Elliptic Problems with Robin boundary coefficient-operator conditions in general Lp Sobolev Spaces and Applications Текст научной статьи по специальности «Математика»

MSC 35J20, 35J40, 47D06

DOI: 10.14529/ mm p 150304

ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY

COEFFICIENT-OPERATOR CONDITIONS

IN GENERAL Lp SOBOLEV SPACES AND APPLICATIONS

M. Cheggag, Polytechnic National School of Oran, Oran, Algeria, [email protected],

A. Favini, University of Bologna, Bologna, Italy, [email protected], R. Labbas, University of Le Havre, Le Havre, France, [email protected], S. Maingot, University of Le Havre, Le Havre, France, [email protected],

A. Medeghri, University of Mostaganem, Mostaganem, Algeria, [email protected]

In this paper we prove some new results on complete operational second order differential equations of elliptic type with coefficient-operator conditions, in the framework of the space Lp(0,1; X) with general p £ (1, X being a UMD Banach space. Existence, uniqueness and optimal regularity of the classical solution are proved. This paper improves and completes naturally our last two works on this problematic.

Keywords: second-order abstract elliptic differential equations; Robin boundary conditions; analytic semigroup.

To the memory of Alfredo Lorenzi.

Introduction and Hypotheses

In this work we study the following operational second order complete elliptic differential Problem

f u''(x) + 2Bu'(x) + Au(x) = f(x), a.e. x £ (0,1), ,,

\ u'(0) - Hu(0) = d0, u(1) = ui, ^

where A, B, H are closed linear operators in X (X being a complex Banach space), d0, ui are given elements in X and f £ Lp(0,1; X), 1 < p <

uu

i) u £ W2'p(0,1; X) n Lp(0,1; D(A)), u' £ Lp(0,1; D(B)),

ii) u(0) £ D(H), (2)

iii) u .

This optimal Lp-regularity is very important to solve many quasilinear parabolic equations corresponding to (1). In fact, the use of the fixed point theorem to solve these nonlinear problems requires necessarily optimal regularities such (2).

The Robin boundary condition u'(0) — Hu(0) = d0 arises in many concrete situations

0

we will see in the applications that we can take H = —A or some fractional power of —A.

We first study the Problem

f u''(x) + (L - M)u'(x) - LMu(x) = f (x), a.e. x e (0,1), ( )

\ u'(0) - Hu(0) = d0, u(1) = u1, ^

where L and M are some closed linear operators in X. Then, in order to solve (1), we LM

L - M C 2B and LM C -A. (4)

Here, when P, Q are two linear operators in X, P C Q, means that D(P) C D(Q) and P = Q on D(P).

u

i) u e W2'p(0,1; X) n Lp(0,1; D(LM)),u' e Lp(0,1; D(L - M)),

ii) u(0) e D(H), (5)

iii) u .

u

u

In order to solve Problems (1) and (3) for any f e Lp(0,1; X), 1 < p < to, we will assume in all this paper that

X

We recall that a Banach space X is a UMD space if and only if for some p > 1 (and thus for all p) the Hilbert transform is continuous from Lp(R; X) into itself (see [1,2]). Many authors have studied the equation

u''(x) + 2Bu'(x) + Au(x) = f (x), a.e. x e (0,1),

with the Dirichlet boundary conditions u(0) = u0, u(1) = u\. When f e Lp(0,1; X), 1 <p < (see for example [3,4]); when f e Ce ([0,1]; X) 0 <0 < 1 (see [5-9]).

0 u (0) -

Hu(0) = do, which contains a general linear closed operator H. Therefore the situation is more complicated because of the different domains for instance. In the particular case B = 0 Problem (1) has been considered in [11] when f e Lp(0,1;X), 1 < p < and in [10] for f e Ce([0,1]; X), 0 < 0 < 1. We recall also the study [12] where B is supposed generating a group.

In this paper we will consider more general situations (see Subsection 2.1). Our techniques are based upon the Dore - Venni Theorem [13], on the sum of two closed linear operators, on the results in Prtiss - Sohr paper [14] and on the reiteration Theorem in the interpolation theory, (see [15,16]).

Let us mention that all the papers quoted above and also our study, deal with the A, B L, M

A recent paper [17] treats one interesting non-commutative framework, for the boundary Dirichlet following problem

J u''(x) + 2Bu'(x) + Au(x) - uu(x) = f (x), a.e. x e (0,1), [ u(0) = u0, u(1) = ui,

(with u > 0 large enough). This new approach will lead us to develop a future work, solving the same equation with operational Robin boundary conditions in non-commutative situations.

The plan of the paper is as follows.

Section 2 is devoted to Problem (3); we first give our assumptions on operators L,M and H, we then give a representation formula of the solution and conclude by analyzing this representation.

In Section 3 we apply the results of section 2 with

L = B - (B2 - A)l/2 and M = -B - (B2 - A)l/2,

to solve Problem (1). We also specify the assumptions on A and B.

In section 4 we study some interesting particular situations in which our assumptions L,M H

Finally in section 5 we give some concrete examples of partial differential equations to which our theory applies.

1. Study of Problem (3) 1.1. Preliminaries

First, define the class BIP (X, a) where a E [0,^) (see [14, p. 430]): U EBIP(X, a) if U is a closed linear densely defined operator satisfying

{

(-<*, 0) c p(U), N(U) = {0} , R(U) = X

and 3C ^ 1 : VA > 0, \\(U + AI^H^ ^ C/A, 1 J

!

(N(U), R (U) and p(U) are respectively the kernel, the range and the resolvent set of U)

For all s E R, Uis E L(X) and

3C ^ 1: Vs E R, \\Uls\\c[X) ^ Cea\s\.

We recall that operator verifying (7) admits a complex power Uz for any z E C (see [18, p. 70]).

On the other hand, let 9 E (0,1),q E [1, +x>],m E N,^ E R and V a closed linear X

]/i, c p(V^d sup \\A (V - AI)-1\\£(X) < Then we consider the interpolation space (X,D(V))e q and define

(X, D(V))m+0qq :={0 E D(Vm) : Vm0 E (X, D(V))0; J . When 9 = 1/2, we can use the well known following reiteration result

(X,D(V2))eqq = (X,D(V))2dqq . (9)

We recall moreover that (D(V),X)0 = (X,D(V))1-e q, so using (9) we get

(D(V2),X)dqq = (X, D(V2)) i-dqq = (X, D(V))2-20q , (10)

(for details on interpolation spaces and reiteration see for instance [19]).

1.2. Hypotheses on Operators

L, M H L, M H

operators such that

D(L) = D(M^d D(ML) = D(LM), (11)

ML = LM,

^0L, 0m e]0, n/2[: -L e BIP (X, 0L) and - M e BIP (X, 0m), (13)

and

{ {

L + M is boundedly invertible, (14)

V С E D(H), VA E p(L), (L - XI)-1 £ E D(H) and (L - XI)-1 H£ = H (L - XI)-1 £,

V£ E D(H), V¡1 E p(M), (M - ¡I)-1 £ E D(H) and (M - ¡I)-1 H£ = H (M - ¡I)-1 £.

(15)

(16)

The previous assumptions allow us to build eL+M E L (X) (see Lemma 1 below) and then we can consider the linear operator Л defined by

D^) = D(L) П D(H^d Л = (M - H) + eL+M (L + H). We will suppose that

Л

This last assumption signifies exactly that the determinant, in some sense, of (3) is invertible. It generalizes the hypothesis (16) (used in the paper [12, p. 526]), since when B = 0 they coincide. This will be discussed further (see section 3).

L, M

if f E Lp(0,1; X) with 1 < p < ж, then Problem (3) has a unique classical solution u in the sense of (5) if and only if

Л-Ч,и1 E (D(LM),X)фрр .

1.3. Consequences of Assumptions Remark 1. Assume (11) and (12). Then

1. D(L2) = D(M2) = D(ML) = D(LM).

2. for 9 E (0,1),q E [1, +ж) we have (X, D(L))e>g = (X, D(M))eq and

(X, D(L2))eqq = (X, D(M2))eqq = (X,D(LM))eq = (X,D(ML))e, ,

and also

(X,D(L))

1+в ,q

= (X,D(M))

1+в, q '

(for the last equality see [17, Remark 5, p. 4970]).

3. VA e p(L), Vi e p(M) : (L - XI)-1 (M - ¡I)-1 = (M - ¡I)-1 (L - XI)-1

Remark 2. Due to Priiss - Sohr [14, Theorem 2, p. 437], assumption (13) implies that L and M generate uniformly bounded analytic semigroups in X : (exL)x>0 , {exM)x>0 •

We then detail some properties of the sum L + M and the product LM.

Remark 3. Under (6) and (11)~(13), we can apply Theorem 4, Theorem 5 and Corollary 3 in [14, p. 441, p. 443 and p. 444]. We obtain the following important results.

1. Operator -L - M with domain D(L) = D(M) is closed and satisfies (7). Moreover, if L or M is boundedly invertible then L + M is boundedly invertible and in this case (14) is satisfied.

2. We can choose e > 0 (arbitrary small) such that

-(L + M) eBIP(X,0) with 0 = max(0L,0M)+ e, (18)

(if 0L = 0M, it is even possible to take e = 0). It follows that L + M generates a

X

way, without applying the BIP operator theory see Lemma 1, statement 6.

3. LM is closable and LM belongs to BIP(X, 0L + 0M). The closability is obtained by a direct application of Corollary 3 in [14], but here, due to the fact that D(L) = D(M), we can apply [4, Lemma 1, p. 168], to show that LM is closed, so LM eBIP(X, 0L + 0 m )•

We now study some commutativity properties. Lemma 1. Assume (6) and (11)^(17).

1. Let C e{M,L,L + M} ,C e{M,L,H} ,x > 0 and£ e D(C), then

exC£ e D(C)) and CexC£ = exCC£.

2. Let C e {M,L,L + M} , C e {M,L} ,x > 0, z e X and X e p (

then

exC (c - XI^ 1 z = (c - XI^ 1 exCz•

3. Let C e {M, L}, then for £ e D(A), X e p(C) we have

(C - XI)-1 £ e D(A) and (C - XI)-1 A£ = A (C - XI)-1 £•

I Let C e {M, L} and £ e D(C) we get CA-1 £ = A-1C£ •

5. For £ e D(A) = D(H) n D(L) we get HA-1£ = A-1H£^

6. Let x > 0 then L + M generates a uniformly bounded analytic semigroup in X satisfying ex(L+M) = exLexM = exMexL.

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

& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 56-77

Proof.

1. Assume x > 0 and £ E D(H). Operator C generates a Co-semigroup, so we can apply the exponential formula (see [20, Theorem 8.3 p. 33]):

exC£ = lim (-I - C)-1)"£, (19)

n^-tt \ x \x / J

and from (12), (15), (16) we deduce that

exC (C£ = lim (- (-I - C VT C£ = Vim C( - (-I - C)-T £, n^tt y x \X / J n^tt \x \x / J

then, since C is closed, we deduce that exC£ E D(C) and CexC£ = exCC£.

2. Setting £ = (d - Aiy z, we deduce, from statement 1, that

CC - AI exC£ = exC CC - AI £, so (C - AI^ exC (C - A^ z = exCz.

3. Since £ E D(H), from (15), (16) we deduce that (C - AI)-1 £ E D(A) and

A(C - AI)-1 £ = ((M - H) + eL+M (L + H)) (C - AI)-1 £

= (M - H)(C - AI)-1 £ + eL+M (L + H) (C - AI)-1 £, now from (12), (15), (16) and statement 2, we deduce

A(C - AI)-1 £ = (C - AI)-1 (M - H)£ + (C - AI)-1 eL+M (L + H) £ = (C - AI)-1 A£.

4. We fix A E p(C) and set y = A-1(C - AI)£, then from statement 3 we have

(C - AI)-1 Ay = A(C - AI)-1 y, that is £ = A (C - AI)-1 A-1(C - AI)£, so

(C - AI)A-1£ = A-1(C - AI)£,

thus

CA-1£ = A-1C£.

5. If £ E D(A) we have AA-1£ = A-1A£, that is

((M - H) + eL+M (L + H)) A-1£ = A-1 ((M - H) + eL+M (L + H)) £,

so

MA-1£ + eL+MLA-1£ - (I - eL+M) HA-1£ = A-1 M£ + A-1eL+ML£ - A-1 (I - eL+M) H£,

(/ _ HA-iç = A-1 ^ - eL+M^ щ = ^ - eL+M^ л-1И£.

then

e-—)H Л Ч =л * [I _ e-—)H£ = [I _ e But I _ eL+M is boundedly inverti ble so НЛ-1£ = Л-1Н£.

6. Applying statement 2, we get that, for x E (0, n E N

exL (n (nj - MY') = (n (nj - MY') exL, x x x x

and by (19), we deduce that exLexM = exMexL. Then, (exLexM) x>0 is a strongly continuous semigroup (see [21, paragraph 5.15, p. 44]). We notice, that due to (14) L + M is closed, then from paragraph 2.7, p. 64 in [21], we deduce that L + M is the generator of the product semigroup {exLexM)x>0.

□

1.4. Representation of the Solution

We assume here (6) and (11)~(17). Suppose that Problem (3) has a classical solution u. Then

u E W2>p(0,1; X) n Lp(0,1; D(LM)),u' E Lp(0,1; D(L - M), u0 := u(0) E D(H) and one can write

{

u''(.) + (L - M)u'(.) + LMu(.) E Lp(0,1; X), a.e. x E (0,1) u(0) = u0, u(1) = u\.

Thus

«(0),«(1)E (D {L2) X) x p = (D {M2) }X) x p , (20)

2p ^

see ( [4, Statement 2, Theorem 5, p. 173]). On the other hand, using (10), one obtains (D (L2) , X) = (X,D(L))2-iqq ={0 E D(L) : L0 E (X,D(L))-iq J .

Then

u(0),u(1) E D(L) = D(M). (21)

As in [3], u satifies, for a.e. x E (0,1)

u(x) = exM £o + e(1-x)L£1 + Ix + Jx, (22)

where

fx !>1 \ — 1 I Ax — s)M£{-\J„ ___1 T tT i Tl/TN—1

x

'0 Jx

Ix = (L + M)—1 e(x—s)Mf (s)ds and Jx = (L + M)—1 e(s—x)Lf (s)ds.

u £o

£1

u'(0) - Hu(0) = do, u(1) = u1. (23)

It is clear that £0, £1 E D (L) = D(M) due to (21), so

u(0) = £o + eL£1 + Jo.

We have, for a.e. x E (0,1)

u'(x) = MexM£o - Le(1-x)L£1 + MIX - LJX\ (24)

g2 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 56-77

so

и'(0) = M£o - LeL£1 - LJo;

therefore

' A-1 u(0) = A-1£o + A-1eL£1 + A-1 Jo

-1LJo.

Since A-1 (X) = D(H) n D(L) = D(H) n D(M) ( due to (17)), we get

!

{

H Л-1и(0) = HЛ-1£o + HЛ-1eL£l + HЛ-1 Jo Л-1и'(0) = Л-^£о - Л-1 LeL£1 - Л-1LJo.

(25)

Then

A-1 do = A-1 [u'(0) - Hu(0)] = A-1u'(0) - HA-1u(0)

= (M - H)A-1£o - (L + H) A-1eL£1 - (L + H) A-1 Jo,

here we have used HA-1u(0) = A-1Hu(0) (see Lemma 1, statement 5 and (21) and MA-1 = A-1M on D(M),LA-1 = A-1L on D(L) (see Lemma 1, statement 4). Now, from u1 = eM£o + £1 + I1; one obtains

A-1 do = [(M - H) + eL+M (L + H)] A-1£o - (L + H)A-1eL (u - I1) - (L + H)A-1 Jo

= £o - (L + H) A-1eL (u1 - I1) - (L + H) A-1 Jo,

[]

£o = A-1 do + (L + H) A-1 [eV - eLI1 + Jo] , (26)

ps

£1 = -eM (L + H) A-1 (eV - eLI1 + Jo) - eMA-1 do + u1 - I1 • (27)

u

1

u(x) = exM [A-1do + (L + H) A-1eLu1] - exM (L + H) A-1eL(L + M)-1 ^ e(1-s)M f (s)ds

o

1

„xM / r , TT\ \ -1?T 1 H/T\~1 „sL

+ ^ (L + H) Л (L + M^ J esLf (s)ds

o

+ e(1-x)L [(I - (L + H^-1eL+M) U1 - Л-^мdo]

o 1

- e(1-x)L(L + H )eM Л-1(L + M )-1 ^ esLf (s)ds

o

1

- e(1-x)L [I - (L + H^-1eL+M] (L + M)-1 ^ e(1-s)Mf (s)ds

o

x 1

+ (L + M)-1 J e(x-s)M f (s)ds + (L + M)-1 у e(s-xX)L f (s)ds,

ox

which can be written as

where

u(x) = 5(x, fo, f, M) + 5(1 - x,fi,f (1 - .) ,L) (28)

+ R(x, Tfi, M) - R(1 - x,fo + TeLf1,L),

T = (L + H) A-1 eC(X), (29)

i

fo = A-1 do + T(L + M)-1 J esLf (s)ds, (30)

o

1

fi = ui - (L + M)-1 i e(1-s)Mf (s)ds, (31)

and for 0 £ X and C = L or M

S(x, 0, f, C) = exC0 + (L + M)-1f e(x-s)C f (s)ds R(x, 0, C) = exCeL+M-C0.

(32)

u

determined by (28).

1.5. Technical Lemmas

We recall an important consequence of the Dore - Venni sum theory.

Lemma 2. Assume (6) and let -C £BIP(X,a) with a £ [0,n/2) and 1 < p < Then

1. For any g £ Lp(0,1; X)

x

x C e(x-s)Cg(s)ds £ Lp(0,1; X). (33)

Jo

2. For any g £ Lp(0,1; X)

1

x CexC j esCg(s)ds £ Lp(0,1; X). (34)

o

C

Statement 2. is a consequence of statement 1 (see [3, p. 200, property (26)]).

□

C

q £]1, m £ N\ {0}

0 £ (D(Cm),X) 1 /mp,q ^ x ^ CmexC0 £ Lq(0,1; X); (35)

(¡. | Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 56-77

it follows, for example, that

ф e (D(C ),X )г/р p ^ CeC ф e L (0} 1; X)

{

ф E (D(C2),X)l/2ptp ^ C2e'Cф e Щ0,1; X), (36)

(see [16, p. 96]).

Lemma 3. Assume (6) and Then, for C = L or M and Ao e p (C) we have

1. ф E (D(C2),X)i/2pp ^ (C - AoI) Ce•Cф e Lp(0,1; X).

2. ф E (D(C2),X)l/2pp ^ (L + M - C) Ce•Cф e Lp(0,1; X).

3. ф e (D(C2),X)i/2pp ^ (L + M - C)2 e•Cф e Lp(0,1; X).

Proof.

1. For the proof, see Lemma 3, p. 171, 172 in [4].

2. We assume that C = L (the proof for C = M is similar). If ф e (D(L2),X)

i./^^^™ since M(L — I) 1 e L (X), we obtain by statement 1 MLe• ьф = M(L - I)-1 (L - I) Le• ьф e Lp(0,1; X).

Conversely, if MLe• ьф e Lp(0,1; X) then

Le•Lф = L (M - I)(M - I)-1 e• ьф

= (M - I)-1 LMe• Lф - L (M - I)-1 e• Lф,

so Le•Lф e Lp(0,1; X), then from

L2e• Lф = L2 (M - I)(M - I)-1 e• Lф

= L (M - I)-1 LMe• Lф - L (M - I)-1 Le• Lф,

we deduce that L2e• Lф e Lp(0,1; X) and by (36) we get ф e (D(L2),X)l/2pp .

3. We assume that C = L.

Иф e (D(L2),X)l/2pp then Me• Lф e Lp(0,1; X) since

Me• Lф = M(L - I)-1 (L - I) e• Lф

= M(L - I)-lLe• Lф - M(L - I)-le• Lф,

by statement 2, we deduce that M2e'e Lp(0, 1; X) since

M2e• Lф = M(L - I)-lM (L - I) e• Lф

= M(L - I)-lMLe• Lф - M(L - I)-lMe• Lф.

Вестник ЮУрГУ. Серия «Математическое моделирование gg

и программирование» (Вестник ЮУрГУ ММП). 2015. Т. 8, № 3. С. 56-77

Conversely, if M 2eL0 £ Lp(0,1; X) the n MeL0 £ Lp(0,1; X) since

MeL0 = M (M - I)(M - I)-1 eL0

= (M - I )-1 M 2eL0 - M (M - I )-1 eL0,

but

LMeL0 = LM (M - I )(M - I )-1 eL0

= L (M - I)-1 M2eL0 - L (M - I)-1 MeL0,

so LMeL0 £ Lp(0,1; X), and statement 2 gives 0 £ (D(L2),X )1/2p p . We are now in position to study the regularity of the terms R, S appearing in (28).

□

Lemma 4. Assume (6), (11)~(17). Let C = L or M and $ a given element in X. Then for the regular term defined above C) we have

LMR(,$,C), L2R(,$,C), M2R(,$,C) G Lp(0,1; X).

For the proof of this lemma see Lemma 2, p. 170, 171 in [4]. Now concerning the singular term S(•, f0, f, M), we have:

Proposition 1. Assume (6) and (11)^(17). Let f G Lp(0,1; X), 1 < p < and P G {LM, M2,L2} then

PS(;fo,f,M) G Lp(0,1; X) ^ A-1 do G (D(LM),X)^ p ,

2p ^

fo

Proof. We set, for a.e. x G (0,1), C G {L, M}

{

L(x, g, C) = LM (L + M )-1 e(x-s)C f (s)ds M(x, g, C ) = LM (L + M )-1exC f^ es(L+M-C }g(s)ds,

(37)

10

in virtue of the commutativity of L, M, one can write for y £ D(L) = D(M) LM(L + M)-1y = (L + M - C) (L + M)-1Cy,

from which we get

{

L(x, g,C) = (L + M - C) (L + M)-1C fQx e(x-s)C f (s)ds M(x, g,C) = (L + M - C) (L + M)-1CexC fQ1 es(L+M-C)g(s)ds.

Since (L + M - C)(L + M)-1 £ L(X), one has

£(•,g,C) £ Lp(0,1; X), (38)

and

x (L + M - C) ex(L+M-C) /1 es(L+M-C)g(s)ds £ Lp(0,1; X),

Jo

thus

T es(L+M-C>g(s)ds E (D (L + M - C) ,X) p p = (D (C) ,X) p p ,

Jo p p

see (36) and again

x CexC f1 es(L+M-C)g(s)ds E Lp(0,1; X), o

from which it follows that

M^,g,C) E Lp(0,1;X). (39)

For example if P = LM then we mite, for a.e. x E (0,1)

PS(x, fo, f, M) = LMexM fo + LM(L + Mj e(x-s)M f (s)ds

o

x

= LMexMA-ld0 + LM(L + Mj e(x-s)M f (s)ds

o

l

xM l l sL

+ LMe (L + H) A~ (L + M)~l J esL f (s)ds

o

= MexMLa-1 do + L(x, f, M) + (L + H) A-lM(x, f, M),

and since LA-1 jHA-1 e L (X), from (38), (39) and (36) we deduce that

PS (•, fo, f,M) e Lp (0,1; X) ^ Me •M LA-ldo e Lp(0,1; X)

^ LA-ldo e (D(M),X)i p = (D(L),X) i ?

p tF pit

^ a-1 do e (X,D(L))2-Kp

p

^ a-1 do e (X, D(L2))l-p p

2p

A-ldo e (D(L ),X) 1

2p

A-ldo e (D(LM),X)x ,

2p "

The cases P = L2 or M are similarly treated.

For the term S(1 - ., fl} f (1 - .) ,L) we have

□

Proposition 2. Assume (6) and (11)~(17). Let f e Lp(0,1; X) and P e {LM,M2,L2} then

PS(1 -;fl,f (1 - .) ,L) e Lp(0,1; X) ^ Ul e (D(LM),X)A

2P 'p

fl

x

Proof. Assume P = LM (the cases P = M2 and P = L2 are similarly treated), one has PS(1 - x,h,f (1 - .) ,L)

1—x

= LMe(1—x)Lf1 + LM(L + M)—1 J e(1—x—s)Lf (1 - s)ds

0

1—x

= LMe(1—x)LU1 + LM(L + M) — 1 J e(1—x—s)Lf (1 - s)ds

0

1

- LM(L + M)—1e(1—x)L J esM f (1 - s)ds

0

= LMe(1—x)LU1 + L(1 - x,f (1 - •) ,L) - M(1 - x, f (1 - •) ,L),

then by Lemma 3, statement 2

PS(1 - .,f1,f (1 - .) ,L) G Lp(0,1; X) ^ LMe(1—• )LU1 G Lp(0,1; X)

^ U1 G (D(L2),X) i p .

2p '

□

1.6. Main Result for Problem (3)

Theorem 1. Assume (6) and (11)~(17). Let f G Lp(0,1;X) with 1 < p < to. Then

U

A—1do,U1 G (D(LM),X) 1/2p ,p .

U

U

U(x) = S(x,fo,f,M) + S(1 - x,f1,f (1 - .) ,L) (40)

+ R(x,Tf1,M) - R(1 - x,fo + TeLf1,L),

where f0, f1 and T are given in (30), (31) and (29). To conclude it is enough to study the regularity of (40). From Lemma 4, one has

LMR(x,Tf1,M) - LMR(1 - x,fo + TeL fl ,L) G Lp(0,1; X),

and Lemmas 1 and 2 give

LMS(., fo, f, M) G Lp(0,1; X) ^ A-1do G (D(LM, X)^ p

2p ^

LMS(1 - ,fhf (1 - .) , L) G Lp(0,1; X) ^ U1 G (D(LM),X^ . Summarizing, we obtain

LMU G Lp(0,1; X) ^^ A-1 doU G (D(LM),X)1/2p p .

gg Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2015, vol. 8, no. 3, pp. 56-77

!

On the other hand

(L - M) u'(x) = (L - M) MS(x, fo, f, M) + (L - M) LR(1 - x,fo + TeLfi,L)

- (L - M) LS(1 - x,fi,f (1 - .) ,L) + (L- M) MR(x, Tfu M)

= (LM - M2) S(x, fo, f, M) + (L2 - LM) R(1 - x,fo + TeLf1,L)

- IL2 - LM)S(1 - x,fi,f (1 - .) ,L) + (LM - M2) R(x,Tfi,M).

Using again Lemma 4 and Propositions 1 and 2 we obtain

(L - M)U(.) E Lp(0,1; X) ^ A-1do,Ui E (D(LM),X)1/2pp .

u

u

has

u''(x) = M2S(x,fo,f,Mu) - L2R(1 - x,fo + TeLfuL) (41)

+ L2S(1 - x,h,f (1 - .) ,L) + M2R(x, Tfu M) + f (x),

using the fact that M2 + (L - M)M - LM = L2 - (L - M)L - LM c 0 and (28), (41) and (41), we obtain

u''(x) + (L - M)U(x) - LMu(x) = [M2 + (L - M)M - LM] S(x, fo, f, M)

- [L2 - (L - M)L - LM] R(1 - x,fo + TeLfi,L) + [L2 - (L - M)L - LM] S(1 - x,h,f (1 - .), L) + [M2 + (L - M)M - LM] R(x, Tfi, M) + f (x) = f (x).

From (28), we have

u(1) = S(1, fo, f, M) - R(0, fo + TeLh,L) + S(0, f1, f (1 - .), L) + R(1,Tf1, M),

so

u(1) = eMfo + (L + M)-1 e(1-s)Mf (s)ds - eM fo + (L + H) A-leLf J

+f1 + (L + H) A-leL+Mfi = fi + (L + M)-1 e(l-s)Mf (s)ds = ui,

and

u(0) = S(0, fo, f, M) + S(1,fi,f (1 - .) ,L) + R(0,Tfi, M) - R(1, fo + TeLfi,L) = fo + eLfi + (L + M)-i } esLf (s)ds + (L + H) A-ieLfi - eL+M (fo + (L + H) A-ieLfi)

= (.I - eL+M) fo + [I + (I - eL+M) (L + H) A-i] eLfi + (L + M)-i } esLf (s)ds,

o

1

1

so

u(0) = A-1 (I - eL+M) do + A-1 (L + M) eL

+ [A + (I - eL+M) (L + H)] A-1(L + M)-1 J esLf (s)ds.

o

Moreover, we have A + (I - eL+M) (L + H) C L + M and

U1 - (L + M)-1 J e(1-s)Mf (s)ds o

(I - eL+M) (L + H) A-1 + I = (L + M) A

1

then

u(0) = A-1 (I - eL+M) do

+ A-1 (L + M) eL

U1 - (L + M)-1 / e(1-s)Mf (s)ds

+ A-1 / esL f (s)ds,

U(0) G D(H)

u'(0) = MS(0, fo, f, M) + LR(1,fo + TeLf1,L) - LS(1, /1,/(1 - .) , L) + MR(0, TfuM). Therefore

u'(0) - Hu(0) = (M - H) S(0, fo, f, M) + (L + H) R(1, fo + TeLf1, L)

- (L + H) S(1,f1,f (1 - .) ,L) + (M - H) R(0,Tf1,M)

= (M - H) fo + (L + H) eL+M (fo + (L + H) A-1eLf^ - (L + H) eLf1

1

- (L + H) (L + M)-1 / e(1-s)Lf (1 - s)ds + (M - H) (L + H) A-1eLf1

= Afo + (L + H) [(M - H) + eL+M (L + H) - A] A-1eLf1

1

- (L + H) (L + M)-1 / e(1-s)Lf (1 - s)ds

o

1 1 = do + (L + H) (L + M)-1 / esLf (s)ds - (L + H) (L + M)-11 / esLf (s)ds = do.

oo

□

Remark 4. Assume (6) and (11)~(17). If

do G (D(M),X)1/p,p ,U1 G (D(M2),X) 1/2p j

then

A-1do,U1 G (D(M2),X)

1/2p p

since A 1 (X) C D(L) = D(M). ^o ^^^^^^m admits a classical solution u.

1

1

1

2. Go Back to Problem (1)

2.1. Hypotheses on Operators A, B and H

Our essential assumption on operators A, B (which means the ellipticity of our equation) is the following

B2 — A is a linear closed operator in X, ] — 0] С p(B2 — A) and

sup

\>0

A (XI + B2 — A)

< +TO,

L(X)

(42)

(it follows that the operator - (B2 - A)1/2 is the infinitesimal generator of an analytic X

D((B2 - A)1/2) C D(B), (43)

Now, set

L = B - (B2 - A)1/2 and M = -B - (B2 - A)1/2, then, we will assume that

30l, 0m G]0, n/2[: -L G BIP (X, 0l) , -M G BIP (X, 9m) , (44)

-1/2 D(D2 a)-1/2

Vy G D(B), (B2 — Ay' By = B(B2 — А)~чу, (45)

{

VC G D(H), VA G p(L), (L — XI)-1 С G D(H) and (L — XI)-1 HC = H (L — XI)-1

{

VC G D(H), VII G p(M), (M — /iI)-1 С G D(H) and (M — iiI)-1 HC = H (M — III)-1 c.

(46)

(47)

In addition, setting D(X) = D(L) П D(H^d Л = (M — H) + eL+M (L + H), we will also suppose that

Л

Remark 5. Assume (42)~(46). Then

1. D(L) = D(M) = D((B2 — A)1/2) so

D(L — M) = D(L + M) = D((B2 — A)1/2) С D (B), thus L — M С 2B,L + M = — 2(B2 — A)1/^d 0 G p (L + M).

2. D(ML) = D(LM) = D(B2 — A) Mid ML = LM С —A.

For the proof, see Lemma 7 in [4].

2.2. Main Result for Problem (1)

Theorem 2. Assume (6) and (42)~(48). Let f e Lp(0,1; X) with 1 < p < œ. Then Problem (1) has a classical solution u satisfying moreover

u e Lp(0,1; D(B2 - A)) and u' e Lp(0,1; D((B2 - A)1/2),

if and only if

A-1do e (D(B2 - A)),X)i/2pp andui e (D(B2 - A)^)^ . u

Proof. If we assume (42)~(48), then hypotheses (11)~(17) are satisfied with L,M, A defined as in subsection 2.1.Thus we can apply Theorem 1.

□

3. Some Cases in Which Assumption (IT) or (48) are Satisfied 3.1. New Assumptions Implying (17) Proposition 3. Assume (6) and (11)~(16). If

M - H is closed and 0 e p (M - H), (49)

and

| (/ - eL+M)-1 (L + M) eL+M (M - H)-1||£(x) < 1, (50)

then assumption (17) is satisfied and we can apply Theorem 1.

Proof. Since I - eL+M is boundedly invertible (see [19, p. 60]), we can write

A = (M - H) - eL+M [(M - H) - L - M]

= (I - eL+M) \l + (I - eL+M)-1 (L + M) eL+M (M - H)-1l (M - H) = G (M - H)

where G := (I - eL+M)

I + (I - eL+M)-1 (_L + M) eL+M (M - H)-1 e L (X).

Now, 0 e p (G) due to (50). Using (49) we deduce that A = G (M - H) is boundedly invertible.

□

Remark 6.

1. Proposition 3 remains true if we replace (50) by

!

for some n1 e N\{0}

(I - eL+M)-1 (L + M) eL+M (M - H)

1

ni

L(X )

< 1.

2. Assumption (49) can be obtained, for instance, in the following manner : under (6) and (11)~(16), if we assume in addition that

-M g BIP (X,0m) H G BIP (X, 0h), with 0h G (0, n) 0 G p (M) U p (H) and 0m + 0h G (0,n),

then (-M) + H is closed and boundedly invertible (see [14, Theorem 4, p. 441 together with the remark at the end of p. 445]), that is (49).

3. The spectral Problem with a parameter u ^ u0 (where u0 ^ 0 is some fixed number)

u''(x) + 2Bu'(x) + Au(x) - uu(x) = f (x), a.e. x G (0,1), , ,

u'(0) - Hu(0) = d0, u(1) = u1, ^ '

{

is studied in [24], as an application of this paper, setting

AM = A - uI,Lu = B - (B2 - AM)1/2 and M^ = -B - (B2 - AM) Problem (51) becomes

!

u''(x) + (Ьш — Ыш ) u'(x) — Ьш Ыш u(x) = f (x), a.e. x G (0,1), u'(0) — Hu(0) = d0, u(1) = ui,

and, under suitable assumptions, we can apply the results of this paper, replacing L, M by , Mu. In particular, the spectral parameter u is used to obtain (50) for u large enough and then (17) by Proposition 3.

Proposition 4. Assume (6) and (11)~(16). If

M - H is closed and 0 G p (M - H),

and, for some n\ G N\{0}

((L + H) (M — H)

-1\ ni

^ 1,

ax )

-Q 73

3.2. Some Particular Cases

Consider Problem 3 and assume (6) and (11)~(16).

1. If H = -L then A = M + L, and (17) is satisfied.

2. If - 1(L - M) C H then A=1(L + M) (I + eL+M) and again (17) is satisfied.

Similarly, consider Problem 1 and assume (42)~(47). If H = -B + VB2 — A or —B then (48) is satisfied.

4. Applications

Example 1. Consider K such that — K has bounded imaginary powers and 0 G p (K).

Take

L = M = -H = -V—K. Then A = M + L = -2\j-K is boundedly invertible and

-L = (-K)1/2, (-L)it = (-K)it/2 (t G R).

{

Moreover L - M C 0 and -ML = K, so we can apply our main result to the following Problem

u''(x) + Ku(x) = f (x), a.e. x G (0,1) u'(0) - ^/-Ku(0) = d0, u(1) = u1.

For example, take X = Lp (Q), with 1 < p < to, Q a bounded domain in Rra with smooth boundary, and, K = A - cI, (c > 0) with Dirichlet boundary conditions or the conditions described in [16, p. 320]. Then the fractional power \j-A + cI is well defined.

Example 2. Consider X = L2 (R) and L, M operators in X defined by

D (L) = D (M) = H2 (R)

d 2u du „ ^ d2 u

Lu = a-7~— + b——+ cu, Mu = a^-— dy2 dy dy2

(a, b, c G R, a > 0,c< 0),

j duu . d2u du

so that (L - M) u = b——+ cu and (L + M) u = 2^—— + b——+ cu with

dy dy2 dy

0 G p (L + M).

1 ( du \

Take Hu = — b— + cu) ,u G D(H) := H1 (R).

2 V dy J

Then our main result applies to

d2u, . f1 d2u du\ . . ( d4u 7d3u d2u\ . .

dx2(x,y) + {bdyd-x + cdx) (x,y) - a\aw + W + cW) (x,y)

= f (x,y), x G (0,1),y G R du bdu c

dx (0, y) + 2 dy (0,y) + 2 u(0, y) = do (y) , y G R

, u(1,y) = ui (y), y G R.

Of course, this example can be generalized to Rra and differential operators

^ д ( д \ ^, du

Lu = — ¿= aXl [aijдХ,) u + ^aX +cu'

г ,7=1 г=1

(see [14,23]).

Example 3. Choose L, M satisfying (11)~(15) and H satisfying (16), assume moreover

M - H Proposition 4, we have

A = [I + eL+M (L + H) (M - H)"1] (M - H),

with (L + H) (M - H)_1 G L (X) and the invertibility of A is guaranted by the smallness of

||eL+M (L + H) (M - H)"1!!^ .

Now, if we replace L by L$ := L - 51 with 8 > 0 large enough, then L$, M and H satisfy (11)~(17), indeed:

||eL*+M (L* + H) (M - H)"1!!^ ^ ||eLaacw ||eM (L + H) (M - H^H +8 ||eL ||L(X) K (M - h)"1|L(X) < |eL||L(X) |eM (L + H) (M - H) L(X) +8e"S ||eL|L(x) hM (M - H)"1|L(x) ,

and then HeLs+M (L$ + H) (M - H)"1|L(X) < 1 for 8 > 0 large enough.

This immediately apply to the differential operators handled in the papers of Priiss and Sohr quoted above.

References

1. Bourgain J. Some Remarks on Banach Spaces in which Martingale Difference Sequences are Unconditional. Ark. Mat, 1983, vol. 21, pp. 163-168. DOI: 10.1007/BF02384306

2. Burkholder D.L. A Geometrical Characterisation of Banach Spaces in which Martingale Difference Sequences are Unconditional. Ann. Probab., 1981, vol. 9, pp. 997-1011. DOI: 10.1214/aop/1176994270

3. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. Complete Abstract Differential Equations of Elliptic Type in UMD Spaces. Funkcialaj Ekvacioj, 2006, vol. 49, pp. 193-214. DOI: 10.1619/fesi.49.193

4. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. A Simplified Approach in the Study of Elliptic Differential Equations in UMD Spaces and New Applications. Funkcialaj Ekvacioj, 2008, vol. 51, pp. 165-187. DOI: 10.1619/fesi.51.165

5. El Haial A., Labbas R. On the Ellipticity and Solvability of Abstract Second-Order Differential Equation. Electronic Journal of Differential Equations, 2001, vol. 57, pp. 1-18.

6. Favini A., Labbas R., Tanabe H., Yagi A. On the Solvability of Complete Abstract Differential Equations of Elliptic Type. Funkcialaj Ekvacioj, 2004, vol. 47, pp. 205-224.

7. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. On the Solvability and the Maximal Regularity of Complete Abstract Differential. Equations of Elliptic Type. Funkcialaj Ekvacioj, 2004, vol. 47, pp. 423-452. DOI: 10.1619/fesi.47.423

8. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. Etude Unifiée de Problèmes Elliptiques dans le Cadre Hôldérien. C.R. Acad. Sei. Paris, Ser. I, 2005, vol. 341, pp. 485-490.

9. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. Necessary and Sufficient Conditions in the Study of Maximal Regularity of Elliptic Differential Equations in Holder Spaces. Discrete and Continuous Dynamical Systems, 2008, vol. 22, pp. 973-987. DOI: 10.3934/dcds.2008.22.973

10. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Abstract Differential Equations of Elliptic Type with General Robin Boundary Conditions in Holder Spaces. Applicable Analysis, 2012, vol. 91, no. 8, pp. 1453-1475. DOI: 10.1080/00036811.2011.635653

11. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Sturm - Liouville Problems for an Abstract Differential Equation of Elliptic Type in UMD Spaces. Differential and Integral Equations, 2008, vol. 21, no. 9-10, pp. 981-1000.

12. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Complete Abstract Differential Equations of Elliptic Type with General Robin Boundary Conditions in UMD Spaces. Discrete and Continuous Dynamical Systems - Series S, 2011, vol. 4, no. 3, pp. 1-16.

13. Dore G., Venni A. On the Closedness of the Sum of Two Closed Operators. Mathematicsche Zeitschrift, 1987, vol. 196, pp. 124-136. DOI: 10.1007/BF01163654

14. Prüss J., Sohr H. On Operators with Bounded Imaginary Powers in Banach Spaces. Mathematicsche Zeitschrift, 1990, vol. 203, pp. 429-452. DOI: 10.1007/BF02570748

15. Lions J.L., Peetre J. Sur Une Classe D'Espaces D'Interpolation. Publications Mathématiques de l'Institut des Hautes Etudes Scientifiques, 1964, vol. 19, no. 1, pp. 5-68. DOI: 10.1007/BF02684796

16. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam, North Holland, 1978.

17. Favini A., Labbas R., Maingot S., Meisner M. Boundary Value Problem for Elliptic Differential Equations in Noncommutative Cases. Discrete and Continuous Dynamical Systems, 2013, vol. 33, no. 11-12, pp. 4967-4990.

18. Haase M. The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, 2006, vol. 169, pp. 19-60.

19. Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995.

20. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Berlin, Heidelberg, Tokyo, Springer-Verlag, 1983.

21. Engel K., Nagel R. One-Parameter Semigroups for Linear Evolution Equtions, New York, Springer Verlag, 2000.

22. Balakrishnan A.V. Fractional Powers of Closed Operators and the Semigroups Generated by Them. Pacific J. Math., 1960, vol. 10, pp. 419-437. DOI: 10.2140/pjm.l960.10.419

23. Prüss J., Sohr H. Imaginary Powers of Elliptic Second Order Differential Operators in Lp-Spaces. Hiroshima Math. J., 1993, vol. 23, pp. 161-192.

24. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Spectral Parameter Problems with Robin Boundary Coefficient-Operator Conditions in UMD Spaces and Applications. (To appear)

Received December 25, 2014

УДК 517.9 DOI: 10.14529/mmpl50304

ЭЛЛИПТИЧЕСКИЕ ЗАДАЧИ С ГРАНИЧНЫМИ ОПЕРАТОРНО-КОЭФФИЦИЕНТНЫМИ УСЛОВИЯМИ РОБИНА В Lp ПРОСТРАНСТВАХ СОБОЛЕВА И ИХ ПРИЛОЖЕНИЯ

М. Чеггаг, А. Фавини, Р. Лаббас, С. Менго, А. Медегри

В статье доказаны некоторые новые результаты о полных операторно-дифференциальных уравнениях эллиптического типа второго порядка с граничными операторно-коэффициентными условиями Робина в пространстве Lp(0,1; X) в случае, когдаp € (1, а X — банахово UMD-пространство. Доказано существование, единственность и оптимальная регулярность классического решения. Статья дополняет и завершает предыдущие исследования авторов по данной проблематике.

Ключевые слова: абстрактные эллиптические дифференциальные уравнения второго порядка; граничные условия Робина; аналитическая полугруппа.

Мустафа Чеггаг, Национальная политехническая школа (г. Оран, Алжир), [email protected].

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

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

Стефан Менго, лаборатория математики, Университет Гавра (г. Гавр, Франция), [email protected].

Ахмед Медегри, лаборатория теоретической и прикладной математики, Университет Мостаганем (Алжир), [email protected].

Поступила в редакцию 25 декабря 2014 г.