Sturm-Liouville abstract problems for the second order differential equations in a non commutative case Текст научной статьи по специальности «Математика»

MSC 35J20, 35J40

DOI: 10.14529/ mm p 180304

STURM LIOUVILLE ABSTRACT PROBLEMS FOR THE SECOND ORDER DIFFERENTIAL EQUATIONS IN A NON COMMUTATIVE CASE

M. Kaid1, K. Ould Melha1

1 Université Abdelhamid Ibn Badis, Mostaganem, Algérie

E-mail: [email protected], [email protected]

In this paper we prove some new results on Sturm-Liouville abstract problems of the second order differential equations of elliptic type in a new non-commutative framework. We study the case when the second member belongs to a Sobolov space. Existence, uniqueness and optimal regularity of the strict solution are proved. This paper is naturally the continuation of the ones studied by Cheggag et al in the commutative case. We also give an example to which our theory applies.

Keywords: second-order abstract elliptic differential equations; Sturm-Liouville boundary conditions in non commutative cases; analytic semigroup; maximal regularity.

Introduction

In a complex Banach space X consider the following second order differential coefficient-operator equation

u''(x) + Au(x) - uu (x) = f (x), x e (0,1), (1)

together with the abstract Robin boundary conditions

u'(0) - Hu(0) = d0, u(1) = u1. (2)

Here A and H are closed linear operators with domains D(A) and D (H) in X, f belongs to Lp(0,1; X) with 1 < p < œ, do and u1 are given elements of X and u is some large positive number.

u

i) u e W2p(0,1; X) n Lp(0,1; D(A)),

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

iii) u

Consider some fixed u0 ^ 0 and for u > u0 set

AM = A - uI.

This paper is a natural continuation of [1] and [2], where the authors have studied (1) - (2) in a commutative framework, when

f e Lp(0,1; X ) with 1 <p< œ and f e C° ([0,1] ; X ) wit h 9 e ]0,1[.

In [1] authors have assumed that

X is a UMD space, (3)

{

ЛШ0 is a linear closed operator in X, [0, С р (ЛШ0) and

sup \\Л (ЛШ0 — XI)_ Il (x) < (4)

л>о ■

(note that the previous assumption implies that, for all u ^ u0, the operator Qu = — (—Au)1/2 is an infinitesimal generator of a bounded analytic semigroup on X).

{

for any s G R, (—ЛШ0)is G L (X) and there exists вА G ]0,^[ such that

sup\\е-вА^(—ЛШ0)iSL(X) < (5)

ser (X )

{

H is a linear closed operator in X, R_ С p(H) and

sup \m+ci г1^ <

00

(ЛШ0 — XI )_ (H + ci Г1 = (H + CI )~члШо — XI Г1, Л ^ 0 С > 0, (7)

3C ^ 1, Зви G]0, п[: Уз G R, (H)is G L (X) and \His\l(x) ^ CeeHlsl,

{

(8)

Y + ви G]0, n[. (9)

Assumptions (3) - (9) imply that

Qu — H is closed and boundedly invertible, and there exists u* ^ such that, for all u ^ u^ the operator Au defined by

!

D (Лш) = D (Q„) П D (H),

= (Qu — H) + e2Q (Qu + H) ,

is also closed and boundedly invertible, see for instance [1, Proposition 7 and Lemma 8, pp. 987-988].

Under these hypotheses, the authors built the representation formula of the solution of (1), (2) in the form

u (x) = (exQ- -e(2-x)Q") A-1 do + e(1-xQщ + (exQ- -e(2-x)Q") (Qu + H)Л-eQ"u+

+ 1 (eQ - e(2-xQ) Q + H) Л-1Q-1 J (eQ - e(2-sQ) f (s) ds-2 0

1 1 1 x 1 1

e(1-xQq-1 J e(1~*)Q» f (s)ds + -Q-1 J e(x-sQ f (s)ds + -Q-1 J e(s-xQ f (s)ds,

2 0 2 0 2 x

and have proved the following result.

Theorem 1. Assume (3) - (9). Let f G Lp(0,1; X) with 1 < p < ж. Then there exists ш* ^ ш0 such that for all ш ^ ш* the two following assertions are equivalent:

1. Л-Чо,щ G (D (A) ,X) 1 .

2p ^

u

u G W2'p(0,1; X) П Lp(0,1; D(A)), u(0) G D(H) andu satisfies (1), (2).

Moreover, in this case, u is uniquely determined by (10).

Recently, in [3], the authors have developed an interesting new approach concerning some general Sturm-Liouville problems with the same Robin boundary condition in 0. They have assumed that:

• Aw is boundedly invertible.

• A"1 is a regularizing operator in the sense that

A J1 (D(QU)) c D(Ql). (11)

1. New Considerations and Main Result

Consider problem (1), (2). In this work we will suppose that

X is a UMD space, (12)

!

!

A^o is a linear closed operator in X, [0, C p (Awo) and

sup IIA (Awo - XI)-1\\(X) < (13)

a>o " 'll(X)

for any s E R, (—AW0)is E L (X) and there exists 0A E ]0,^[ such that

sup\\e-dAlsl(—AUo)isL(X) < (14)

seR " Jl(x)

and

3v e]0, 1 [, 3C > 0 : V/> 0, Vw ^ uo, D(Au) C D (H) and \\H (Aw — /I)-1\\ (X) <

-i\ < C (15)

l(X) < |w + i\l/2+v'

Q

HA This article is organized as follows. In Section 2 we do some consequences and present preliminary technical results. In Section 3 we recall the representation formula of the

u

some examples of application to boundary value problems.

2. Consequences

Let us write some remarks which follow from the above assumptions. Remark 1.

1. (12) implies that X is reflexive, moreover the operator AWo is sectorial, thus D(A)

X

2. From (13) and (14) we get for any s E R

sup

seR

-(0a/2)\S\

(V-A0)

< +K>.

L(X)

3. By using Lemma 2.6, statement b, p. 103 in G. Dore and S. Yakubov [4], we get 3ko, K1 > 0, 3co > 0 such that Vw ^ 0, || eQ ||L(X)< Koe-2co^

{

and || QMeQ Hc(x)< Kxe-2co^.

(16)

4. Under (13), for all u ^ uo, the operator I — e2Q^ is boundedly invertible (see A. Lunardi [5, Corollary 2.3.7, p. 62]).

5. Suppose that problem (1), (2) has a strict solution u. Then, from above

u e W2'p (0,1; X) n Lp (0,1; D(Ql)) , 1 <p<

which implies

u (0), u (1) e DQ),X) = (X,D(Ql)) 1-± , (17)

2p 2p

(see Grisvard [6, Teorema 2, p. 678]). Recall that, for m e N\ {0}, we have

(D Q), X)i/mpp = Dqrn (1 — 1/mp,p), in virtue of [6]. So, by the well known reiteration Lions theorem, we get

DQZl (1 — 1/mP,P) = dq„ (m — 1/P,P) = dQq (m — 1 + (1 — 1/P) ,P) = = {V e D (Qm-1) : Qm-1<f e (D (Qu) , X)i/pp} = (D (Qu) ,X)(n-i)+i/pp •

In particular, for m = 2:

(D (Ql), X) 1 /2pp = DqZ (1 — 1/2p,p) = (D (Qu) ,X)l+l^p = = { V e D (Qi) : Qi V e (D Q) , X\/pp} C D Q),

from which it follows that

u (0), u (1) e D(Qi).

6. Assumptions (4) and (5) involve that, for u > u0, — Au belong to the class BIP(X, 9) [7, 8, Definition 1, p. 431].

Lemma 1. Assume that (13) and (15) hold. Then there exist constants C > 0 and u1 > u0 such that, for all u ^ u1 operator Qu ± H is boundedly invertible and

-HI C

l(x) л/ш

IIQ ±H) L*) <

Proof. For all ш > ш0 we have

Q-1 = (—V—A—un)-1 = 1 J (A — uIu ~,I>' d,

which implies

n J l1

0

Q-1|I < C

r»^ f(X) ^

ш ) ^ ш1/2'

On the other hand HQ-1 is well defined for all ш > ш0 since, by (15), we have

lHQ- IL(X)

1 f H (A - ш1 - iI)-1 ,

W -W2-dl

0

dl C

<

L(X)

^ C -CZ—.— < —

l1/2 \ш + i\1/2+v шv

0

so there exists u1 > uo such that for all u ^ u1 we have

\\HQ- \\l(x) < 2'

which implies that operator

Qw ± H = (I ± HQ-1) QM

for u ^ u1 is invertible and

-1 -1

± H)-1\L(x) = Q-1 (I ± HQ-1)-1

< \Q-1 \ _1_ < C

^ r(X) 1 II ur\-1 II ^

L(X)

- i|l(X) 1 — HHQ-1^) ^ U1/2

□

1

Lemma 2. Assume that (13) and (15) hold. Then (Qw — H) for u ^ u1 is a linear bounded operator from (D(QU),X)e into itself for all 9 e]0, 1[,q E [1, to].

Proof. Since QM is closed then QM (Qu ± H) 1 is bounded, whence we deduce that

Q ± H)-1 EL(D(QU),d(qu)),

(here D(QW) is a Banach space endowed with the graph norm). So, by the well known interpolation property we get

±H)-1 G C

(Qu ± H )-1 EL ((D(Qw ), X), , q)

where 9 e]0, q E [1, to]. Therefore we deduce the result.

□

Remark 2. Observe that when Qw and (Qu ± H)-1 are commuting for large u, the same proof as above implies that

Qw Q ± H)-1 el ((d(qw), X), , ^ .

Therefore in this work, instead of assumption (11), we suppose

Qw Q ± H)-1 [(D Q) ,X)1/pp] C (D Q) ,X^ , (18)

since operators QM and (Qu ± H)-1 do not commute. This hypothesis is better in some sense than (11). We recall that

(D Q) , X)/p = (x E X : t t1-1/pQu (Qu — tI)-1 x E Lp(R+] X)} ,

see P. Grisvard [6] and [9].

Lemma 3. Assume that (13) and (15) hold. Then there exists u2 ^ uo such that for u ^ u2 the opera tor Aw defined by

{

D (Aw) = D Q) n D (H) = D Q) Aw = Q — H) + e2Q (Qu + H),

is closed, boundedly {-avertible and

{

K1 = Q - H)-1 (I + W) with

W eL (X) an dW (X) с П D(Ql). (19)

Ш

)k\ ' Ш)

k&N\{0}

Proof. Let u ^ uo. In virtue of assumption (15) and Lemma 1 operator A. can be written as follows

Лш = (I- e2Q") Q - H) + 2QUe22Quj = (I - e2Q"}) I + 2 (I - e2Q"})-1 QweQ Q - H)-1] Q - H)

Set

>-11ы e2Q^ rr,-1

Тш = 2 (I - e2Q")~ QMe2Q- Q - H)"

Due to (16) and Lemma 1, we have for ш ^ ш1

UTJ£(X) = || 2 (I — e2Q^~ Q.e2Q (Qu — H)-1 Hcw< < 2 || (I — e2Q^-1 HtWH Q^e2Q- H^H (Qu — H)-1 Kw<

< T—^t^ " ^^ Q — H)-1 h(X)< U1/2 (1 —CZ-2^) ^

which implies the existence of u2 ^ u1 such that for u ^ u2

-CK1 -e-2co^ < 1,

ш1/2 (1 - Koe-2c0yfc)

hence I + T. E L (X) is boundedly invertible. Then from Remark 1, statement 4 and

()

A. = (I — e2Q") (I + T.)(Q. — H) is closed and boundedly invertible with

л-1 = Q - H)-1 (I + Тш)-1 (I + Бш)-1 = = Q - h)-1 [I - тш (I + тш)-1] [I - S„ (I + Бш)-1]

Moreover, for ш ^ ш2.

тт\-1

(20)

Тш = 2e2Q (I - e2Q")~ QM Q - H)-1 е L (X) Бш = -e2Q^ е L (X) and

Тш (X) (X) с П D(Qt),

k&i\{0}

from which we deduce (19).

□

Remark 3. Assume that (13) and (15) hold. It is natural to consider for u ^ uo an operator n. (instead of A.) defined by

{

D (П) = D Q) П D (H) = D Q) , П = (Qu - H) + Q + H) e2Q^, [ZL)

since e2Q-(X) C D(Qu)• And again we obtain that for u ^ u2 the operator nu is closed and boundedly invertible.

Indeed, let u ^ u0. The operator nu can be written as follows

ni = (Qu — H) [I — e2Q-) +2Qie2Q- =

= (Qu — H) [i + 2 (Qu — H)-1 Que2Q- (I — e2Q-)-1j (I — e2Q-)

similarly to Au for u ^ u2 we have

2(Qu — H)-1 Que2Q- (I — e2Q-)

1

L(X)

< 1•

Remark 4. We can use the fact that (Qu — H) and (Qu + H) are invertible by writing

Au

(Qu — H) + e2Q- (Qu + H) =

(Qu — H) (Qu + H)-1 [I + (Qu + H) (Qu — H)-1 e2Q-] Q + H) •

Due to (16), we have

\(Qu + H) (Qu — H )-1 e2Q- \\C{X) < || (Qu + H) (Qu — H y^^) ||e2Q- \\C{X) <

<

I + 2HQ-1 (I — HQ-1)

1 -1

L(X)

K0e-2co^,

and due to Lemma 1, for u ^ u1 we get

2HQ-1 (I — HQ-1)-< 2 WHQ-1!

^2 W^JL (X)

L(X) 11 u nL(X) 1

1

-1)-1

(I — HQ-1) 2C 1

<

L(X)

C(X) 1 —\\HQ-1\\C{X) uv 1 — C

moreover, for u ^ u2

2C 1

u 1 - c

< 1,

whence we deduce that Au is boundedly invertible. Similarly, for u ^ u0, we have

nu = (Qu — H) + (Qu + H) e2Q- =

[I + (Qu + H) e2Q- (Qu — H)-1] (Qu — H)

and similarly to Au for u ^ u2 we obtain

(Qu + H) e2Q- (Qu — H)

1

(I + HQ-1) e2Q- (I — HQ-1)

L(X) 1

L(X)

< 1.

Let us compare Au and nu. First we have D (Au) = D (nu) = D (Qu), moreover

n

Qu — H + (Qu + H) e2Q- =Au — e2Q- (Qu + H) + (Qu + H)

,2Q-

Au — e2Q-H + He2Q- = Au + [H; e2Q-]

(22)

математическое моделирование

where, for all £ е D(H) = D (Qu)

[H; e}Quj] £ = He2Q"£ - e2Q" H£.

Then

П = (I + [H; e2Q»] Л-1) Лш. Using (20) and due to Lemma 1, for ш ^ ш2 we have

11Л- ^C(X) — ||(Q - H) IIl(x) II (I + T) ^C(X) II (I + SШ) ^C(X) —

— ||(Q - h)-1Il(x) t-WJLX) t-ШХ —

— ||(Q - H) 11 L(X) 1 2CK1 e-2c0y^ 1 _ Kne-2coVШ —

1 ui/2(1-Koe-2°oV*)e 1 K0e

C

— ||(Q - H)-1IIC(X) — Ш/2 ■

Due to (16), we get

llHeQл-111 <IIhq-1II IIq e2Q^ll 11л-111 <

llHe ЛШ Wc(X) — ||HQw Wc(X) IlQe \\c(X) I1ЛШ Wc(X) —

— C-, K1e-2c0 — C

ш1/2 - ш"+1/2'

In the other hand,

lleQ H Л-1|| <

lle H Лш \\c(X) — < lle2Q" 1C(x) 1 H (Q - H) 11 L(X) |(I + Тш) 11L(X) 11 (I + S'Ш) 11L(X) —

— Koe-2c°^ ||H Q - H)-1H£ < Koe-2c°^ ||HQ(I - HQ-1 )-1"

\C(X) — 0 11 L(X)

11 1 C

— CK0e~2co^--,, ^ 1M-— CK0e-2co^— — —.

- 0 ш" 1 - WHQ^Wlx) - 0 W- шv

<

L(X)

Hence

||[H• e2Q*>] Л-1| <<CII H • e J ЛШ \\c(X) — ши ' so there exists ш3 ^ ш2 such that for all ш ^ ш3 we have

C — 1,

which confirm the invertibility of ПШ whith

П-1 = л-1 (I + [H• e2Q] Л-1)-1. (23)

Similarly, due to (22), we obtain

Л-1 = П-1 (I - [H; e2Q] П-1)-1. (24)

Lemma 4. [10, Theorem, p. 96] Assume that (13) holds. Let p е ]1, е X and

n е N*. Then, we have

1. eQ- V e LP (0,1,X),

2. QZ eQ- v e Lp (0,1,X) if and only if v e (D (Qnu) ,X) i •

np '"

Lemma 5. Assume that (12) - (14) hold. Then for f e Lp (0,1,X) with 1 < p < we have

1. x — L (x,f ) = Quf e(x-s)Q- f (s) ds e LP (0,1,X),

0

i

2. x — L (1 — x,f (1 — •)) = Quf e(s-x)Q- f (s) ds e LP (0,1,X),

x

i

3.x — L (x, f ) = Quf e(x+s)Q- f (s) ds e LP (0,1,X),

0

4. f esQ- f (s)ds, f e(1-s)Q- f (s)ds e (D(QU),X) 1

Proof. For statements 1, 2 and 3, see [3, 11, 12, pp. 167, 168], and also [13, (24), (25) and (26)]. Statement 4 is an easy consequence of statements 1 and 2, we proceed as in

x 1

Remark 1 by using the fact that x — f e(x-s)Q- f (s)ds and x — f e(s-x)Q- f (s)ds belong

0x

to W1 'p (0,1,X) n Lp (0,1, D (Qu)).

□

3. Representation of the Solution

Assume that (13) and (15) hold. Consider for a.e. x E (0,1) the following formula

u (x) = (exQ- - e(2-x)Q") A-1 do + e(1-x)Q-ui + +Q-1 (exQ- - e(2-x)QQ + H)A-1QUIeQ-ui+

1

+2Q-1 (exQ- — e(2-x)Q-) (Qu + H) A-1 J (esQ- — e(2-s)Q-) f (s) ds—

0

1 x 1

— 2e(1-x)Q-Q-1 J e(1~s)Q- f (s) ds + 2Q-1 J e(x-s)Q- f (s) ds + ±Q-1 J e(s~x)Q- f (s) ds,

00x

used in Cheggag et al [1] in the commutative case.

The main idea is in searching a solution u to (1), (2) for a.e. x e (0,1), in the following form:

u (x) = (exQ- — e(2~x)Q-) A-1d* + e(1-x)Q-u\ +

+Q-1 (exQ- — e(2~x)Q-) (Qu + H) A-1QueQ-u\ +

1

+ 2Q-1 (exQ- — e(2~x)Q-) (Qu + H) A-1 J (esQ- — e(2~s)Q-) f * (s) ds— ^

0

1 x 1

— 1 Q-1e(1-x)Q- Je(1~s)Q- f *(s) ds + 1Q- Je(x-s)Q- f *(s)ds + 1Q- J e(s-x)Q- f *(s) ds.

0 0 x

Taking into account the boundary conditions (2) deduce d0,u\ and f *.

It is easy to obtain

u*1 = u(1) = щ. (26)

Now, from (25) for a.e. x E (0,1), we have

u' (x) = (exQ- + e(2-x)Q-) Л-Ч*0 - Que(1-x)Q-ui + + (exQ- + e(2-x)Q- ) (Qi + H) Л-1дшeQ-ui+ 1 i + - (exQ- + e(2-x)Q-) (Qi + H) Л-V (esQ- - e(2-s)Q-) fl (s) ds+ (27)

2 о

1 1 1 x 1 1

+ _e(1-x)Q-f e(1-s)Q- fl (s) ds + - / e(x-s)Q- fl (s) ds — / e(s-x)Q- fl (s) ds,

2 о 2 о 2 x

and

u'' (x) = Qi (exQ- - e(2-x)Q-) + Qie(1-x)Q-u1+ +Qi (exQ- - e(2-x^Q-) (Qi + H) Л-1QШeQ-щ + 1 1 +ÖQi (exQ- - eß~x)Q-) (Qi + H) Л-11 / (esQ- - e(2-s)Q-) fl (s) ds-

r, -V l

2о

u (0) = (I - e2Q-) Л-Ч*о + Q-1 [I + (I - e2Q-) (Qi + H) Л-1] QweQ-u1 +

яe

1~..........2Q^ , ^,-Ц1'

+ÖQ-1 [I + (I - e2Q") (Qi + H) Л-1] / (esQ- - e(2-s')Q-) f (s) ds.

2 0 Note that

I + (i - e2Q-) (Qi + H) Л-1 = [Лш + Qi + H - e2Q- (Qi + H)] Л-1 = [Qi - H + e2Q- (Qi + H) + Qi + H - e2Q- (Qi + H)] Л-1 = 2QWЛ,

u

(28)

1 1 1 x

—Q .e(1-x)Q" f e(1-sQ f * (s) ds + -Q. J e(x-sQ f * (s) ds+

2 o 2 o

+1 Q.I e(s-xQ f * (s) ds + f * (x).

2x

Since A. = —Q. and in virtue of (25), it is easy to see that

u" (x) + A.u(x) = f * (x) , therefore, for a.e. x E (0,1), we deduce

f * (x) = f (x). (29)

Now, for dO we have

then

1

(0) = (I - e2Q") Л-Ч*0 + 2Л-^ШeQ-щ + Л-11 / (esQ- - e(2-s)Q-) f (s) ds. (30)

In virtue of assumption (15) and due to Lemma 3, we deduce that u (0) E D(H). Applying H

1

Hu (0) = H (I-e2Q") Л-Ч0 + 2HЛ-QieQ-u + HЛ- $ (esQ--e(2-s)Q-) f (s) ds. (31)

о

x=0

u' (0) = Qu (I + e2Q-) A-1d0 + [(I + e2Q-) Q + H) A-1 — I] QueQ-u1+ 1 1

+ r(I + e2Q-) (Qu + H) A-1 — I] J (esQ- — e(2~s)Q") f (s) ds. 20

On the other hand, we have

(I + e2Q-) (Qu + H)A-1 — I = [Qu + H + e2Q- (Qu + H) — AJ A-1 = --[Qu + H + e2Q- (Qu + H) — (Qu — H) — e2Q- (Qu + H)] A-1 = 2HA-1.

Then we get

u' (0) = Qu (I + e2Q-) A-1d*0 + 2HA-1QueQ-u1+ 1

1

(32)

+HA-1 f (esQ- — e(2-s)Q-) f (s) ds. 0

Therefore, using (31) and (32), we conclude that

u' (0) — Hu (0) = [Qu (I + e2Q-) — H (I — e2Q-)] A-1d00 = nuA-1d*0 = d0• Then, due to Eemak 3 and (23), we obtain

d00 = Aun-1d0 = (I + [H; e2Q-] A-1)- d0• (33)

u

u (x) = (exQ- —e(2-x)Q-) A- (I+ [H; e2Q-] A-1)-1 d0 + e(1-)Q-u+

+Q- (exQ- — e(2-x)Q-) (Qu + H)A-1QueQ-u+

1 ( ) 1 ( )

+ oQ- (exQ- — eS2-x)Q-) (Qu + H)A- J (esQ- — e(2~s)Q-) f (s) d— 20

1 1 1 x 11

— 0Q-1e(1-x)Q^ e(1-s)Q- f (s) ds + -Q-1 f e(x-s)Q- f (s) ds + -Q-1 J e(s-x)Q- f (s) ds.

2 0 2 0 2 x

H; e2Q-

becomes zero and then the representation formulas of solution (10) and (39) coincide.

u

u (•) = S1(;d0,u1) + S2(• ,u1, f) + D(,f) — R(;d0,u1,f), (34)

x e (0, 1)

S1(x,d0,u1) = exQ-A-1 (I + [H; e2Q-] A-1)-1 d0 + e(1-x)Q-uu (35)

S2(x,u1,f) = Q-1exQ- (Qu + H) A-1QueQ-u1+

+1 Q-1exQ- (Qu + H) A-1 f (esQ- — e(2-s)Q-) f (s) ds, (36)

20

(37)

1 1 1 x

D(x, f) = — e,(1~x)Q-Q-1 J e(1-s)Q- f (s) ds + -Q-1 J e(x-s)Q- f (s) ds+ 2 0 2 0 1 1 +2Q-7 e(s-xQ f (s) ds,

2x

and

R(x, do,uu f) = e(2-xQA-1 (I + [H; e2Q^ A-1)-1 do+

+e(2-xQ Q-1 (Qu + H) A-1QMeQui+ 1 i + -e(2~xQQ-1 Q + H) A~Jf (eQ - e(2~s)Q") f (s) ds. 2 0

On the other hand, due to (24), we have

т-1

(38)

u (x) = )xQ-eJ2~x)Q-) n-1d0+e(1-xQu+ +Q- (exQi°-eS2-xQ) (Qu + H )I- (I - [H; eQ ] П-1)"1 Qu eQ- u+

+ 1Q-1 (exQ- — e(2-x)Q-) (Qu + H) n-1 (I — [H; e2Q-]n-1)-1 f(esQ- —e(2-s)Q-) f (s)d+ (39) 20

1 1 1 x 11

— -Q-1e(1-x)Q- J e(1-s)Q- f (s) ds + -Q-1 J ex-s)Q- f (s) ds + -Q-1 J e(s-x)Q- f (s) ds.

2 0 2 0 2 x

4. Main Result

Under assumptions (12) - (15) and (18), we focus on the study of the optimal regularity of the strict solution given by (39).

Lemma 6. Assume that (13), (15) and (18) hold. For any v e (D (Qu) , X) 1 p, p e ]0,

p 'p

and u ^ u2, we have

(Qu ± H)A- v e (D (Qu) , X) ip .

p

Proof. We have a representation

(Qu + H) A-1 = 2QuA-1 + e2Q- Q + H) A-1 — I, which, by Lemma 3, can be written as follows

(Qu + H) A-1 = 2Qu (Qu — H)-1 (I + W) + e2Q- (Qu + H) A-1 — I,

where W (X) C f| DQ) C (D (Qu) ,X) i p.

fcen\{0} p'

Then, due to assumption (18), for any v e (D (Qu) ,X) i p, we deduce that

p 'p

(Qu + H)A-1v e (D (Qu), X) i p.

p

On the other hand, we also have

(Qu - H)A-1 = I - eQ (Qu + H) A-1

Thus, for any ф E (D (Qu) ,X) i we get (Qu - H) А-1ф E (D (Qu) ,X) i

p ^ p^

□

Lemma 7. Assume that (13) and (15) hold. Let d0, u1 E X, и ^ ш2 and f E Lp(0,1;X), 1 <p< то. Then

AuR(,do,uuf) E Lp(0,1; X).

Proof. Since A = -Qi, e(2-Q = eQ-e(1~)Q- and eQ-£ E Q™=1 D (Qi) for all £ E X taking into account (38), the term

AiR(x,do,u1,f) = -Qie(2-xQQiA-1 (/ + [H; e2Q-} A-1)-i d0--Qie(2-x)Q- (Qi + H) A-1QieQ-u-

1 i ^ (40)

-Qie(2-xQ (Qi + H) A-1 _f (esQ- - e(2-s)Q-) f (s) ds = QieQ-e(1-x)Q-£,

0

is bounded and thus belongs to Lp(0,1; X), 1 < p < to for all £ E X.

□

Remark 5. As we have seen, the term R(-, d0,u1, f) is regular, but this can not be applied to the term eQ- (Qi + H) A-1eQ-Qiu1 in S2 (•,u1,f) since operators (Qi + H) A-1 and eQ- are not considered to be commutative.

Remark 6. Assume that (12) - (14) hold and f E Lp (0,1; X) with 1 < p < to. Then, by using Lemma 5, we can easily see that

AiD(,f) E Lp (0,1; X),

see for instance [1].

Lemma 8. Assume that (12) - (15) hold and (18) and f E Lp (0,1; X), 1 < p < to. Then, for all u ^ u2

AiS2(.,u1, f) E Lp (0,1; X) , 1 <p< to.

Proof. Taking into account (36) and A^^ = -Qi, for a.e. x E (0,1) we can write

AiS2(x,u1,f) = -QiexQ- (Qi + H)A-1eQ-Qu -

1

-1 QiexQ- (Qi + H) A-1 J (esQ- - e(2-s)Q-) f (s) ds.

0

Since

1

eQ- Qiu1, J (esQ- - e(2-s)Q-) f (s) ds E (D (Qi), X) 0

then, due to Lemmas 4 and 6, we deduce that

1 ,P p

AS2(.,f) E Lp (0,1,X), 1 <p< to.

□

Theorem 2. Assume that (12) - (15) and (18) hold. Let f E Lp (0,1; X) with 1 <p< to and u ^ u2. Then, the following assertions are equivalent

u

u E W 2'p (0,1; X) n Lp (0,1; D (A)) , 1 <p< to,

u

2. n-ldo, ui Е (D(A),X)ip where

2p '

П = Qi - H + Q + H) e2Qi^

Proof, a) Let us begin with a uniqueness result. Let u, u be strict solutions of (1), (2), then v = u — U is a strict solution of

{

v"(x) + Ашv(x) = 0, a.e. x Е (0,1), V(0) - Hv(0) = 0, v(1) = 0.

Then v E C 1([0,1]; X) and there exist yo,z1 E D(Qu) such that, for any x E [0,1]

v (x) = exQ-yo + e(1-x)Q-zi, see Section 3, in [1]. Moreover, for any x E [0,1]

v' (x) = QuexQ- yo - Que(1-xQ zi.

{

Note that v (0) = y0 + eQ- z1 E D (Qu) n D (H) and using the boundary conditions (2), we get

v' (0) - Hv (0) = Qyo - QueQ-zj - H(yo + eQ-zj = 0, v (1) = eQ-y0 + z1 = 0.

From z1 = -eQ- y0 we deduce

0 = (Quyo + Que2Q-yo) - H (yo - e2Q-yo) = = [Qu (I + e2Q-) - H(I - e2Q-)] yo = nuyo,

and due to Remark 3, for all u ^ u2 we get yo = 0 and then z1 = 0. So, v = 0 and thus u = U.

b) Consider u given by (39). Let us show that Auu E Lp (0,1; X) with 1 < p < to. In fact, due to Lemmas 7 and 8 and from Remark (6), it is enough to prove that AuS1(.,do,u1) E Lp (0,1; X). From (35) for almost every x E (0,1) one has

AuS1(x,do,u1) = -QlexQ-n-1do - Qle(l-x)Q-un. From (17), we have u1 E (D(Q2u),X)and thus, due to Lemma 4, statement 2 we get

Que(1~)Q"u1 E Lp(0,1;X), 1 <p< to. Using again Lemma 4, statement 2, we conclude that

AuS^J) E Lp(0,1; X)

if and only if

QuexQ-n-1do E Lp(0,1; X),

and thus

n-ldo Е (D(Ql),X),p = (D(A),X)

□

5. Applications

In this section we give some applications for our abstract results. Example 1. Let X = L2(]0,1[). Consider operators Qu and H in X, defined by

D(Qu) = D(H) = {V e H2(]0, 1[):v (0) = V (1) = 0} , (QuV) (y) = V' (y) + a (y) v' (y) — V^V (y), y e (0,1) , (Hv)(y) = —V'' (y), y e (0,1) •

Suppose that a e C2 ([0,1]) with a (0) = a (1) = 0 and u > 0 (large enough).

1. It is not difficult to prove that Qu, Qu — H and H are boundedly invertible. Therefore

{

2(

(Qu — H)v (y) = 2v'' (y) + a (y) v' (y) — V^V (y), y e (0,1)

D(Qu — H) = {V e H2(]0,1[) : V (0) = V ( 1) = 0}

u

(see for instance Cheggag et al. in [3]).

2. By using A. Lunardi [5, Theorem 3.1.3, p. 73], we obtain that the operator Qu — (—Au)l/2 is well defined and generates an analytic semigroup. We also have

D(Au) = D(Q2U) = {v e H4(]0,1[) : v'' (0) = v'' (1) = V (0) = V (1) = 0} ,

and

AuV (y) = —Q2uV (y) = —V(4) (y) — 2a (y) v(3 (y) — (a2 (y) + 2a' (y) — 2/u) v'' (y) — — [a'' (y) + a (y) (a' (y) — 2^u)] v' (y) — uV (y) •

On the other hand, there exist conxtants 5, C > 0 such that for z e S$ {z e C\ {0} : |arg z| < n/2 + 5} we have

* -111 . C

\(Au zI) uL2(]0,m <

lL2(]0'1[)~ ^Vu + zl' y e (0, 1)

y 1

(H(y) = (1 — y) i s^(s)ds + y i(1 — s)^(s)ds,

so that

then we get

(QuH) (y) = Qu (H) (y) = (H(y) + a (y) (H(y) — y/u (H-1 ^) (y),

y 1

1

(QuH(y) = —<$(y) — a(y) f s^(s)ds + a(y)J(1 — s)^(s)ds—

0 y

i y 1

— J (1 — y) s^(s)ds + f y(1 — s)^(s)ds ) .

0y

Indeed, we have

H-iQ.^ (y) = H-i Q(y) = y 1

= (1 - y) J s Q(s)ds + y j(1 - s) Q(s)ds =

0 y

y y y

= (1 — y) J stf' (s) ds + J a (s) (s) ds — y/u J ^ (s) ds+

0 0 0 i i i

+y J(1 — s)^" (s) ds + J a (s) (s) ds — y/u J ^ (s) ds =

y y y

y i

= (y) + (1 — y) J sa (s) ^ (s) ds + y J(1 — s)a (s) ^ (s) ds—

o y

(y i

(1 — y) J s^ (s) ds + y J(1 — s)-0 (s) ds

0y

and deduce that

E D(QU), QuH-1^ = H-1QU^. Therefore, all our assumptions are satisfied. We have the following: Proposition 1. Letp E ]1, œ[, f E Lp(0,1; L2(]0,1[)) and

n-ldo, ui E (H4(]0,1[),L2(]0,1[))x p ,

2p

where

nw = Qu — H + Q + H) e2Q-. Then, there exists u* > 0 such that for all u > u* the problem

' d2u d4u d3u d2u

Sx2 (x,y) — (x,y) — 2a (y) dyfi (x,y) — (a2 (y) +2a' (y) — 2vU) dy2 (x,y)—

du

- [a" (y) + a (y) (a' (y) - 2^/ш)] — (x,y) - u-u(x,y) = f (x,y), x,y Е (0,1), du д 2u

^(0,y) + (0,y) = do (y)' y Е (0'1)' (41)

u(1,y) = ui (y) , y Е (0,1), u(x, 0) = u(x, 1) = 0, д2u d2u

w(x, 0) = W(x, 1) = 0'

u

u Е W2'p (0, 1; L2 (]0, 1[)) n Lp (0, 1; H4(]0, 1[)) , 1 <p< то,

u

References

1. 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.

2. 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: an International Journal, 2012, vol. 91, no. 8, pp. 1453-1475.

3. Cheggag M., Favini A., Labbas R., Maingot S., Ould Melha K. New Results on Complete Elliptic Equations with Robin Boundary Coefficient-Operator Conditions in Non Commutative Case. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2017, vol. 10, no. 1, pp. 70-96. DOL 10.14529/mmpl70105

4. Dore G., Yakubov S. Semigroup Estimates and Noncoercive Boundady Value Problems. Semigroups Forum, 2000, no. 60, pp. 93-121.

5. Lunardi A. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Basel, Birkhaüser, 1995.

6. Grisvard P. Spazi di tracce ed applicazioni. Rendiconti di Matematica. Serie VI, 1972, vol. 5, pp. 657-729. (in Italian)

7. Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A. Elliptic Problems with Robin Boundary Coefficient-Operator Conditions in General Lp Sobolev Spaces and Applications. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 56-77. DOI: 10.14529/mmpl50304

8. Prüss J., Sohr H. On Operators with Bounded Imaginary Powers in Banach Spaces. Mathematische Zeitschrift, 1990, vol. 203, no. 3, pp. 429-452.

9. Grisvard P. Commutativité de deux foncteurs d interpolation et applications. Journal de mathématiques pures et appliquées, 1966, no. 45, pp. 143-290.

10. Triebel H. Interpolation Theory, Functions Spaces, Differential Operators. Amsterdam, N.Y., Oxford, North-Holland, 1978.

11. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. A Simplified Approach in the Study of Elliptic Differential Equation in UMD Space and New Applications. Funkcialaj ekvacioj, 2008, vol. 51, no. 2, pp. 165-187.

12. Dore G., Venni A. On the Closedness of the Sum of Tow Closed Operators. Mathematische Zeitschrift, 1987, vol. 196, no. 2, pp. 189-201. DOI: 10.1007/BF01163654

13. Favini A., Labbas R., Maingot S., Tanabe H., Yagi A. Complete Abstract Differential Equation of Elliptic Type in U.M.D Spaces. Funkcialaj ekvacioj, 2006, vol. 49, no. 2, pp.193-214.

Received June 3, 2018

УДК 517.9

БО!: 10.14529/ ттр180304

АБСТРАКТНЫЕ ЗАДАЧИ ШТУРМА-ЛИУВИЛЛЯ

ДЛЯ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ВТОРОГО ПОРЯДКА

В НЕКОММУТАТИВНОМ СЛУЧАЕ

Хенд Магомет1, Хеллаф Ульд Мелха1

^Университет имени Абдул Хамид Ибн-Б ЭДИС сЦ г. Мостаганем, Алжир

В статье доказываются некоторые новые результаты о задаче Штурма - Лиувилля для дифференциальных уравнений эллиптического типа второго порядка в некоммутативном случае. Исследование выполнено при условии, что второй член принадлежит пространству Соболева. Доказано существование, единственность и оптимальная регулярность строгого решения. Работа является продолжением исследований в коммутативном случае М. Чегага, А. Фавини, Р. Лаббаса, С. Менго и А. Медегри. В работе рассматривается пример приложения построенной абстрактной теории.

Ключевые слова: эллиптическое дифференциальное уравнение второго порядка; задача Штурма - Лиувилля в некоммутативном случае; аналитическая полугруппа; максимальная регулярность.

Хенд Магомет, лаборатория чистой и прикладной математики, Университет имени Абдул Хамид Ибн-Бадиса (г. Мостаганем, Алжир), [email protected].

Хеллаф Ульд Мелха, лаборатория чистой и прикладной математики, Университет имени Абдул Хамид Ибн-Бадиса (г. Мостаганем, Алжир), [email protected].

Поступила в редакцию 3 июня 2018 г.