Energy integral of the Stokes flow in a singularly perturbed exterior domain

Dedicated to Professor Semyon Yakubovich on the occasion of his 50th birthday

ENERGY INTEGRAL OF THE STOKES FLOW

IN A SINGULARLY PERTURBED EXTERIOR DOMAIN

Matteo Dalla Riva

Abstract.We consider a pair of domainsΩb andΩs

inRn_{and we assume that the closure of} Ωb

does not intersect the closure ofǫΩsforǫ∈(0, ǫ0). Then for a fixedǫ∈(0, ǫ0)we consider

a boundary value problem inRn_\(Ωb_∪ǫΩs_{) which describes the steady state Stokes flow}

of an incompressible viscous fluid past a body occupying the domainΩb

and past a small impurity occupying the domainǫΩs. The unknown of the problem are the velocity fieldu

and the pressure fieldp, and we impose the value of the velocity field uon the boundary

both of the body and of the impurity. We assume that the boundary velocity on the impurity displays an arbitrarily strong singularity whenǫtends to0. The goal is to understand the

behaviour of the strain energy of(u, p)forǫsmall and positive. The methods developed aim

at representing the limiting behaviour in terms of analytic maps and possibly singular but completely known functions ofǫ, such asǫ−1,logǫ.

Keywords: boundary value problem for the Stokes system, singularly perturbed exterior domain, real analytic continuation in Banach space.

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 and for all

n∈N_{\ {0,1}, α∈(0,1).}

Here N_{denotes the set of natural numbers including 0. Then we fix two sets Ω}b _and

Ωs_{in then-dimensional Euclidean spaceR}n_{. The letter “b” stands for “body” and the}

letter “s” stands for “small impurity”. We assume thatΩb _andΩs_{satisfy the following}

condition:

Ωb _{and Ω}s_{are open bounded connected subsets of R}n _{of (1.1)}

class C1,α with connected exterior, the origin 0 of Rn

belongs to Ωs but not to the closure clΩb of Ωb.

For the definition of the functions and sets of the usual Schauder classC0,α_,C1,α _we

refer for example to Gilbarg and Trudinger [8, §§4.1,6.2] (see also [3, §2]). We note that condition (1.1) implies thatΩb _andΩs_{have no holes and that there exists a real}

numberǫ0such that

ǫ0∈(0,1) and clΩb_∩(ǫclΩs_{) =∅for allǫ∈(0, ǫ0). (1.2)}

Then we denote byΩe_{(ǫ)the exterior domain defined by}

Ωe_(ǫ)≡Rn_{\ {clΩ}b_∪(ǫclΩs_{)}, ∀ǫ∈(0, ǫ0).}

Here the letter “e” stands for “exterior domain”. A simple topological argument shows that Ωe_{(ǫ)is connected, and that R}n_\clΩe_{(ǫ) has exactly the two connected}

com-ponents Ωb _{and ǫΩ}s_{, and that the boundary ∂Ω}e_{(ǫ) of Ω}e_{(ǫ) has exactly the two}

connected components∂Ωb _andǫ∂Ωs_{, for allǫ∈(0, ǫ0).}

Now,

letγ be a function from(0, ǫ0)to(0,+∞) (1.3) such thatγ0≡lim

ǫ→0γ(ǫ) exists in [0,+∞).

Let(fb_{, f}s_)∈C1,α_(∂Ωb_,Rn_)×C1,α_(∂Ωs_,Rn_{). Letǫ∈(0, ǫ0). We consider the}

follow-ing boundary value problem for a pair(u, p)ofC_loc1,α(clΩe_(ǫ),Rn_)×C0,α

loc(clΩe(ǫ),R),



   

   

∆u− ∇p= 0 inΩe_(ǫ),

divu= 0 inΩe_(ǫ),

u=fb _on∂Ωb_,

u(x) =γ(ǫ)−1fs_{(x/ǫ) forx∈ǫ∂Ω}s_.

(1.4)

(For the definition ofC_loc1,αandC_loc0,αwe refer, e.g., to [3, §2].) As is well known problem (1.4) admits a unique solution which satisfies the decay condition

sup

|x|>R n

|x|n−2_|u(x)|,|x|n−1_|Du(x)|,|x|n−1_|p(x)|o_{<+∞ (1.5)}

withR≡sup_x∈{Ωb_∪(ǫΩs_)}|x|. The uniqueness of the solution of (1.4), (1.5) can be

We denote by (uǫ, pǫ) the unique solution in the space Cloc1,α(clΩe(ǫ),Rn)× C_loc0,α(clΩe_{(ǫ),R) of (1.4), (1.5) with R ≡ sup}

x∈{Ωb_∪(ǫΩs_)}|x|. Then we denote by

Eǫ the strain energy integral of the Stokes flow(uǫ, pǫ). Namely, we set

Eǫ≡

1 2

Z

Ωe_(ǫ)

tr

(Duǫ+Dtuǫ)·(Duǫ+Dtuǫ)dx, ∀ǫ∈(0, ǫ0). (1.6)

The aim of this paper is to understand the behaviour of Eǫ when ǫ shrinks to0. If

γ0= 0then the velocity fielduǫon the boundary ofǫΩsdisplays a singular behaviour

asǫ→0due to the presence of the factorγ(ǫ)−1 _{in the right hand side of the fourth} equation in (1.4). As we shall see in our main Theorem 4.1, the rate of convergence ofγ asǫ→0and its limit valueγ0determine the limiting behaviour ofEǫ. In particular, if

γ1≡lim

ǫ→0 ǫn−2

γ(ǫ)2 exists in[0,+∞), (1.7)

then Theorem 4.1 implies that

lim

ǫ→0Eǫ=E

b_+γ1Es_{, (1.8)}

where Eb _{denotes the energy integral of the unique solution (u}b_{, p}b_{) in the space}

Cloc1,α(Rn\Ωb,Rn)×C 0,α

loc(Rn\Ωb,R)of the “unperturbed boundary value problem”



 

 

∆ub_{− ∇p}b_{= 0 inR}n_\clΩb_,

divub_{= 0 inR}n_\clΩb_,

ub_=fb _on∂Ωb_,

(1.9)

which satisfies the decay condition in (1.5) for R≡supx∈Ωb|x|, andEs denotes the

energy integral of the unique solution(us_{, p}s_)∈C1,α

loc(Rn\Ωs,Rn)×C 0,α

loc(Rn\Ωs,R) of the “limiting boundary value problem”



 

 

∆us_{− ∇p}s_{= 0 in R}n_\clΩs_,

divus_{= 0 in R}n_\clΩs_,

us_=fs _on∂Ωs_,

(1.10)

which satisfies the decay condition in (1.5) forR≡sup_x∈Ωs|x|. Namely, we set

Eb≡1 2

Z

Rn_\Ωb

tr

(Dub+Dtub)·(Dub+Dtub)

dx, (1.11)

Es_≡1

2

Z

Rn_\Ωs

tr

(Dus+Dtus)·(Dus+Dtus)

dx. (1.12)

when the velocity on the boundary of the impurity displays a singular behaviour for ǫ→0. Moreover, if n = 2, then the limit in (1.8) differs from Eb _{also for γ0 > 0}

(cf. condition (1.7)). If insteadγ0= 0and

lim

ǫ→0 ǫn−2

γ(ǫ)2 does not exist inR, (1.13)

thenEǫ may display a singular behaviour asǫ→0. In this case we can show that

lim

ǫ→0 γ(ǫ)2

ǫn−2(Eǫ− E

b_{) =E}s_{. (1.14)}

However, our main interest is focused on the description of the behaviour ofEǫ when

ǫ is near0, and not only on the limiting behaviour. Actually, we pose the following question:

What can be said of the function which (1.15) takesǫto Eǫ whenǫis 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 ǫ. It is perhaps difficult to provide a complete list of the contributions. Here, we mention the books of Kozlov, Maz’ya and Movchan [14] and Maz’ya, Nazarov and Plamenewskii [20] 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 of singularly perturbed problems in fluid mechanics we refer to the book of Van Dyke [22]. For the asymptotic analysis of a Navier-Stokes flow with low Reynold number we mention the work of Kevorkian and Cole [11], Hsiao and MacCamy [10], Hsiao [9], Fischer, Hsiao and Wendland [7]. 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 [12,13]. 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 ǫ plus a remainder which is smaller than a positive known function ofǫ. Instead, the approach adopted in this paper aims to express the dependence uponǫ in terms of real analytic maps and in terms of possibly singular but completely known functions of ǫ such as ǫ−1_, logǫ,γ(ǫ)−1_{. In particular, our main Theorem 4.1 shows that forǫsmall and positive} Eǫ can be expressed by means of a real analytic map of four variables defined in a

neighborhood of (0,0,1−δ2,n, γ0) and evaluated at (ǫ, ǫ(logǫ)δ2,n,(logǫ)−δ2,n, γ(ǫ))

The paper is organized as follows. Section 2 is a section of preliminaries, where we introduce some notation and we show the validity of a suitable Green formula for an exterior Stokes flow(u, p)which satisfies the decay condition in (1.5). In Section 3 we collect some results of [2] and [3]. In particular, in Theorem 3.2 we consider the map which takes ǫto(uǫ|clΩM, pǫ|clΩM)forǫsmall and positive. Here ΩM is an open

bounded subset ofRn_{contained inR}n_\Ωb_{such that0∈/ clΩ}

M. The letter “M” stands

for “macroscopic”. Instead in Theorem 3.3 we investigate the “microscopic” behaviour of (uǫ, pǫ)in the proximity ofǫΩs. To do so, we consider the function which takes ǫ

to(uǫ(ǫ·)|clΩm, pǫ(ǫ·)|clΩm)forǫsmall and positive. HereΩmis an open bounded set

contained in Rn_\Ωs _{and the letter “m” stands for ‘microscopic”. In the Section 4,}

we prove our main Theorem 4.1 which answers the question in (1.15). Moreover, Theorem 4.1 implies the validity of the limits in (1.8) and (1.14) under conditions (1.7) and (1.13), respectively (cf. equalities (4.2) and (4.3)).

2. SOME PRELIMINARIES

We denote bySV,n≡(Si,j_V,n)i,j=1,...,n,SP,n≡(SP,ni )i=1,...,nthe functions fromRn\{0}

toMn(R)and to Rn, respectively, defined by

S_V,ni,j(x)≡ 1

2sn(δ2,n+ (2−n))

δ2,nlog|x|+ (1−δ2,n)|x|2−nδi,j−

−(δ2,n+ (2−n))

xixj

|x|n

,

SP,ni (x)≡ −

1 sn

xi

|x|n

for all i, j ∈ {1, . . . , n} and allx∈ Rn_{\ {0}. Here M}

n(R) denotes the space of real

n×n-matrices, and sn denotes the(n−1) dimensional measure of the unit sphere

∂B_{n in}Rn_{, andδ}

i,j is defined byδi,j≡0ifi6=j andδi,j≡1ifi=j, for alli, j∈N.

We also find convenient to set

Sj_V,n≡(S_V,ni,j)i=1,...,n,

which we think of as column vectors for allj∈ {1, . . . , n}. For each scalarρ∈R_and

each matrixA∈Mn(R)we set

T(ρ, A)≡ −ρ I+ (A+At_),

where I denotes the unit matrix in Mn(R)and At denotes the transpose matrix to

boundary ofΩand byνΩ the outward unit normal to∂Ω. We set

wV[µ](x)≡ − 



 Z

∂Ω

µt(y)T(S_P,nj (x−y), DxSV,nj (x−y))νΩ(y)dσy 



 j=1,...,n

,

wP[µ](x)≡ −2 div Z

∂Ω µt_(y)S

P,n(x−y)νΩ(y)dσy

, ∀x∈Rn_,

for allµ∈C1,α_(∂Ω,Rn_).

Then we have the following Lemma 2.1, where we show that the decay condition of a solution (u, p)of problem (1.4), (1.5) can be improved in the case n= 2. In the sequel B_{n denotes the open unit ball in}Rn_{, namely we set B}

n≡ {x∈Rn : |x|<1}.

Lemma 2.1. Let Ω be an open bounded subset ofR2 _{of class C}1,α_{. Let (u, p) be a}

pair of function of C_loc1,α(R2\Ω,R2₎×C_loc0,α(R2\Ω,R_{) such that}

∆u− ∇p= 0and divu= 0 inR2_\clΩ.

Assume that (u, p) satisfies the decay condition in (1.5) for R ≡ supx∈Ω|x|. Then

there exists a vectorλ∈R2 _{such that}

sup

|x|>R

|x| |u(x)−λ|, |x|2|Du(x)|, |x|2|p(x)| <∞.

Proof. LetR≡sup_x∈Ω|x|. LetZ2 denote the skew-symmetric matrix defined by

(Z2)ij ≡δi,1δj,2−δi,2δj,1, ∀i, j∈ {1,2}.

Then there exists µ∈C1,α_(R∂B2,R2_{)such that}

u(x) =wV[µ](x) +

1 4π

Z

R∂B2

µ dσ+Z2·x |x|2

Z

R∂B2

µt_{(y)Z2·y dσ} y,

p(x) =wP[µ](x,) ∀x∈R2\RB2 (cf. Power [21, §2], see also [3, Prop. 3.2]). We setλ≡(4π)−1R

R∂B2µ dσ. Then the validity of Lemma follows by a straightforward calculation.

Now we have the following Lemma 2.2, where we introduce a suitable Green for-mula for an exterior Stokes flow(u, p).

Lemma 2.2. LetΩbe an open bounded subset ofRn _{of classC}1,α _{such thatR}n_\clΩ is connected. Let (u, p) be a pair of function of C_loc1,α(Rn_\Ω,Rn_)×C0,α

loc(Rn\Ω,R)

such that

∆u− ∇p= 0 and divu= 0in Rn_\clΩ.

Assume that(u, p)satisfies the decay condition in (1.5)forR≡supx∈Ω|x|. Then the

following equality holds:

1 2

Z

Rn_\Ω

tr

(Du+Dtu)·(Du+Dtu)

dx=−

Z

∂Ω

Proof. Let ρ > 0 be such that clΩ ⊆ρB_{n. By the Divergence Theorem we deduce}

that

1 2

Z

(ρBn)\Ω

tr

(Du(x) +Dtu(x))·(Du(x) +Dtu(x))

dx=

=−

Z

∂Ω

ut_{(x)T(p(x), Du(x))νΩ(x)dσ} x+

+

Z

ρ∂Bn

ut(x)T(p(x), Du(x))νρBn(x)dσx.

(2.1)

By the decay conditions in (1.5) and by Lemma 2.1 we verify that

lim

ρ→∞

Z

ρ∂Bn

ut_{(x)T(p(x), Du(x))ν}

ρBn(x)dσx= 0.

Hence the validity of the lemma follows by taking the limit asρ→ ∞in equality (2.1) and by a standard argument based on the Monotone Convergence Theorem.

3. REAL ANALYTIC REPRESENTATION THEOREMS FOR THE SOLUTION(uǫ, pǫ)

In Theorem 3.2 we describe the limiting behaviour of(uǫ, pǫ)whenǫ→0. To do so,

we need the following Lemma 3.1 where we introduce the auxiliary pair of functions (gV

Ω, gPΩ). The validity of Lemma 3.1 can be verified by a standard argument based on the ellipticity properties of the Stokes system (see also [3, Lemma 5.2] for a proof).

Lemma 3.1. Let Ωbe an open bounded subset ofRn _{of classC}1,α _{such that0}∈_/ clΩ. Then there exists a unique pair of functions(gV

Ω, gΩP)in the spaceL1loc(Rn\Ω, Mn(R))×

L1

loc(Rn\Ω,Rn)such that ∆gV

Ω− ∇gΩP =δ0I anddivgVΩ = 0in the sense of distributions inRn\clΩ

and such that the following conditions hold:

(i) IfB ⊆Rn_{\clΩis open and bounded and0∈/clB, theng}V

Ω|Bextends to a function ofC1,α_{(clB, M}

n(R))and gPΩ|B extends to a function of C

0,α_(clB,Rn_),

(ii) gV

Ω|∂Ω= 0, (iii) gV

Ω andgΩP satisfy the decay condition in (1.5)forR≡supx∈Ω|x|.

We are now ready to state our Theorem 3.2, where we consider the behaviour of (uǫ, pǫ)whenǫshrinks to0. For a proof we refer to [3, Th. 5.1, 5.3]. To simplify our

notation we introduce the following definition. Let ǫ0 be as in (1.2). Letγ, γ0 be as in (1.3). Then we denote byΨn[·]the function from(0, ǫ0)to R4defined by

We note thatlimǫ→0Ψn[ǫ] = (0,0,1−δ2,n, γ0).

Theorem 3.2. Let Ωb _{and Ω}s _{be as in (1.1). Let ǫ0 be as in (1.2). Let γ, γ0 be} as in (1.3). Let (fb_{, f}s_{) ∈ C}1,α_(∂Ωb_,Rn_)×C1,α_(∂Ωs_,Rn_{). Let (g}V

Ωb, g_ΩPb) be as in Lemma 3.1 withΩ = Ωb_{. Let(u}b_{, p}b_{)denote the unique solution inC}1,α

loc(Rn\Ωb,Rn)× C_loc0,α(Rn _{\ Ω}b_,R_{) of the boundary value problem in (1.9) which satisfies the decay}

condition in (1.5) for R ≡ sup_x∈Ωb|x|. Let (us, ps) denote the unique solution in

C_loc1,α(Rn_\Ωs_,Rn_)×C0,α

loc(Rn\Ωs,R) of the boundary value problem in (1.10) which

satisfy the decay condition in (1.5)forR≡supx∈Ωs|x|. Letτs, λs∈Rn be defined by

τs_≡ Z

∂Ωs

T(ps_{, Du}s_)νΩ

sdσ, λs≡ lim

|x|→∞u

s_(x)

(see also Lemma 2.1). Let ΩM ⊆Rn\clΩb be open bounded and such that 0∈/clΩM. Then, there exist ǫΩM ∈(0, ǫ0), and an open neighborhoodUΩM of(0,1−δ2,n, γ0)in

R3_{, and real analytic maps}

UΩM: (−ǫΩM, ǫΩM)× UΩM →C

1,α_(clΩ M,Rn),

PΩM: (−ǫΩM, ǫΩM)× UΩM →C

0,α_(clΩ M,R) such that

clΩM ∩(ǫclΩs) =∅andΨn[ǫ]∈(−ǫΩM, ǫΩM)× UΩM, ∀ǫ∈(0, ǫΩM) (3.2)

and

uǫ|clΩM =u b

|clΩM +

ǫn−2

γ(ǫ)(logǫ)δ2,nUΩM[Ψn[ǫ]],

pǫ|clΩM =p b

|clΩM +

ǫn−2

γ(ǫ)(logǫ)δ2,nPΩM[Ψn[ǫ]], ∀ǫ∈(0, ǫΩM).

Here (uǫ, pǫ)denotes the unique solution inCloc1,α(clΩ(ǫ),Rn)×C 0,α

loc(clΩ(ǫ),R)of the

boundary value problem in (1.4) which satisfies the decay condition in (1.5)forR≡ supx∈(Ωb_∪(ǫΩs₎₎|x|. Moreover,

UΩM[0,0,1−δ2,n, γ0] =−g V

Ωb_{|cl ˜Ω}·(δ2,nλ

s_{+ (1−δ2} ,n)τs),

PΩM[0,0,1−δ2,n, γ0] =−g P

Ωb_{|cl ˜Ω}·(δ2,nλ

s_{+ (1−δ2} ,n)τs).

(3.3)

In Theorem 3.3 we consider the “microscopic” behaviour of (uǫ, pǫ) near the

im-purityǫΩs_{as ǫ→0. For a proof we refer to [2, Th. 4.1].}

Theorem 3.3. Let Ωb _{and Ω}s _{be as in (1.1). Let ǫ0 be as in (1.2). Let γ, γ0 be as} in (1.3). Let(fb_{, f}s_)∈C1,α_(∂Ωb_,Rn_)×C1,α_(∂Ωs_,Rn_{). Let(u}s_{, p}s_{)denote the unique} solution inC_loc1,α(Rn_\Ωs_,Rn_)×C0,α

loc(Rn\Ωs,R)of the boundary value problem in (1.10)

be open and bounded. Then there exists ǫΩm ∈ (0, ǫ0), and a neighborhood UΩm of

(0,1−δ2,n, γ0)in R3, and real analytic maps

UΩm: (−ǫΩm, ǫΩm)× UΩm →C

1,α_(clΩ m,Rn),

PΩm: (−ǫΩm, ǫΩm)× UΩm →C

0,α_(clΩ m,R) such that

(ǫclΩm)∩clΩb=∅ andΨn[ǫ]∈(−ǫΩm, ǫΩm)× UΩm, ∀ǫ∈(0, ǫΩm) (3.4)

and such that

uǫ(ǫx) =

1

γ(ǫ)UΩm[Ψn[ǫ]](x),

pǫ(ǫx) =

1

ǫγ(ǫ)PΩm[Ψn[ǫ]](x), ∀x∈clΩm, ǫ∈(0, ǫΩm).

Here (uǫ, pǫ)denotes the unique solution inCloc1,α(clΩ(ǫ),Rn)×C 0,α

loc(clΩ(ǫ),R)of the

boundary value problem in (1.4) which satisfies the decay condition in (1.5)forR≡ sup_x∈(Ωb_∪(ǫΩs₎₎|x|. Moreover,

UΩm[0,0,1−δ2,n, γ0] =u s

|clΩm,

PΩm[0,0,1−δ2,n, γ0] =p s

|clΩm.

(3.5)

4. REAL ANALYTIC REPRESENTATION THEOREM FOR THE ENERGY INTEGRALEǫ

We are now ready to prove our main Theorem 4.1.

Theorem 4.1. Let Ωb _{and Ω}s _{be as in (1.1). Let ǫ0 be as in (1.2). Let γ, γ0 be as} in (1.3). Let(fb_{, f}s_)∈C1,α_(∂Ωb_,Rn_)×C1,α_(∂Ωs_,Rn_{). Let(u}b_{, p}b_{)denote the unique} solution inC_loc1,α(Rn_\Ωb_,Rn_)×C0,α

loc(Rn\Ωb,R)of the boundary value problem in(1.9)

which satisfies the decay condition in (1.5)forR≡supx∈Ωb|x|. LetEbbe defined as in

(1.11). Let(us_{, p}s_{)denote the unique solution inC}1,α

loc(Rn\Ωs,Rn)×C 0,α

loc(Rn\Ωs,R)

of the boundary value problem in (1.10)which satisfy the decay condition in (1.5)for

R≡supx∈Ωs|x|. Let Es be defined as in(1.12). Then there existǫ_E∈(0, ǫ0), and an open neighborhoodUE of(0,1−δ2,n, γ0) inR3, and a real analytic map

E : (−ǫE, ǫE)× UE→R,

such that

Ψn[ǫ]∈(−ǫE, ǫE)× UE, ∀ǫ∈(0, ǫE) (4.1) and such that

Eǫ=Eb+

ǫn−2

(see also (3.1)). Moreover,

E[0,0,1−δ2,n, γ0] =Es+ (1−δ2,n)γ0 Z

∂Ωb

fb_T(gP

Ωb·τs, DgV_Ωb·τs)νΩbdσ. (4.3)

Here Eǫ denotes the energy integral of the solution (uǫ, pǫ) ∈ Cloc1,α(clΩ(ǫ),Rn)× C_loc0,α(clΩ(ǫ),R_{) of the boundary value problem in (1.4) which satisfies the decay}

con-dition in (1.5)forR≡sup_x∈(Ωb_∪(ǫΩs₎₎|x| (see definition (1.6)).

Proof. Let ǫ ∈ (0, ǫ0). Let (uǫ, pǫ) denote the solution in Cloc1,α(clΩ(ǫ),Rn) × C_loc0,α(clΩ(ǫ),R_{)of the boundary value problem in (1.4) which satisfies the decay}

con-dition in (1.5) forR≡sup_x∈(Ωb_∪(ǫΩs₎₎|x|. By Lemma 2.2 and by definition (1.6), we

deduce that

Eǫ=− Z

∂Ωb

utǫ(x)T(pǫ(x), Duǫ(x))νΩb(x)dσx−

−

Z

ǫ∂Ωs

ut

ǫ(x)T (pǫ(x), Duǫ(x))νǫΩs(x)dσ_x=

=−

Z

∂Ωb

ut

ǫ(x)T(pǫ(x), Duǫ(x))νΩb(x)dσx−

−ǫn−1 Z

∂Ωs

ut ǫ(ǫx)T

pǫ(ǫx),

1

ǫDxuǫ(ǫx)

νΩs(x)dσ_x.

(4.4)

Now letΩ˜M be an open bounded neighborhood ofclΩb. LetΩ˜mbe an open bounded

neighborhood ofclΩs_{. Let ǫΩ}

M, UΩM, UΩM, PΩM be as in Theorem 3.2 with ΩM ≡

˜

ΩM\clΩb. LetǫΩm,UΩm,UΩm,PΩm be as in Theorem 3.3 withΩm≡Ω˜m\clΩ s_{. Let}

ǫE≡inf{ǫΩM, ǫΩm}. LetUE≡ UΩM∩ UΩm. Then the validity of (4.1) follows by (3.2)

and (3.4). Moreover, if ǫ∈(0, ǫE), then Theorems 3.2, 3.3 and the equality in (4.4)

imply that

Eǫ=− Z

∂Ωb

(ub(x))tT(pb(x), Dub(x))νΩb(x)dσx−

− ǫ

n−2

γ(ǫ)(logǫ)δ2,n Z

∂Ωb

(UΩM[Ψ[ǫ]](x))

t_T(pb_{(x), Du}b_(x))ν

Ωb(x)dσ_x+

+

Z

∂Ωb

(fb(x))tT(PΩM[Ψn[ǫ]](x), DUΩM[Ψn[ǫ]](x))νΩb(x)dσx

−

− ǫ

n−1

γ(ǫ)2

Z

∂Ωs

(fs_(x))t_{T (PΩ}

m[Ψ[ǫ]](x), DUΩm[Ψ[ǫ]](x))νΩs(x)dσx.

Thus it is natural to set

E[e]≡ −ǫ2ǫ3

Z

∂Ωb

(UΩM[e](x))

t_T(pb_{(x), Du}b_(x))νΩ

b(x)dσ_x+

+

Z

∂Ωb

(fb_(x))t_T(PΩ

M[e](x), DUΩM[e](x))νΩb(x)dσx

−

−

Z

∂Ωs

(fs(x))tT (PΩm[e](x), DUΩm[e](x))νΩs(x)dσx

(4.6)

for alle≡(ǫ, ǫ1, ǫ2, ǫ3)∈(−ǫ_E, ǫ_E)× U_E. By the real analyticity ofUΩ

M,PΩM,UΩm

and PΩm and by standard calculus in the Banach space we deduce that E is real

analytic from (−ǫE, ǫE)× UE to R. The validity of (4.2) follows by the definitions

in (3.1) and (4.6), by the equality in (4.5), and by Lemma 2.2 (see also (1.11)). The validity of (4.3) follows by the definition in (3.1) and (4.6), and by the equalities in (3.3) and (3.5), and by the equality

UΩM[0,0,(1−δ2,n), γ0]|∂Ωb=−g V

Ωb_|∂Ωb·(δ2,nλs+ (1−δ2,n)τs) = 0

(cf. Lemma 3.1 (ii)).

We observe that the equalities in (4.2) and (4.3) immediately imply the validity of the limits in (1.8) and (1.14) under conditions (1.7) and (1.13), respectively (note that condition (1.13) implies thatγ0= 0).

Acknowledgements

This work was supported by FEDER funds through COMPETE–Operational Pro-gramme Factors of Competitiveness (“Programa Operacional Factores de Compet-itividade”) and by Portuguese funds through the Center for Research and De-velopment in Mathematics and Applications (University of Aveiro) and the Por-tuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. The research was also supported by the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnolo-gia”)with the research grant SFRH/BPD/64437/2009.

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Matteo Dalla Riva

[email protected]

CIDMA

Center for Research and Development in Mathematics and Applications University of Aveiro

3810-193 Aveiro, Portugal