] + Dt vV− [µ]) dx 2 Ω− Z Z − − − t t µ dσ = 0 . =− vV [µ] T (vP [µ], DvV [µ])νΩ dσ = −ρ ∂Ω

∂Ω

29

Thus Dv − [µ] + Dt v − [µ] = 0 and vV− [µ] belongs to RRn \clΩ,loc and, by the decay properties of Stokes single layer potentials, vP− [µ] = 0. We conclude that T (vP− [µ], DvV− [µ]) = 0 in Ω− , and that accordingly µ = 0 by (52). Next we note that (RnΩ )|∂Ω is contained in Ker(− 21 I + WV ). Indeed, if ρ ∈ (RnΩ )|∂Ω , then the Green formula in (48) implies that wV [ρ](x) = 0 for all x ∈ Ω− , and thus that wV− [ρ](x) = 0 for all x ∈ ∂Ω, and accordingly ρ ∈ Ker(− 12 I + WV ). By Theorem 6.3, we have 1 1 dim Ker(− I + WV∗ ) = dim Ker(− I + WV ) . 2 2 Clearly, we have Ker(− 21 I + WV∗ ) ≤ n. dim Ker(− 12 I + WV∗ )0 Hence, statement (ii) implies that dim

Ker(− 12 I + WV ) ≤ n. vV [(Ker(− 12 I + WV∗ ))0 ]

Since the dimension of (RnΩ )|∂Ω is n, and 1 (RnΩ )|∂Ω ∩ vV [(Ker(− I + WV∗ ))0 ] = {0} , 2 we conclude that statement (iv) holds. We now prove statement (v). If ρ ∈ Ker(− 21 I + WV ), then (iv) implies that ρ is the sum of an element b of (RnΩ )|∂Ω and of a simple layer vV [µ]|∂Ω with µ in (Ker(− 21 I + WV∗ ))0 . Clearly, T (vV+ [µ], DvV+ [µ])νΩ = 0 on ∂Ω, and thus the Divergence Theorem imply that Z
1 tr (DvV+ [µ] + Dt vV+ [µ]) · (DvV+ [µ] + Dt vV+ [µ]) dx = 0 2 Ω which in turn implies that DvV+ [µ] + Dt vV+ [µ] = 0 and vV+ [µ]|∂Ω ∈ (RΩ,loc )|∂Ω . Hence, ρ = b + v + [µ]|∂Ω ∈ (RΩ,loc )|∂Ω . Conversely, let ψ ∈ RΩ,loc . Let ρ be the trace of ψ on ∂Ω. Then we clearly have ∆ψ − ∇p = 0 in Ω and T (p, Dψ)νΩ = 0 on ∂Ω with p = 0. Then by the Green representation formula, we have ψ(x) = wV [ρ](x) for all x ∈ Ω, and thus ρ = wV+ [ρ] on ∂Ω. Hence, wV− [ρ] = −ρ+wV+ [ρ] = 0 on ∂Ω. The proof of (v) is now complete. Finally, we have the following. Theorem 6.5. Let Ω be an open bounded subset of Rn of class C m,α . Let µ ∈ C 1,α (∂Ω, Rn ). If − 21 µ + WV [µ] ∈ C m,α (∂Ω, Rn ), then µ ∈ C m,α (∂Ω, Rn ). 30

Proof. Let R > supx∈Ω |x|. By the jump properties of the Stokes double layer potentials, the pair of functions (wV [µ]|Bn (0,R)\Ω , wP [µ]|Bn (0,R)\Ω ) is the solution of a Dirichlet type boundary value problem for the Stokes system in the domain Bn (0, R) \ Ω with boundary data of class C m,α . Then by the classical regularity theory for elliptic systems we deduce that wV [µ]|Ω− m,α extends to a function wV− [µ] of Cloc (clΩ− , Rn ) and wP [µ]|Ω− extends to a m−1,α − function wP [µ] of Cloc (clΩ− ) (cf. e.g. Agmon, Douglis and Niremberg [1, Theorem 9.3].) Thus T (wP− [µ], DwV− [µ])|∂Ω νΩ ∈ C m−1,α (∂Ω, Rn ) and since we have T (wP+ [µ], DwV+ [µ])|∂Ω νΩ = T (wP− [µ], DwV− [µ])|∂Ω νΩ by Proposition 6.2, it follows that T (wP+ [µ], DwV+ [µ])|∂Ω νΩ ∈ C m−1,α (∂Ω, Rn ). Then, by the regularity properties of the solutions of a Neumann type boundary value problem for the Stokes system, wV+ [µ] ∈ C m,α (clΩ, Rn ). Finally, by the equality µ = wV+ [µ] − wV− [µ] we deduce that µ ∈ C m,α (∂Ω, Rn ). Now, by Theorems 6.5 and 6.3 and by a standard argument, one can easily deduce the following. Theorem 6.6. Let Ω be an open bounded subset of Rn of class C m,α . Then − 21 I + WV is a Fredholm operator of index 0 from C m,α (∂Ω, Rn ) to itself.

References [1] S. Agmon, A. Douglis, and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. [2] R. B¨ohme and F. Tomi. Zur Struktur der L¨osungsmenge des Plateauproblems, Math. Z. 133 (1973), 1–29. [3] I-D. Chang and R. Finn. On the solutions of a class of equations occurring in continuum mechanics, with application to the Stokes paradox, Arch. Rational Mech. Anal. 7 (1961), 388–401. [4] A. Cialdea. Completeness theorems for elliptic equations of higher order with constants coefficients, Georgian Math. J. 14 (2007), 81–97. [5] M. Dalla Riva and M. Lanza de Cristoforis. A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functional analytic approach, Analysis (Munich) 30 (2010), 67–92. [6] M. Dalla Riva and M. Lanza de Cristoforis. Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem:

31

a functional analytic approach, Complex Var. Elliptic Equ. 55 (2010), 771–794. [7] M. Dalla Riva and M. Lanza de Cristoforis. Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem. A functional analytic approach, Complex Anal. Oper. Theory 4 (2010), to appear. [8] M. Dalla Riva and M. Lanza de Cristoforis. Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functional analytic approach, Eurasian Mathematical Journal 1 (2010), 31–58. [9] K. Deimling. Nonlinear functional analysis. Springer-Verlag, Berlin, 1985. [10] T. M. Fischer, G. C. Hsiao, W. L. Wendland. Singular perturbations for the exterior three-dimensional slow viscous flow problem, J. Math. Anal. Appl. 110 (1985), 583–603. [11] G. B. Folland. Introduction to partial differential equations. Princeton University Press, Princeton, NJ, 1995. [12] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. [13] N. M. G¨ unter. Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der mathematischen Physik. B. G. Teubner Verlagsgesellschaft, Leipzig, 1957. [14] D. B. Henry. Topics in nonlinear analysis, volume 192 of Trabalho de Matematica. Universidade de Brasilia, Brasilia, 1982. [15] G. C. Hsiao. Integral representations of solutions for two-dimensional viscous flow problems, Integral Equations and Operator Theory 5 (1982), 533–547. [16] G. C. Hsiao and R. C. MacCamy. Solution of boundary value problems by integral equations of the first kind, SIAM Review 15 (1973), 687–705. [17] G. H. Hsiao and W. L. Wendland. Boundary integral equations, volume 164 of Applied Mathematical Sciences. Springer-Verlag, Berlin, 2008.

32

[18] J. Kevorkian and J. D. Cole. Perturbation methods in applied mathematics, volume 34 of Applied Mathematical Sciences. Springer-Verlag, New York, 1981. [19] M. Kohr and I. Pop. Viscous incompressible flow for low Reynolds numbers, volume 16 of Advances in Boundary Elements. WIT Press, Southampton, 2004. [20] M. Kohr, G. P. R. Sekhar and W. L. Wendland. Boundary integral method for Stokes flow past a porous body, Math. Methods Appl. Sci. 31 (2008), 1065–1097. [21] M. Kohr, G. P. R. Sekhar and W. L. Wendland. Boundary integral equations for a three-dimensional Stokes-Brinkman cell model, Math. Models Methods Appl. Sci. 18 (2008), 2055–2085 [22] V. Kozlov, V. Maz’ya, and A. Movchan. Asymptotic analysis of fields in multi-structures. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1999. [23] R. Kress. Linear integral equations, volume 82 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. [24] V. D. Kupradze, T. G. Gegelia, M. O. Bashele˘ıshvili, and T. V. Burchuladze. Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, volume 25 of North-Holland Series in Applied Mathematics and Mechanics. North-Holland Publishing Co., Amsterdam, 1979. [25] O. A. Ladyzhenskaya. The mathematical theory of viscous incompressible flow. Gordon and Breach Science Publishers, New York, 1969. [26] M. Lanza de Cristoforis. Properties and pathologies of the composition and inversion operators in Schauder spaces. Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 15 (1991), 93–109. [27] M. Lanza de Cristoforis. Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole, and relative capacity. In Complex analysis and dynamical systems, volume 364 of Contemp. Math., pages 155–167. Amer. Math. Soc., Providence, RI, 2004. [28] M. Lanza de Cristoforis. A singular perturbation Dirichlet boundary value problem for harmonic functions on a domain with a small hole. In Finite or infinite dimensional complex analysis and applications, pages 205–212. Kyushu University Press, Fukuoka, 2005. 33

[29] M. Lanza de Cristoforis. Asymptotic behaviour of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach, Complex Var. Elliptic Equ. 52 (2007), 945–977. [30] M. Lanza de Cristoforis. Singular perturbation problems in potential theory and applications. In Complex analysis and potential theory, pages 131–139. World Sci. Publ., Hackensack, NJ, 2007. [31] M. Lanza de Cristoforis. Asymptotic behaviour of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Analysis (Munich) 28 (2008), 63–93. [32] M. Lanza de Cristoforis and L. Preciso. On the analyticity of the Cauchy integral in Schauder spaces, J. Integral Equations Appl. 11 (1999), 363– 391. [33] M. Lanza de Cristoforis and L. Rossi. Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density, J. Integral Equations Appl. 16 (2004), 137–174. [34] V. Maz0 ya, S. Nazarov, and B. Plamenevskij. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I, II, volume 112 of Operator Theory: Advances and Applications. Birkh¨auser Verlag, Basel, 2000. [35] C. Miranda. Sulle propriet`a di regolarit`a di certe trasformazioni integrali, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 7 (1965), 303–336. [36] P. Musolino. A functional analytic approach for a singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. In Numerical analysis and applied mathematics, pages 928–931, AIP Conf. Proc. 1281, Amer. Inst. Phys., Melville, NY, 2010. [37] H. Power. The completed double layer boundary integral equation method for two-dimensional Stokes flow, IMA J. Appl. Math. 51 (1993), 123–145. [38] H. Power and G. Miranda. Second kind integral equation formulation of Stokes’ flows past a particle of arbitrary shape, SIAM J. Appl. Math. 47 (1987), 689–698.

34

[39] G. Prodi and A. Ambrosetti. Analisi non lineare. I quaderno. Pisa: Scuola Normale Superiore Pisa, Classe di Scienze, 1973. [40] J. Schauder. Potentialtheoretische Untersuchungen, Math. Z. 33 (1931) 602–640. [41] J. Schauder. Bemerkung zu meiner Arbeit ”Potentialtheoretische Untersuchungen I (Anhang)”, Math. Z. 35 (1932) 536–538. [42] T. Valent. Boundary value problems of finite elasticity, volume 31 of Springer Tracts in Natural Philosophy. Springer-Verlag, New York, 1988. [43] M. Van Dyke. Perturbation methods in fluid mechanics. Applied Mathematics and Mechanics, Vol. 8. Academic Press, New York, 1964. [44] W. Varnhorn. The Stokes equations, volume 76 of Mathematical Research. Akademie-Verlag, Berlin, 1994. Matteo Dalla Riva CMUP, Departamento de Matem´atica, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal.

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