An impedance effect of a thin adhesive layer in some boundary value and transmission problems governed by elliptic differential equations Текст научной статьи по специальности «Математика»

MSC 34K30, 35C15, 35J05, 35J25, 35Q60

DOI: 10.14529/ mmp150405

AN IMPEDANCE EFFECT OF A THIN ADHESIVE LAYER IN SOME BOUNDARY VALUE AND TRANSMISSION PROBLEMS GOVERNED BY ELLIPTIC DIFFERENTIAL EQUATIONS

A. Favini, University of Bologna, Department of Mathematics, Bologna, Italy, [email protected],

R. Labbas, Laboratory of Mathematics Applied, University of du Havre, Le Havre, France, [email protected],

K. Lemrabet, Laboratory AMNEDP, Faculty of Mathematics, University of Sciences and Technology - Houari Boumediene, Algiers, Algeria, [email protected]

In this paper we consider a problem of two bodies bonded through a thin adhesive layer (a third material) of thickness S. Leting S go to zero, one obtains a boundary value transmission problem set on a fixed domain. We then give new results for the study of this problem in the framework of Holder spaces: an explicit representation of the solution and necessary and sufficient conditions at the interface for its optimal regularity are obtained using the semigroups theory and the real interpolation spaces.

Keywords: boundary value problem of elliptic type; transmission problems; impedance effect; thin layer.

To the memory of Alfredo Lorenzi.

Introduction

Consider the boundary value and transmission second-order operational problem (us)'' (x) + Aus (x) = gs (x) on ]-1,0[ U ]0,ô[ U ]ô, 1 + ô[

U (-1) = fm) (f) ' (1 + = f+ s s s s (1) (PM us (0-) = us (0+) + us (ô-) = us (ô+ ) + 3s W

p- (u) (0-) = po (us)' (0+) + as k po (us)' (ô-)= p+ (us) (ô+) + bs,

set in some complex Banach space E; here A is a closed linear operator of domain D(A) C E (not necessarily dense in E) which verifies the Krein's ellipticity condition (see Section 2, (16)) f— f+, as, 3s, as, bs are given in E and satisfy some necessary and sufficient conditions which will be specified later. The function gs is such that

g- = gs 1-1,0] G Cn ([-1,0]; E) g0 = gsG Cn ([0, ô] ; E) g+ = gs |[S,1+S] G Cn ([ô, 1 + ô]; E ),

(with 0 < n < 1). It is not difficult to prove that the holderianity of g-7 g0 and g+ imply the global hôlderianity of g^n [-1,1 + ô] if and only if

g- (0) = g0 (0)

We do not assume these two conditions. Set

and gt (5) = g+(5).

u_

U\

—1,0['

ut = ut U0 = u|]0,t['

u

+

u

s

|]t,1+t[>

then problem (Pt) writes

(u— )" (x) + Au— (x) = g— (x) on ]-1,0[ Ы0) (x

(EQS) I (u0)" (x) + Au0 (x) = g0 (x) on ]0,5[

I / ..Л/ч Л/ч т,--.

(Pt H

(BC )

(TC )

(u+) (x) + Au+ (x) = g+ (x) on ]5, 1 + 5[

u— (-1) = f—

(u+)' (1 + 5) = f+,

u— (0) = u0 (0) + a0

u0 (5) = u+ (5) + ßs

p— (u)' (0)= p0 (u0)' (0) + a&

P0 Ц) ' (5)= p+ K) ' (5) + bs.

The numerical solution of this problem is usually very difficult to compute. In fact, the small thickness of the thin layer generates difficulties in the meshing. As 8 ^ 0, the interval ]0,8[ degenerates into the point {0} and we can no more have an equation on it. The interval ]8,8 + 1[ becomes ]0,1[.

Therefore, the main question is: what will be the appropriate transmission conditions {0}

]0,8[as 8 ^ 0?

We will answer formally to this question in the most interesting case, characterizing so the effect of the small bond ]0, 8[ (as 8 ^ 0).

Let us begin by giving a formal derivation of the effect of the small bond ]0,8[ (as 8 ^ 0).

In order to deal with our problem ), we solve the scalar equation on the small interval ]0,8[ and write down relations between the Cauchy data

u0(0), (u0)' (0)) and (u0(6), (u0)' (5))

Then making use of the transmission condition at {0} and {$}, we will deduce relations linking

u— (0),p— (u—У (0)) ,

u+(5),p+ (us+)' (5)) ,

to

which allow us, as 8 ^ 0 to obtain the limiting transmission conditions at {0}. Therefore, we will see that the interesting limit problem writes in the form

(u-)'' (x) + Au- (x) = g- (x) on ] —1,0[ (w+)'' (x) + Aw+ (x) = h+ (x) on ]0,1[ (Pi^ u (—1) = f-, (w+)' (1) = f+

u— (0) = w+(0) + ф k P— (u— )' (0) - p+ (w+)' (0) = qAu— (0) + Ф,

see the details in Subsection 1.1.

Our main results concerning problem (P1A) are summarized in the following Theorem.

Theorem 1. Let g- e Cn ([-1, 0]; E), h+ e Cn ([0,1]; E) with 0 <n< 1 an d f- e D (A), f+ G D ((-A)1/2); y G D(A), ^ e E. Assume (16), see below. Then problem, (pia) has a unique solution

f u-(x) :] - 1,0H- E \ w+(x) :]0,E

such that

1. u- G C([-1, 0]; D(A)) n C2([-1, 0]; E), w+ e C([0,1]; D(A)) n C2([0,1]; E) if and

only if

(*s)

{

Af- - g- (-1)eD (A) (-A)1/2f+ e D(A)

(* * s)

qg-(0)+ ^ e D(A) _

h+(0) - g-(0) + Ay e D(A).

2. Au-(.), u'L e Cn ([-1, 0]; E), Aw+(.), w+ e Cn ([0,1]; E) if and only if

i Af- - g- (-1) e Da (n/2, +») ( (-A)1/2f+ e Da (n/2, +»)

(* * r)

qg-(0)+ ^ e Da (n/2, h+(0) - g-(0) + Ay e Da (n/2,

In this main result, note that (*s^d (* * s) are respectively the necessary and sufficient compatibility conditions at the boundary and the necessary and sufficient compatibility conditions at the interface {0} to obtain a strict solution u = (u_,w+). Similarly, (*r) and (* * r) are those to obtain optimal regularities on u.

The definition and the properties of the interpolation space DA (n/2, are given, for instance, in ( [1])

Many authors have worked on analogous problems, see [2 4] in hilbertian spaces. In [5,6], a study is given for a similar problem respectively in the framework of Holder spaces and Lp-spaces. These two last studies have considered only two materials. In our work we will use some techniques of these approachs which are based on the theory of semigroups, the Dunford functional calculus and the interpolation spaces.

This paper is organized as follows.

In Section 1, one gives the formal calculus for the limiting transmission problem and a concrete problem which motivates our study. In section 2, we give the basic hypothesis and some technical lemmas useful to the study of our problem (P1A). Section 3 is devoted to the derivation of an explicit representation of the solution of (P1A). In Sections 4 and 5 we study the solution and give in addition necessary and sufficient compatibility conditions on the data in order to obtain the above Theorem. In a last Section 6, we go back to the main physical example given in Subsection 1.2 and apply our results in the case of the space E = C(Q) of the ^-Holder continuous functions vanishing on the boundary dQ.

1. Formal Derivation of the Limiting Effect of the Thin Junction

1.1. Derivation of the Transmission Conditions

In order to have an idea at least formally of the limiting problem, let us first consider the case when operator — A is replaced % a complex scalar — z (with z E C\R+) and for the simplicity

{

9\]0_[

0

a_ = ß_ = a_ = b_ = 0.

Define functions w+ and h+ on the fixed intervall ]0,1[ by

w+(x) = u+(8 + x), h+(x) = g+(8 + x).

For simplicity, we have supposed that these functions do not depend on 8. The equation on the intervall ]0, 8[ writes

U) " + zU0 = 0,

which gives

Mo(x) = C\e~

+ C2e

-^/-z(S-x)

where ^ and C2 are constants to be fixed by the boundary conditions. We thus have

u0(O) = Ci + C2e-^zS u0(8) = Cie-^s + C2 (u0)' (0) = —Ci^—Z + C2^—~ze-^s (ug)' (8) = —C\^—Ze-^s + ^—ZC2.

Now, the transmission conditions

u- (0) = u0 (0)

us0 (8) = u+ (8) = w+ (0)

p- (u-)' (0)= po(u0)' (0)

Po (u0) ' (8) = p+ (u+) ' (8) = p+ (w+)' (0)

lead to

2e'

-V^s^-Zw+(0) = (l + e-2V-zs) V-Zu-(0) + (l - e-2V-zs) PP- {u-) (0)

and

p-Po

(2)

2P+ e-V-Zs (w+)' (0) = (l - e-2V-Zs) ^-Zus_(0) + p- (l + e-2V-Zs) (us_)' (0). (3) po - po -

These two last relations link the Cauchy data (w+(0), (w+)' (0)) (at the interface {0}) of

the function w+ (defined on [0,1]) to the Cauchy data ^u-(0), (u(0)j (at the interface

{0}) of the function u— (defined on [—1, 0]).

The limiting transmission conditions as 8 ^ 0, which are obtained from the analysis of (2) and (3) depend on the behavior of p—1 p0 and p+ with respect to 8. So we must

assume some conditions on p-, rpo Mid p+. The most interesting case we will consider is the following

q

p- and p+ are independent of ^d p0 = -, (4)

o

q

This problem may model an electrostatic potential u5 in an heterogeneous material (see

next subsection). The heterogeneity of the material is translated by the discontinuity of the

p

thus the continuity of the potential through the sheet, but the normal component of the electric induction field is no longer continuous through the interface {0}, it has a jump proportional to the potential at {0}. In that case one has

(l _ e-2^-5) 1 = (1 _ e-2^5) ° - o as 0 - 0, V / p0 v / q

p0 (l _ e-2^ = 0 (l _ e-2^-5) - 2q/_Z as 0 - 0, and then the limiting transmission conditions are

!

w+(0) = u-(0)

p+ (w+)' (0) = q/_z/_Zu-(0) + p- (u-)' (0),

and the limiting scalar problem becomes

(u-)'' (x) + zu- (x) = g- (x) on ] — 1, 0[ (w+) (x) + zw+ (x) = h+ (x) on ]0,1[ (PuW u (_1) = f-, (w+)'(1) = f+ (5)

u-(0) = w+ (0)

p- (u-)' (0) _ p+ (w+)'(0) = qzu-(0).

p- p+

0 tat p0 = 5q. In that case the small bond is not sufficiently conductive, implying thus

{0}

{0}

{0}

(1 _ e-2^5) - = (1 _ e-2^5) 1 - 21 /_Z as0 - 0

p0 q0 q

(1

po ^ " J q00 ' ^q

po (1 _ e-2^-5^ = q0 (1 _ e— 0 as 0 — 0

and then the transmission conditions are

{

qw+(0) = qu-(0) + p+ (u-)' (0) p+ (w+)'(0) = p- (u-)'(0)-

The limiting problem becomes

(Pa)

(u-) (x) + Au- (x) = g- (x) on ]_1, 0[ (w+)''(x) + Aw+ (x) = h+ (x) on ]0,1[ u (_1) = f-, (w+)'(1) = f+, p- (u-)' (0) = q (u-(0) _ w+(0)) = p+ (w+)' (0)).

In the general case where the data g^06[, as, ß6, a^d Ъ6 are different from 0, one

obtains the following relations between (w+(0), (w+)' ^u_(0), (u(0)

2e"

(l + e/-Zu6_(0) + (l - ep- (и-)' (0) + ф ^g[j0>6[,a6, ß6,a6)

(6)

and

2e_^_6 (w+)' (0)

4 +' w Po

(l - \f—zuU_(0) + (l + (u_y (0)p— + ф6 (g6 0>6[,a6,a6,Ъ6^

(7)

where

Ф6 (g(o,6[' a6,ß6V) =

= - (l - f06 e_^sg0 (s) ds - (l - e_2^_z6^j

'-za° + ^ -

po I

ze

-J^z6 06

and

Ф6 (g6]o,6[,a6=

= (l - f06 e_^_zsg60 (s) ds + (l - e_2^_z6^j (\

-Za6 + ^ - ^.

po po

When we assume (4), we obtain the following limiting scalar problem with non homogeneous transmission conditions

(u_)'' (x) + zu_ (x) = g_ (x) on ] — l, 0[ (w+) (x) + zw+ (x) = h+ (x) on ]0, l[

(PlzЦ u (-l) = f_, (w+)' (l) = f+

u_(0) = w+ (0) + ip k p_ (u_)' (0) - p+ (w+)' (0) = qzu_ (0) + ф.

(8)

where

V = lim/ (gjja5,a5) , ^ = lim^5 g a5,a5,b5,

Therefore, in this work we will focus ourselves on the complete analysis of the following problem

(u-)'' (x) + Au- (x) = g- (x) on ] —1, 0[ (w+)'' (x) + Aw+ (x) = h+ (x) on ]0,1[ u (—1) = f-, (w+)' (1) = f+ (9)

u- (0) = w+(0) + V

p- (u-)' (0) — p+ (w+)' (0) = qAu-(0) + ^.

(Pia) <

1.2. Electrostatic Potential in a Heterogeneous Cylinder

Consider the cylinder G5 = ] —1,1 + 8[ x Q constituted by the junction of two homogeneous cylinders G- = ] —1, 0[ x Q and G+ = ] 8,1 + 8[ x Q bonded together by the thin cylinder G0 = ]0, 8[ x Q (here Q is a bounded domain of Rra, n ^ 1, with a regular boundary r). Denote by (x,y) the generic variable in G5.

The transmission problem

V. (pVu5) = pg5 in G5 u5 = 0 on ]-1,1 + 5[ x r 5 = f— on {-1} x Q

u

(10)

p

d_u5 dx

f+ on {1 + 5} x Q,

models an electrostatic problem in G5. The function u5 is the electrostatic potential, _Vu5

is the electric field and _pVu5 is the electric induction field. The heterogeneity of the

p

( p- in ]_1,0[ x Q p = < p0 in ]0,0[ x Q

[ p+ in ]0,1 + 0[ x Q.

p- p0 p+ g f-

f+

Set

u

u

—1,0['

g—

U0 = ^^ |]0,5 [, U+ = Uf]5,1+5['

gl]—1,0[> g0 = g|]0,5[' g+ = g!]S, 1+5[;

then, the equation

V. (pVu5) = pg5 in G5, is equivalent to the following equations

Au- = g- in ]_1,0[ x Q Au0 = g0 in ]0,0[ x Q Au+ = g+ in ]0,1 + 0[ x Q,

(H)

with the transmission conditions

' u

U

5 = u0 on {0} x Q

0

u+ on {5} x Q

p— p0

du— du0

-= on {0} x Q

dx

du+

-8X-p+^0n {5}x Q-

dx s 0

(12)

du5

w+ h+ ]0, 1[ x Q

w+ (x,y) = u+(0 + x,v), h+(x,y) = g+(0 + x,v),

0

du0

The approximations for u0(x,.) and-7—0 (x, .) as 0 — 0, give

dx

du5 02 d2u5

u0(0,y) * u0(0,y) + 5Cdx(0,y) + du(0,y) =

du5

52

u

(0,y) + 5^X0(0,y) - t (A„u0 + g„) (0,y)

^(5,у) - (».у) + = IX<°'У>"s " »)

Using these relations with (11) and (12), we get the following problem set on the fixed domain (not depending on 5):

(eqs){AW;==g-X]—1№ <i3>

under the boundary conditions

f u- = 0 on ]-1, 0[ x r w+ = 0 on ]0,1[ x r (BCW u- = f- on { —1} x Q (14)

dw+ mo

p+~x = f+on {1} x Q'

and the following transmission conditions (depending on 5)

Qu_ 52

w+ = u- + 5-----— (Ayuo — £o) on {0} x Q

(TC) J Po ox 2

dw+ ди_ /л ч г^т ^

p+1x = - у и- - g°) on {^ x S

]0' 5[ x Q

]0' 5[ x Q {0} x Q

through these transmission conditions.

There are two limiting cases of particular interest. The first case is

q

Po = 5'

(q is a positive constant) which is considered in this work assumes that the thin layer is

50

w+ = и- on {0} x П

dw + ди_ ~ ~ p+-7— = p--q (Дуи- + g0) о» {0} x П

dw+ = On. _ (Д г , , _ _ (15)

dx P- dx q У

which corresponds to the fact that the potential is continuous through the sheet, but the normal component of the electric induction field has a jump proportional to the potential.

The second case is p0 = q5 and corresponds to the fact that the thin layer is poorly conductive. We get, as 5 goes tо 0:

1 ди_ w+ = и_ +—P-^— оn {0} x q dx dw+ ди_

P+_dX = P-1X ОП {0} x S'

here, the normal component of the electric induction is continuous through the sheet but the potential has a jump proportional to the normal componant of the electric induction field.

Therefore, using the classical operational notations

u-(x)(y) := u-(x,y), w+ (x)(y) := w+(x,y),--

the concrete problem (13), (14), (15) writes exactly in the form (9) with

0 = qg0 and v = 0

and

{

D (A) = {v G W2'p (0,1) : v (0) = v (1) = 0} Av (y) = v" (y)

in the case E = Lp (0,1), or

{

D (A) = {v G C2[0,1] : v (0) = v (1) = 0} Av (y) = v" (y)

in the case E = C [0,1].

2. Hypotheses and Technical Lemmas

We assume in all this work the following ellipticity hypothesis: for any 9 E ]0, n[ p (A) d Sn-e U {0} and

^C > 0 : VA E S- U {0} ||(A/ — A)-1\\l{x) ^ , ^

where q (A) denotes the resolvent set of A and

Se = {z E C\{0} : |argz\ <9} . (17)

This assumption implies that there exist a ball B (0, r0), r0 > 0, such that

q (A) d B (0,ro), (18)

and the estimate in (16) is still true in Sn-e U B (0, r0).

It is well known that the above assumption implies that the square root V—A is well defined and B = — V—A generates an analytic semigroup

(«*)>, (19)

0

B

Lemma 1. Let 0 E E and x E Cn ([0,T]; E) with T > 0. Then

1. eBÎ0 — 0 as£ — 0+ iff 0 G D(B) = D(A), (see [7, p. 20, Proposition 1.2]),

2. £ — eBÎ0 G Cn ([0,T] ,E) iff 0 G Db (n; = Da (n/2;+ro), (see [7, p. 29, Proposition 1.12]),

5. £ - fBeJZ-t)B [х (t) - X (£)] dt G Cn ([0,T] ,E) П B (0,T; db (n; +^)), (see [7, 0

p. 55, Theorem 4-. 5]).

Let 0 G]0,n/2[ and set

5(ro,0) = {z G C\{0} : \z\ ^ ro sin0 and |argz\ < 0} .

Note that for all w G S(r0, 0), one clearly has

Re w ^ ro sin 0. (20)

We will also use the following result proved in [6, p. 1880, Proposition 4.10]. Lemma 2. For any w G S(ro,0), one has

1. |arg (1 - e-w) - arg (1 + e-w)| < 0,

2. |1 + e-w | ^ Ce = 1 - e-2 tan в > 0,

s- U - е-"'| — T^ > ТГ^ =

1 + Re w 1 + r0 sin 0

Now, consider the following space

H^(S(ro,0)) = {f : f is an holomorphic and bounded function on S(ro,0)},

then, under our assumptions on Д if f G H^(S(ro,0)) is such that 1/f G H^(S(ro,0)) and (1/f)(-B) G L(X), then f (-B) is invertible with a bounded inverse and

[f (-B)]-1 = (1/f)(-B),

see, for instance [8] or [9, p. 45, Remark 2.5.1].

On the other hand we recall that operator I - e2^d I + e2B are boundedly invertible; see for instance [10, p. 60, Proposition 2.3.6].

Let us now apply this result to the following operator

A* = I - ^B-1 (I - e2B)-1 (I + e2B) - B-1 (I + e2B)-1 (I - e2B) .

Lemma 3. The operator A* : E — E is boundedly invertible. Proof. Let 0 < 0 < n/2. Consider the function f

f (w) = 1 + --1 + e-2w + -+ 1 - e-2w f (w) qw 1 - e-2w qw 1 + e-2w '

It is clear that f and 1/f are well defined and holomorphic on S(ro,0) in virtue of the above Lemma. The function f is clearly bounded on S(ro, 0). Moreover there exists C > 0 such that for all w G S(ro ,0) one has

p- 1 + e

— 2w

qw 1 — e

—2w

+

P+ 1 — e

2w

qw 1 + e

2w

<

C_

R

Therefore one can find xq > 0 such that

p- 1 + e-2w + p+ 1 — e-2w

qw 1 — e

2w

qw 1 + e

2w

1

< -- 2

for w E S(r0, 0) with Re w > xq\ consequently

\f (w)| =

p_ 1 + e-2w p+ 1 — e-2w 1 + --^ +

qw 1 — e

2w

qw 1 + e

2w

(21)

1

> 1/2

p- 1 + e-2w + p+ 1 — e-2w

qw 1 — e

2w

qw 1 + e

2w

for w E S (r0, 0) wit h Rew > x^. In the compact sector

Kxg = {w E C\ {0} : r0 sin 0 ^ Re w ^ x^d |arg(w)| ^ 0} ,

there is at most a finite number of roots of f (w) (not belonging to [0,x^]), (see [11, the remark on Proposition 4.1 , p. 41]); so there exists 0* E]0,0] such that f (w) does not vanish on

Sr0,e* := {w E C\ {0} : r0 sin0 ^ Re w ^ x^d \arg(w)\ ^ 0*} .

(22)

From (21), we conclude that f (w) does not vanish on S(r0, 0*). Hence f E H^(S(ro,0*)), 1/f E H~(S(ro, 0*)) and consequently (1/f )(—B) E L(X) and

A* = f (—B ) = I — q B-1 (.I — e2B)-1 (.I + e2B) — B-1 (I + e2^-1 (/ — e2B) is boundedly invertible.

e2B)-1 (t i e2B\ p+ u-i

q

□

3. Representation of the Solution of (P\a)

It is well known that, the solution of the second order following equation

in E, writes with a,ß E E and

U'(x) — B2u(x) = g(x), x E]a, b[ u(x) = e(x-a)Ba + e(b-x)Bß + v (g) (x)

x b

V (g) (x) = 2 / e(x-t)BB-1g (t) dt + 1 f e(t-x)BB-1g (t) dt.

Therefore

u- (x) = e-xBa- + e(1+x)Bв- + v (g-) (x), x g] - 1,0[ w+ (x) = e xBa+ + e(1-x)Bв+ + v (h+) (x), x g]0, 1[

with а-,в-,a+, в+ G E and

x o

V (g-) (x) = 1 / e(x-t)BB-1g- (t) dt + 2 J e(t-x)BB-1g- (t) dt,

-1 x

x 1

v (h+) (x) = 1 J e(x-t)BB-1h+ (t) dt + 2 J e(t-x)BB-1h+ (t) dt.

ox

The boundary conditions

{

u- (-1) = eBa- + в- + v (g-) (-1) = f-w+ (1) = Be Ba+ - Вв+ + v' (h+) (1) = f+,

give

в- = -eBa- - v (g-)(-1) + f-в+ = e Ba+ + B-1v' (h+)(1) - В-1f+;

{

and the transmission conditions

!

u- (0) = w+ (0) + ф

p- (u-)' (0) - p+ (w+)' (0) = qAu-(0) + ф,

give

a- + eBв- - a+ - eBв+ = ф - v (g-)(0)+ v (h+) (0)

-- (-a- + eBв-) - -+( a+ - eBв+) + qB [a- + eBв-) = -p-B-1v' (g-) (0) + -+B-1v' (h+) (0) - qBv (g-) (0) + B-1ф.

Using (23), one obtains the system

!

(I - e2B) a- - (I + e2B) a+ = F*

-p- (I + e2B) + qB (I - e2B)} a- - p+ (I - e2B) a+ = G

where

(23)

(24)

F* = ф - eBf- - eBB-1 f+ + eBv (g-) (-1) -v (g-) (0) + eBB-1v' (h+) (1) + v (h+) (0)

G* = B-1ф - (p- + qB) eB f- + p+eBB-1f+ (25)

+ (p- + qB) eBv (g-) (-1) - p+ eBB-1v' (h+) (1) -p-B-1v' (g-) (0) + p+B-1v' (h+) (0) - qBv (g-) (0).

The abstract determinant of this system (acting on D(B)) is

A(B) = (26)

= p+(I - e2B) (I - e2B) - (I + e2B) [-p- (I + e2B) + qB (I - e2B)] =

= p+ (I - e2B)2 + p- (I + e2B)2 - qB (I - e2B) (I + e2B) .

One has

A(B) : D (B) ^ E

with

—A(B) = —p+ (.I — e2B(2 — p- (.I + e2B)2 + qB(l — e2B) (I + e2B) = qB (I — e2B)(I + e2B)

I — b-1 (I — e2B)-1 (I + e2B) — p+ B-1 (I + e2B)-1 (I — e2B) := qB (I — e2B) (I + e2B) A*,

which is boundedly invertible by Lemma 4. It follows that

a- = [A(B)]-1 [p+ (I — e2B) F* — (I + e2B) G*] , (27)

a+ = [A(B)]-1 [(—p- (I + e2B) + qB (I — e2B)) F* — (I — e2B) G*] , (28)

and then

P- = —eB [A(B)]-1 [p+ (I — e2B) F* — (I + e2B) G*] — v (g-) (—1) + f-, (29)

3+ = e B [A(B)]-1 [ —p- (I + e2B) + qB(I — e2B)) F* — (I — e2B) G*] +B-1v (h+)(1) — B-1f+.

Therefore, the functions

{

u- (x) = e-xBa- + e(1+x)B+ v (g-) (x), x g] — 1, 0[ w+ (x) = e xBa+ + e(1-x)B/+ + v (h+) (x), x g]0, 1[

with

and

x o

V (g-) (x) = 1 / e(x-t)BB-1g- (t) dt + 2 J e(t-x)BB-1g- (t) dt

1x

x 1

v (h+) (x) = 1 J e(x-t)BB-1h+ (t) dt + 2 J e(t-x)BB-1h+ (t) dt

ox

are completely determined.

(30)

4. Analysis of v (g—) and v (h+) and Their Derivatives 4.1. Analysis of v (g-) on (—1, 0)

One has

v (g-)(x)

x 0

= 1 J e(x-t)BB-1 [g- (t) — g- (x)] dt + 2 J e(t-x)BB-1 [g- (t) — g- (x)] dt

-i x

x 0

+ 2 J e(x-t)BB-1g- (x) dt +2 J e(t-x)BB-1g- (x) dt

-1 x

x 0

= 2 / e(x-t)BB-1 [g- (t) — g- (x)] dt + 2 J e(t-x)BB-1 [g- (t) — g- (x)] dt

-1 x

1 (x+l)BD-2 , 1 -xBr>-2„ t\ d-2.

+ _e^x+l1BB-2g- (x) + -e-xBB-2g- (x) - B-2g- (x) ,

and

v (g-) (x)

x o

= 1 J e(x-t)BB-1 [g- (t) - g- (x)] dt + - J e(t-x)BB-1 [g- (t) - g- (x)] dt

-1 x

+-e^x+l1BB-2 [g- (x) - g- (-1)] + -e-xBB-2 [g- (x) - g- (0)] - B-2g- (x)

+1 e'yX+l1BB-2g- (-1) + 1 e-xBB-2g- (0) = vR (g-) (x) + vs-1 (g-) (x) + vs,o (g-) (x) where

vs,-1 (g-) (x) = 1 e(x+1)BB-2g- (-1); vs,o (g-) (x) = 1 e-xBB-2g- (0).

!

vR ( g- )

vr (g-) G C([-1, 0]; D (B2)) B2vr (g-) G Cn([-1, 0]; E),

while for vS,-1 (g-^d vS,o (g-) (x) we only have

vs,-1 (g-) G C(]-1, 0]; D (B2)); B\s,-1 (g-) G Cn(] - 1, 0]; E) vs,o (g-) G C([-1,0[; D (B2)); B2vs,o (g-) G Cn([-1, 0[; E).

The behaviour of B2vS,-1 (g-) (x) in ^te neighbourhood of -1 is that of

-e^x+l1Bg- (-1) , (31)

and the behaviour of B2vS,o (g-) (x) in ^te neighbourhood of 0 is that of

-e^x+1')Bg- (0) . (32)

4.2. Analysis of v (h+) on (0,1) Recall that

x 1

v (h+) (x) = 1 i e(x-t)BB-1h+ (t) dt + 2 i e(t-x)BB-1 h+ (t) dt,

0

which can be written as

v (h+)(x)

x 1

= 1 J e(x-t)BB-1 [h+ (t) - h+(x)] dt + 2 J e(t-x)BB-1 [h+ (t) - h+(x)] dt

0x

+ 2exBB-2h+ (x) + 2e(1-x)BB-2h+(x) - B-2h+(x)

x 1

= 2 i e(x-t)BB-1 [h+ (t) - h+(x)] dt + 2 f e(t-x)BB-1 [h+ (t) - h+(x)] dt

0

1 r, / S , /„M 1

-B-2h+(x) + 2exBB-2 [h+(x) - h+(0)] + 2e(1-x)BB-2 [h+(x) - h+(1)] + 2exBB-2h+ (0) + 1 e(1-x)BB-2h+(1) = vR (h+) (x) + vs,o (h+ ) (x) + vs,1 (h+) (x)

with

vs,o (h+) (x) = 2exBB-2h+(0), v^ (h+) (x) = 2e(1-x)BB-2h+(1). (33) Due to Lemma 1, vR,o (h+) has the following maximal regularity properties

!

vR (h+ ) E C([0,1]; D (B2)) B2vr (h+) E Cn([0,1]; E),

while for vs>0 (h+^d vs>1 (h+) we only have

{

vs,o (h+) E C(]0,1]; D (B2)); B2vs,o (h+) E Cn(]0,1]; E) vs,1 (h+) E C([0,1[; D (B2)); B2vs,1 (h+) E Cn([0,1[; E).

Remark 2. We have in view the study of the regularity of u- on [-1, 0^d w+ on [0,1]. Since

u- (x) = e-xBa- + e(1+x)B/3- + v (g-) (x), x e] - 1, 0[ w+ (x) = e xBa+ + e(1-x)B/+ + v (h+) (x), x e]0, 1[

the singular parts vs>-1 (g-) ,vs,0 (g-) ,vs,0 (h+^d vs>1 (h+) must be associated with the

u- w+

f-, f+, g- and h+ in order to get optimal regularity for u- on [-1,0^d w+ on [0,1].

4.3. Analysis of the Derivatives

In order to study the regularity of the solution of our problem, we also need to analyze the behaviour of the derivative of v (g-) and v (h+). We have, for all x E ] —1, 0[

x 0

v ' (g-) (x) = 2 / e(x-t)Bg- (t) dt — IJ e(t-x)Bg- (t) dt + 2B-ig- (x) — 2B-lg- (x)

-1 x

x 0

= 2 i e(x-t)Bg- (t) dt — 2 / e(t-x)Bg- (t) dt,

then

-i

0 0 0 1 ' „-tB„ J, _1 /„-tB г CnM ^ , 1 /„-tB,

^ (g-) (0) = - J e g- (t) dt = -J e [g- (t) - g- (0)] dt + -J e-tB g- (0) dt

-i -i -i

0

11 r 1

= -B-lg- (0) + 2 у e-tB [g- (t) - g- (0)] dt = -B-1g- (0) + Д* (g-) (0)

-i

Similarly, for all ж G ]0,1[, one has

x 1

v' (h+) (x) = 1 J e(x-t)Bh+ (t) dt - 1 J e(t-x)Bh+ (t) dt,

0x

then

1 1 v (h+) (0) = -1 J etBh+ (t) dt = 1B-1h+ (0) - 1 J etB [h+ (t) - h+ (0)] dt 00 U-i

-B-1 h+ (0) + (h+) (0)

and

1

v (h+) (1) = -B-1h+ (1) + - J eSl-t)B [h+ (t) - h+ (1)] dt (34)

0

= 1 B-1h+ (1) + (h+)(1).

5. Study of the Regularity of the Solution of Problem (Pia) 5.1. Necessary Conditions on the Transmission Data

Assume that u- and w+ are strict solutions, that is

u- G C([-1, 0]; D(A)) П C2([-1, 0]; E) and w+ G C([0,1]; D(A)) П C2([0,1]; E),

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then

y = u-(0) — w+(0) G D(A)'

and for any t1 G [—1, 0]

/ \> / \ 1- u_ (T1) — u_ (t') TTT1T (u-)' (T1) = lim -G D(A);

r'^T! T1 — T'

we also deduce that

(u-)''(T1) G D(A).

Similarly, for any t2 G [0,1], (w+)' (t2), (w+)'' (t2) G D(A). It follows that

!

(u-)" (0) = g- (0) — Au- (0) G D(A), (w+)'' (0) = h+(0) — Aw+(0) G D(A).

Using the first transmission condition

y = u-(0) — w+(0) G D(A),

one has

(u-)'' (0) = g-(0) — Au-(0) = g-(0) — Ay — Aw+(0) = g-(0) — Ay — h+(0) + (w+)'' (0), from which we deduce the first following necessary compatibility condition

h+(0) — g-(0) + Ay = (w+r (0) — (u_)" (0) G D(A). (35)

The second transmission condition gives

p- (u-)' (0) — p+ (w+)' (0) = ^ + qAu-(0) = ^ + q [Au-(0) — g-(0)] + qg-(0) G D(A) which implies the second following necessary compatibility condition

p- (u-)' (0) — p+ (w+)' (0) — q [Au-(0) — g-(0)] = ^ + qg-(0) G D(A). (36)

The two conditions (35) - (36) are equivalent to

!

^ + qg-(0) G D(A)

^ + q [h+(0) + Ay] G D(A), due to the identity

h+(0) — g-(0) + Ay = 1 [^ + qh+(0) + qAy] — 1 [^ + qg-(0)] G DA).

qq

In the case ^ = ^d y = 0, the necessary conditions on the transmission data are

g-(0) G D(A), h+(0) G D(A).

5.2. Analysis of u- (x^ar —1

Assume the necessary condition f- G D (B2) = D (A). Recall that

u- (x) = e-xBa- + e(1+x)B/- + v (g-) (x), where, due to (23), one has

3- = —eBa- — v (g-)(—1) + f-. Then the behaviour of B2u- (x^ar — 1 is the same as that of the function

e(1+x)Bb2 [—v (g-) (—1) + f-] + B2v (g-) (x).

One has

v (g-)(—1) =

o o

= 1 J e(t+1)BB-1g- (—1) dt + 2 J e(t+1)BB-1 [g- (t) — g- (—1)] dt -1 -1

= — 2B-2g- (—1) + R(g-)(—1) ,

and the term R(g-) (—1) is regular since

B2 [R(g-)(—1)] G Db (n;+rc)'

see Lemma 1.

Now from the study and the results on the behaviour of B2v (g-) (x^ar —1, see (31), one concludes that the behaviour of B2u- (x^ar —1 is the same as that of

e(1+x)B

2g- (-1) + Bf-

+ 2e(x+1)Bg- (-1) = e(1+x)B [g- (-1) + B2f-\

Due to Lemma 1, B2u- (.) has the following maximal regularity properties near —1

(

B2u- (.) e C([—1,0[; E) iff [g- (—1) + B2f-] e D(B2) = D(B) B2u- (.) e Cn([—1, 0[; E) iff [g- (—1) + B2f-] e Db (n, .

w+ 1

Let us assume the necessary condition f+ G D (B). Recall that

w+ (x) = e xBa+ + e(1-x)B/+ + v (h+) (x),

then the behaviour of

B2w+ (x) = B2 [e xBa+ + e(1-x)Bß+ + v (h+) (5 1

B2e(1-x)Bß+ + v (h+) (x).

One has seen in (30), (34) that

в+ = -e BA-1 [(-p- (I + e2B) + qB (I - e2B)) F* + (I - e2B) G*] + B-1v' (h+)(1) - B-1f+,

with

v' (h+) (1) = -B-1h+ (1) + 5* (h+)(1).

Since the behaviour of B2v (h+) (x^ar 1 is that of 1 e(1-x^Bh+ (1), as mentioned in (33),

B2w+ (x) 1

-e(1-x)B( - h+ (1) + Bf^j + 1 e^1-x^B h+ (1) = -e(1-x'lB Bf+. Due to Lemma 1, B2w+ (.) has the following maximal regularity near 1

!

В2w+ (.) G C(]0,1]; E) iff f+ G D (B) et Bf+ G D (B) В2w+ (.) G Cn(]0,1]; E) iff Bf+ G Db (n, .

5.4. Analysis of u- at the Inter face 0

The behaviour of

B2u- (x) = B2e-xBa- + B2e(l+x)B[- + B2v (g-) (x),

0

B2e-xBa- + B2v (g-)(x), and due to (32), it is the same as that

B2e-xBa- + 1 e-xBg- (0).

Now, in virtue of (27), one has

a- = [A (B)]-i [p+ (I — e2B) F* — (I + e2B) Gj

= [A (B)]-i [p+F* — G*] — e2B [A (B)]-i [p+F* + G*]

Write

A (B) = p+ (I — e2BY + p- (I + e2BY — qB(I — e2B) (I + e2B) (37)

= p+I + p-I — qB + T*,

where, clearly T* is a very regularizing operator. Then, it suffices to analyze

" 1 ]

B e-xB [A (B)]■

(p+F* - G*) + 2B-2 (p+I + p-I - qB ) g- (0)

0.

1

(p+F* - G*) + 2B-2 (p+I + p-I - qB ) g- (0)

by using the expressions in (24). One has F, = у - eB f- - eBB-lf+ + eBv (g-) (-1) - v (g-) (0) + eBv' (h+) (1) + v (h+) (0)

[у - v (g-) (0) + v (h+) (0)] + R, = у + 2B-2g- (0) - 2B-2h+ (0) + R,

and

G, = B-1ф - (p- + qB) eB f- + p+eBB-1f+ + (p- + qB) eBv (g-) (-1)

- p+ eBB-1v' (h+) (1) - p-B-1v' (g-) (0) + p+B-1v' (h+) (0) - qBv (g-) (0)

= B-1ф - 2p-B-2g- (0) + 2P+B-2h+ (0) + 2qB-1g- (0) +

Then

(p+F, - G,) + 2B-2 (p+I + p-I - qB ) g- (0) = p+ (у + 2B-2g- (0) - 2B-2h+ (0))

2 u-r * v , ,-r ^r 2 w 2

- (B-1^ - 2p-B-2g- (0) + 2p+B-2h+ (0) + 2qB-1g- (0)) +2B-2 (p+ + p- - qB ) g- (0) + P,

= -B-1 (ф + q g- (0)) + p+y + B-2 [(p+ + p-) g- (0) - p+h+ (0)] + P,. Now, as у e D (B2^d B-2 [(p+ + p-) g- (0) - p+h+ (0)] e D (B2) , the behaviour of

1

B e-xB [A (B)]"

is that of

(p+F, - G,) + 2B-2 (p+ + p- - qB) g- (0)

B2e-xB [A (B)]-1 B-1 (ф + qg- (0))

The determinant

A (B) = -qB (I - e2B) (I + e2B) A,

being invertible of inverse

[A (B)]-1 = -2A-1 (I + e2B)-1 (I - e2B)-1 B-1, (38)

A,-1 I + e2B -1 I - e2B -1 e-xB

that the behaviour of B2u- (ж) near 0 is that of

e-xBBB-1 [ф + qg- (0)] = e-xB [ф + qg- (0)] .

u-

0

[ф + qg- (0)].

One gets the following Theorem.

Theorem 2. Assume (16). Let f+ e D ((-A)1/2), у e D(A), ф e E and let g- e Cn ([-2, 0]; E), h+ e Cn ([0, 2]; E) with 0 <n< 2 Then

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1. u- e C([-1, 0]; D(A)) n C2([-1,0]; E) if and only if

[g- (-1) + B2f-] e D(A) and qg-(0) + ^ e D(B) = D(A),

2. Au-(.), (u-)" e Cn([-1, 0]; E) if and only if

[g- (-1) + B2f-] e Da (n/2, and qg-(0) + ^ e Da (n/2, .

We have used tha fact that

D(B) = D(A) and Db (n, = Da (n/2, .

5.5. Study of w+ at the Interface 0

Recall that

B2w+ (x) = B2 [e xBa+ + e(1-x)B3+ + v (h+) (x)] ,

where

a+ = [A (B)]-1 -p- (I + e2B) + qB(l - e2B)] F* - (I - e2B) G*, (see (28)). Therefore, the behaviour of B2w+ (x) near 0 is the same as that of

B2e xBa+ + B2v (h+) (x).

Now as (see (33)) the behaviour of B2v (h+) (x) near 0 is that of 1 exBh+ (0) and

A (B) = p+I + p-I - qB + T*,

it suffices to analyze the behaviour of

?2 xB [\ / Dm-i

B2e xB [A (B)]■ where as we have seen

(-p-I + qB) F* - G* - 1 (p+1 + p-I - qB) B-2h+ (0)

{

F* = y + 1B-2g- (0) - 1 B-2h+ (0) + R** G* = B- 1p-B-2g- (0) + 2p+B-2h+ (0) + 1 qB-1 g- (0) + 5*

with regular terms R*^d S*. Then

(-p- + qB) F* - G* - 2 (p+I + p-1 - qB) B-2h+ (0) = (-p- + qB) (y + 2B-2g- (0) - 2B-2h+ (0)

- (- 2p-B-2g- (0) + 1 p+B-2h+ (0) + 1 qB-1g- (0)) + 1 (p+I + p-I - qB) B-2h+ (0) + Q* = -B-1 [fy - qB2y + qh+ (0)] + Q*,

J() Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 4, pp. 50-75

where Q* G D (B2). Therefore, the behaviour of B2w+ (i^ar 0 is that of

B2exB [A (B)]-1 B-1 - qB2y + qh+ (0)] .

as mentioned in the previous section, see (38). One then deduces that the behaviour of B2w+ (i) near 0 is that of

exB - qB2y + qh+ (0)] .

Then, using again [7], the necessary and sufficient conditon to obtain the regularity of w+ 0

^ - qB2y + qh+ (0) . We conclude by the following Theorem.

Theorem 3. Assume (16). Let f+ G D ((-A)1/2), y G D(A) and ^ G E and let g- G Cn ([-1, 0]; E), h+ G Cn ([0,1]; E) with 0 <n< 1- Then

1. w+ G C([0,1]; D(A)) n C2([0,1]; E) if and only if

(-A)1/2f+ G D(A) and ^ - qB2y + qh+ (0) G D(A),

2. Aw+(.), (w+)" G Cn [0,1]; E) if and only if (-A)1/2f+ G Da (n/2, and ^ -qB2y + qh+ (0) G Da (rj/2, .

By observing that

1 [^ - qB2y + qh+ (0)] - 1 [^ + qg- (0)] = h+ (0) - g- (0) + Ay, qq

one deduce the complete Theorem announced in the Introduction.

6. Going Back to the Concrete Example

Let us go back to our concrete limiting problem

{

with the boundary conditions

A(Xty)U- = g- in ]—1,0[ x Q A(x,y)w+ = h+ in ]0,1[ x Q

u- = 0 on ] —1, 0[ x Г w+ = 0 on ]0,1[ x Г u- = f- on {—1} x Q u = f- on { —1} x Q

dw+

(39)

(40)

— = 0 от {1} x Q

dx

and the transmission conditions

w+ = u- on {0} х П

дп_ dw+ (41)

p-~dx - = yu- + qg° on {0} х п.

Here Q is a bounded domain of Rra, n ^ 1, with a regular boundary r.

In view to illustrate our abstract analysis, we are going, in this section, to explicit and interpret our impedance compatibility conditions

{

qg-(0)+ ^ e D(A) _

h+ (0) - g- (0) + Ay e D(A),

{

qg- (0)+ ^ e Da (n/2, +œ) h+ (0) - g- (0) + Ay e Da (n/2, +œ),

in the case of the following vector valued Banach spaces

C([-1, 0]; C°0(H)) and C([0,1]; C^(H)),

provided that

g- e Cn([-1, 0]; Co (H)) and h+ e C([0,1]; CO(H))

and all the compatibility boundary conditions are satisfied. Consider the following operator defined in C°(H) as

I

D(A) = {V e C2(H) : v, Ayv e Cf° [Av] (y) = Ayv(y).

Then A verifes (16) and D(A) coincides with the well known little Holder continuous functions space hQ (Q), see [12, p. 497].

Let us point out that the boundary condition in space CQ(Q) is essential. Otherwise the estimate in (16) is not verified, see [10, p. 110, Example 3.1.33]. One also has

D(A) = da(1 + 3/2, = {v e C2+Q(Q) : v = AyV = 0 on dQ} ;

see [10, p. 110, Corollary 3.1.32 and Corollary 3.1.35]. The interpolation DA(1+ 3/2, is intended in the Banach space C0(Q). Therefore, in our subspace E = CQ(Q) C C0(Q), one has

DA(n/2, = (D(A),CQ (Q))i

= (d(a),cq (q))i-v/2,+«

= ({v e C2+Q(Q) : v = Av = 0 on dQ} , CQ(Q)) = C(Q), {v e C2+Q(Q) : v = Av = 0 on dfi}) ^

= cQ+n/2[2+Q-Q] (Q)

= ch (Q),

see [10, p. 31, Corollary 1.2.18].

The conditions ^ = qg0 e E, f- e D (A) become

{

qgo e CO (H)

f- e {v e C2(H) : V, AyV e C0

and Af- — g- (-1), qg-(0) + ^, qh+(0) + ^ E D(A) signifies

y Ayf-(y) — g-(—1,y) Ehi(H) y g-(0,y)+ go(y) E hl(H) y h+(0,y)+ go(y) E hl(H),

similarly Af- — g- (—1), qg-(0) + qh+(0) + ^ E DA (n/2, mean

y Ayf-(y) — g-(—1,y) E Cj+n(H) y g-(0, y) + go(y) E C^(H) y h+(0,y) + go(y) E Ci+n(H).

Note that Col+n (H) C hi (H).

One can conclude by the following result

Theorem 4. Assume that g- E Cn([—1,0]; Col(H)), h+ E C([0,1]; Col(H)) and

qgo E Cl(H), f- E [v E C2(H) : v, AyV E C0(H)} := C20+l(H). T/ien f/tere exists a unique solution u of Problem (39) - (4-1) defined as

] - 1,o[u]o, 1[ —^ c0(П)

u-(x,.) on ] — 1, 0[ w+(x,.) on ]0,1[

x —-> u(x, .) = I

such that

l.u- E C(—_1, 0]; Cl+ß(П)) П C2([—1, 0]; Cß0 (П)), w+ E C([0,1]; C%ß(П)) П C2([0,1]; C0e(П)) г/ and only if

y Af-(y) — g-(—1,y)Ehß0 (П), y g-(0, y) + go(y) E h0(fi), y h+(0,y) + go(y) E he(П),

Ayu-, ^T E Cn ([—1, 0]; Ce(П)) , A,w+(.), ^Г E Cn ([0,1] ; Cв(П)) г/and only г/

y Ayf-(y) — g-(—1,y) E Cj+n(П) y g-(0, y) + go(y) E Ce+n(П) y h+(0,y) + go(y) E C0+v(П).

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Received January 14, 2015

УДК 517.96+517.93 Б01: 10.14529/ттр150405

ЭФФЕКТ ВОЗМУЩЕНИЯ ТОНКОГО КЛЕЕВОГО СЛОЯ В НЕКОТОРЫХ КРАЕВЫХ ЗАДАЧАХ ПЕРЕНОСА НА ОСНОВЕ ЭЛЛИПТИЧЕСКИХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

А. Фавини, Р. Лаббас, К. Лемрабе

В данной работе рассматривается задача о двух телах, скрепленных тонким клеевым слоем (третий материал) толщины 6. При 6, стремящемся к нулю, получается краевая задача переноса на фиксированной области. Получены новые результаты по исследованию данной задачи в пространствах Гельдера, а именно, явное представление решения. С помощью теории полугрупп и вещественных интерполяционных пространств получены необходимые и достаточные условия на границе раздела при которых существует единственное решение задачи.

Ключевые слова: эллиптическая краевая задача; задача переноса; эффект возмущения; тонкий слой.

Анджело Фавини, кафедра математики, Болонский университет (г. Болонья, Италия ), [email protected].

Раба Лаббас, лаборатория прикладной математики, Университет Гавра (г. Гавр, Франция), [email protected].

Кеддуа Лемрабе, факультет математики, Университет науки и технологии - Бу-медьен (г. Алжир, Алжир), [email protected].

Поступила в редакцию Ц января 2015 г.