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 AM  Vol.12 No.6 , June 2021
Quasilinear Degenerated Elliptic Systems with Weighted in Divergence Form with Weak Monotonicity with General Data
Abstract: We consider, for a bounded open domain Ω in Rn and a function u : Ω → Rm, the quasilinear elliptic system: (1). We generalize the system (QES)(f,g) in considering a right hand side depending on the jacobian matrix Du. Here, the star in (QES)(f,g) indicates that f may depend on Du. In the right hand side, v belongs to the dual space W-1,P’(Ω, ω*, Rm), , f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ, but with only very mild monotonicity assumptions.

1. Introduction

In this paper, the main point is that we do not require monotonicity in the strict monotonicity of a typical Leray-Lions operator as it is usually assumed in previous papers. The aims of this text are to prove analogous existence results under relaxed monotonicity, in particular under strict quasi-monotonicity. The main technical tool we advocate and use throughout the proof is Young measures. By applying a Galerkin schema, we obtain easily an approximating sequence u k . The Ball theorem [1] and especially the resulting tool mode available by Hungerbühler to partial differential equation theory give them a sufficient control on the gradient approximating sequence D u k to pass to the limit. This method is used by Dolzmann [2], G. J. Minty [3], H. Brezis [4], H. E. Stromberg [5], Muller [6], J. L. Lions [7], Kristznsen, J. Lower [8], M. I. Visik [9] and mainly by Hungurbühler to get the existence of a weak solution for the quasi-linear elliptic system [10]. This paper can be seen as generalization of Hungurbühler and as a continuation of Y-Akdim [11].

This kind of problems finds its applications in the model of Thomas-Fermis in atomic physics [12], and also porous flow modeling in reservoir [13].

2. Priliminaries

Let ω = { ω i j ; 0 i n ; 1 j m } the weight function systems defined in Ω and satisfying the following integrability conditions:

ω i j L l o c 1 ( Ω ) , ω i j 1 p 1 L l o c 1 ( Ω ) , for some p ] 1, [ and s > 0 such that ω i j s L 1 ( Ω ) . (2.1)

with ω * = { ω i j * = ω i j 1 p , 0 i n , 1 j m } , σ = ( σ r s ) with 1 s n , 1 r m and which satisfies some hypotheses (see below).

We denote by I M m × n the real vector space of m × n matrices equipped with

the inner product M : N = i j M i j N i j .

The Jacobian matrix of a function u : Ω I R m is denoted by D u ( x ) = ( D 1 u ( x ) , D 2 u ( x ) , , D n u ( x ) ) with D i = / ( x i ) .

The space W 1, p ( Ω , ω , I R m ) is the set of functions

{ u = u ( x ) / u L p ( Ω , ω 0 ¯ , I R m ) } , D i j u = u i x j L p ( Ω , ω i j , I R m ) , 1 i n , 1 j m .

with

L p ( Ω , ω i j , I R m ) = { u = u ( x ) / | u | ω i j 1 p L p ( Ω , I R m ) }

The weighted space W 1, p ( Ω , ω , I R m ) can be equipped by the norm:

u 1 , p , ω = ( j = 1 m Ω | u j | p ω 0 j d x + 1 i n , 1 j m Ω | D i j u | p ω i j d x ) 1 p

where ω 0 ¯ = ( ω 0 j ) and 1 j m . the norm . 1, ω , p is equivalent to the norm | | | | | | , on W 0 1, p ( Ω , ω , I R m ) , such that, | | | u | | | = ( 1 i n , 1 j m Ω | D i j u | p ω i j d x ) 1 p .

Proposition 2.1 The weighted Sobolev space W 1, p ( Ω , ω , I R m ) is a Banach space, separable and reflexive. The weighted Sobolev space W 0 1, p ( Ω , ω , I R m ) is the closure of C 0 ( Ω , ω , I R m ) in W 1, p ( Ω , ω , I R m ) equipped by the norm . 1, p , ω .

Proof: The prove of proposition is a slight modification of the analogous one in [14] [Kufner-Drabek].

Definition 2.1 A Young measure ( ϑ x ) x Ω is called W 1, p -gradient young measures ( 1 p < ) if it is associated to a sequence of gradients D u k such that u k is bounded in W 1, p ( Ω ) . The W 1, p -gradient young measures ( ϑ x ) x Ω is called homogeneous, if it doesnt depend on x, i-e, if ϑ x = ϑ for a.e. x Ω .

Theorem 2.1 (Kinder Lehirer-Pedregal) let ( υ x ) x Ω , be a family of probability measures in ( C ( M m × n ) ) . Then, ( υ x ) x Ω is W 1, p Young measures if and only if:

1) There is a u W 1, p ( Ω , I R m ) such that D u ( x ) = M m × n A d ϑ x ( A ) , a.e in Ω .

2) Jensens inequality: ϕ ( D u ( x ) ) M m × n ϕ ( A ) d ϑ x ( A ) hold for all ϕ X p quasi-convex, and.

3) The function: ψ ( x ) = M m × n | A | p d ϑ x ( A ) L 1 ( Ω ) . Here, X p denotes the (not separable) space: X p = { ψ C ( M m × n ) : | ψ ( A ) | c × ( 1 + | A | p ) , forall A M m × n } .

proof: see [15].

Theorem 2.2 (Ball) Let Ω I R n be Lebesgue measurable, let K I R m be closed, and let u j : Ω I R m , j I N , be a sequence of Lebesgue measurable functions satisfying u j K , as j , i.e. given any open neighborhood U of K I R m , lim j | x Ω : u j ( x ) U | = 0 . Then there existsa subsequence u k of u j and a family ϑ x , x Ω , of positive measures on I R m , depending measurablyon x, such that

1) ϑ x M = I R m d ϑ x 1 , for a.e x Ω .

2) s u p p ϑ x K for a.e x Ω .

3) f ( u k ) ϑ x , f = I R m f ( λ ) d ϑ x ( λ ) in L ( Ω ) for each continuous functions f : I R m I R satisfying

lim f ( λ ) = 0, | λ | [1].

Theorem 2.3 (vitali) Let Ω I R n be an open bounded domain and let u n be a sequence in L p ( Ω , I R m ) with 1 p < .

Then u n is a cauchy sequence in the L p -norm if and only if the two following conditions hold:

1) u n is cauchy in measure (i.e. ε > 0 , | { x Ω , | u n ( x ) u m ( x ) | ε } | = 0 as m , n .

2) | u n | p is equiintegrable i.e.:

( s u p n Ω | u n | p d x < and ε > 0 , δ > 0 such that E | u n | p d x < ε for all n whenever E Ω and | E | < δ ). Note that if u n converges pointiest, then u n is cauchy in measure.

Hypotheses (H0) (Hardy-Type inequalities): There exist some constant c > 0 , some weighted function γ and some real q ( 1 < q < ) such that,

( j = 1 m Ω | u j ( x ) | q γ j ( x ) d x ) 1 q c ( 1 i n , 1 j m Ω | D i j u | p ω i j ) 1 p ,

for all u W 0 1, p ( Ω , ω , I R m ) , with γ = { γ j / 1 j m } .

The injection W 0 1, p ( Ω , ω , I R m ) ↪↪ L q ( Ω , γ , I R m ) is compact, and W 0 1, p ( Ω , ω , I R m ) ↪↪ L r ( Ω , I R m ) is compact, (by [14] ) with

{ 1 r n p s n ( s + 1 ) p s if p s n ( s + 1 ) r 1 if n ( s + 1 ) p s

(H1) Continuity: σ : Ω × I R m × I M m × n I M m × n is a Carathéodory function (i-e x σ ( x , u , F ) is measurable for every ( u , F ) I R m × I M m × n and ( u , F ) σ ( x , u , F ) is continuous for almost every x Ω ). (H2) Growths and coercivity conditions: There exist c 1 0 , c 2 > 0 , λ 1 L p ( Ω ) , λ 2 L 1 ( Ω ) , λ 3 L ( p / α ) ( Ω ) , 0 < α < p , 1 < q < and β > 0 such that for all 1 r n , 1 s m , we have:

| σ r s ( x , u , F ) | β w r s 1 / p [ λ 1 ( x ) + c 1 j = 1 m | γ j | 1 / p | u j | q / p + c 1 1 i N , 1 j m ω i j 1 / p | F i j | p 1 ] (2.2)

and

σ ( x , u , F ) : F λ 2 ( x ) j = 1 m ω 0 j ( x ) α / p λ 3 ( x ) | u j | α + c 2 1 i n , 1 j m ω i j ( x ) | F i j | p (2.3)

(H3) Monotonicity conditions: σ satisfies one of the following conditions:

1) For all x Ω , and all u I R m , the map F σ ( x , u , F ) is a C 1 -function and is monotone (i-e, ( σ ( x , u , F ) σ ( x , u , G ) ) : ( F G ) 0 , for all x Ω , all u I R m and all F , G I M m × n ).

2) There exists a function W : Ω × I R m × I M m × n I M m × n such that σ ( x , u , F ) = W F ( x , u , F ) and F W ( x , u , F ) is convex and C 1 function.

3) For all x Ω , and for all u I R m the map F σ ( x , u , F ) is strictly monotone (i.e., σ ( x , u ,. ) is monotone and: [ ( σ ( x , u , F ) σ ( x , u , G ) ) : ( F G ) = 0 ] F = G ).

4) σ ( x , u , F ) is strictly p-quasi-monotone in F, i.e.,

I M m × n ( σ ( x , u , λ ) σ ( x , u , λ ¯ ) ) : ( λ λ ¯ ) d ϑ ( λ ) > 0,

for all homogeneous W 1, p , w -gradient young measures ϑ with center of mass λ ¯ = ϑ , i d which are not a single Dirac mass.

The main point is that we do not require strict monotonicity or monotonicity in the variables ( u , F ) in (H3) as it is usually assumed in previous work (see [15] or [16] ).

( F 0 ) : (continuity) f : Ω × I R m × I M m × n I R m is a Carathéodory function i-e: x f ( x , u , F ) is measurable for every u I R m , and F I M m × n , ( u , F ) f ( x , u , F ) is continuous for almost every x Ω .

( F 1 ) : (growth condition): The exist: b 1 L p ( Ω ) , c 1 > 0 , c 2 > 0 such that:

| f j ( x , u ) | [ b 1 ( x ) + c 1 γ j 1 p | u j | q p + c 2 r , s ω r s 1 P | F r s | p 1 ] ω 0 j 1 P ;

1 j m ( G 0 ) : (continuity) the map g : Ω × I R m I M m × n is a Carathéodory function. ( G 1 ) : (growth condition) There exist: b 2 L p ( Ω )

| g r s | ω r s 1 p [ b 2 + j γ j 1 p | u j | q p ]

For all 1 r n and 1 s m .

Our aim of this paper is to prove the existence of the problem ( Q E S ) f , g in the space W 0 1, P ( Ω , ω , I R m ) .

Remark 2.1 -The condition ( F 0 ) and ( G 0 ) ensure the measurability of f and g for all measurable function u.

- ( F 1 ) and ( G 1 ) ensure that growths conditions, in particularly: if u W 0 1, P ( Ω , ω , I R m ) then f ( ., u ( . ) , D ( . ) ) u ( . ) and g ( ., u ) : D u is in L 1 ( Ω , ω ) .

- Exploiting the convergence in measure of the gradients of the approximating solutions, we will prove the following theorem.

Theorem 2.4 If p ( 1, ) and σ satisfies the conditions (H0)-(H3), then the Dirichlet problem ( Q E S ) f , g has a weak solution u W 0 1, p ( Ω , ω , I R m ) , for every v W 1, p ( Ω , ω , I R m ) , f satisfies ( F 0 ) and ( F 1 ) and g satisfies ( G 0 ) and ( G 1 ) .

In order to prove theorems, we will apply a Galerkin scheme, with this aim in view, we establish in the following subsections, the key ingredient to pass to the limit for this, we assume that the conditions: (H0)-(H3), ( F 0 ) , ( F 1 ) , ( G 0 ) and ( G 1 ) .

Lemma 2.1 For arbitrary u W 0 1, p ( Ω , ω , I R m ) and v W 1, p ( Ω , ω , I R m ) , the functional

F ( u ) : W 0 1, p ( Ω , ω , I R m ) I R φ Ω σ ( x , u ( x ) , D u ( x ) ) : D φ ( x ) d x v , φ Ω f ( x , u , D u ) : φ d x + Ω g ( x , u ) : D φ d x .

is well defined, linear and bounded.

Proof For all φ W 0 1, p ( Ω , ω , I R m ) , we denote

F ( u ) ( φ ) = I 1 + I 2 + I 3 + I 4

with

I 1 = Ω σ ( x , u ( x ) , D u ( x ) ) : D φ ( x ) d x ,

and

I 2 = v , φ .

I 3 = Ω f ( x , u , D u ) : φ d x

I 4 = Ω g ( x , u ) : D φ d x

We define

I r s = Ω σ r s ( x , u ( x ) , D u ( x ) ) : D r s φ ( x ) d x

Firstly, by virtue of the growth conditions (H2) and the Hölder inequality, one has

| I r s | Ω | σ r s ( x , u ( x ) , D u ( x ) ) | : | D r s φ ( x ) | d x Ω β ω r s 1 / p ( x ) [ λ 1 ( x ) + c 1 j = 1 m | γ j ( x ) | 1 / p | u j ( x ) | q / p + c 1 1 i n , 1 j m ω i j 1 / p | D i j u | p 1 ] | D r s φ | d x β [ ( Ω | λ 1 ( x ) | p d x ) 1 / p ( Ω | D r s φ ( x ) | p ω r s d x ) 1 / p + ( Ω | D r s φ ( x ) | p ω r s ) 1 / p ( j = 1 m Ω | u j | q γ j d x ) 1 / p + ( 1 i n , 1 j m Ω | D i j u | p ω i j d x ) 1 / p ( Ω | D r s φ | p ω r s d x ) 1 / p ]

with ( p = p ( p 1 ) ) , and thanks to Hardy inequality we have:

| I r s | c β [ λ 1 p φ 1, p , ω r s + c 1 D φ p , ω r s ( Ω | u | q γ d x ) 1 / p + c 1 i j D φ p , ω i j D u p , ω r s ] c β [ λ 1 p φ 1, p , ω r s + φ 1, p , ω r s u q , γ + u 1, p , ω φ 1, p , ω r s ]

with c = max ( c ,1 ) . Which gives

| I 1 | c β [ λ 1 p + u 1 , p , ω q / p + u 1 , p , ω ] φ 1 , p , ω < .

and

| I 2 | Ω | v | | φ | d x v 1 , p , ω * φ 1 , p , ω < .

I 3 = j Ω f j ( x , u , D u ) φ j ( x ) d x

We denote I 3, j = | Ω f j ( x , u , D u ) φ j ( x ) d x | .

I 3, j Ω | f j ( x , u , D u ) | | φ j ( x ) | d x Ω b 1 ( x ) | φ j ( x ) | ω 0 j 1 P d x + c 1 Ω γ j 1 p | u j | q p | φ j ( x ) | ω 0 j 1 P + c 2 Ω r s | φ j ( x ) | | D r s u | p 1 ω 0 j 1 P d x ( Ω | b 1 ( x ) | p ) 1 p ( Ω | φ j ( x ) | p ω 0 j d x ) 1 p + ( Ω γ j ( x ) | u j | q d x ) 1 p ( Ω | φ j ( x ) | p ω 0 j d x ) 1 p + r s ( Ω | φ j | p ω 0 j ) 1 p ( Ω | D r s u | p ( p 1 ) ω r s ) 1 p b 1 p φ 1 , p , ω + c 1 ( j Ω γ j ( x ) | u j | q d x ) 1 p φ 1 , p , ω + c 2 φ 1 , ω , p D u 1 , p , ω p p

b 1 p φ 1 , p , ω + c 1 D u 1 , p , ω φ 1 , p , ω + c 2 φ 1 , ω , p D u 1 , p , ω p p ( b 1 + c 1 D u 1 , p , ω + c 2 D u 1 , p , ω p p ) φ 1 , p , ω .

I 4 = r s Ω g r s ( x , u ) D r s φ d x

Ω | g r s | : | D r s φ | d x Ω b 2 ω r s 1 p D r s φ d x + j Ω γ j 1 p ( x ) | u j | q p ω r s 1 p D r s φ d x ( Ω | b 2 | p d x ) 1 p ( Ω | D r s φ | p ω r s d x ) 1 p + j ( Ω | u j | q γ j ( x ) d x ) 1 p ( Ω | D r s φ | p ω r s ( x ) d x ) 1 p b 2 p D r s φ 1 , p , ω r s + u q , γ q p ( Ω | D r s φ | p ω r s d x ) 1 p

I 4 b 2 p D r s φ 1 , p , ω r s + u q , γ q p ( Ω | D r s φ | p ω r s d x ) 1 p b 2 p D φ 1 , p , ω + u q , γ q p D φ 1 , p , ω c φ 1 , p , ω

Hence I c 4 φ 1, p , ω . With c 4 < .

Finally the functional F ( . ) is bounded.

Lemma 2.2 The restriction of F to a finite dimensional linear subspace V of W 0 1, p ( Ω , ω , I R m ) is continuous.

Proof Let d be the dimension of V and ( e 1 , e 2 , , e d ) a basis of V. Let u j = 1 i d a j i e i be a sequence in V which converges to u = 1 i d a i e i in V. The sequence ( a j ) converge to a I R d , so u j u and D u j D u a.e., on the other hand u j p and D u j p are bounded by a constant c. Thus, it follows by the continuity conditions (H1), that

σ ( x , u j , D u j ) : D φ σ ( x , u , D u ) : D φ

for all φ W 0 1, p ( Ω , ω , I R m ) and a.e. in Ω . Let Ω be a measurable subset of Ω and let φ W 0 1, p ( Ω , ω , I R m ) .

Thanks to the condition (H2), we get

Ω | σ ( x , u j , D u j ) : D φ | d x < ,

By the continuity conditions ( F 0 ) and ( G 0 ) we have:

f ( x , u j , D u j ) φ f ( x , u , D u ) φ

And

g ( x , u j ) D φ g ( x , u ) D φ

almost everywhere. Moreover we infer from the growth conditions ( F 1 ) and ( G 1 ) that the sequences:

( σ ( x , u j , D u j ) : D φ ) , ( f ( x , u j , D u j ) φ ) and ( g ( x , u j ) D φ )

Are equi-integrable. Indeed, if Ω Ω is a measurable subset and φ W 0 1, p ( Ω , ω , I R m ) then:

Ω | f ( x , u j , D u j ) φ | d x < (by ( F 1 ) and Hölder inequality).

Ω | g ( x , u j ) D φ | d x < (by ( G 1 ) and Hölder inequality).

Ω | σ ( x , u j , D u j ) : D φ | d x < (by Hölder inequality).

which implies that σ ( x , u j , D u j ) : D φ is equi-integrable. And by applying the Vitali’s theorem, it follows that

Ω σ ( x , u j , D u j ) : D φ d x Ω σ ( x , u , D u ) : D φ d x ,

for all φ W 0 1, p ( Ω , ω , I R m ) .

Finally

lim j F ( u j ) , φ = F ( u ) , φ ,

which means that

F ( u j ) F ( u ) in W 1, p ( Ω , ω , I R m ) .

Remark 2.2 Now, the problem ( Q E S ) f , g is equivalent to find a solution u W 0 1, p ( Ω , ω , I R m ) such that F ( u ) , φ = 0 , for all φ W 0 1, p ( Ω , ω , I R m ) .

In order tofind such a solution we apply a Galerkin scheme.

3. Galerkin Approximation

Remark 3.1 (Galerkin Schema)

Let V 1 V 2 W 0 1, p ( Ω , ω , I R m ) be a sequence of finite dimensional subspaces with k I N V k dense in W 0 1, p ( Ω , ω , I R m ) . The sequence V k exists since W 0 1, p ( Ω , ω , I R m ) is separable.

Let us fix some k, we assume that V k has a dimension d and that ( e 1 , e 2 , , e d ) is a basis of V k . Then, we define the map,

G : I R k I R k ( a 1 , , a k ) ( F ( u ) , e 1 , , F ( u ) , e k ) ; u = i = 1 d a i e i .

Proposition 3.1 The map G is continuous and G ( a ) a tends to infinity when a I R k tends to infinity.

Proof. Since F restricted to V k is continuous by Lemma 2.2, so G is continuous.

Let a I R d and u = 1 i d a i e i in V k , then G ( a ) a = F ( u ) , u and which implies that a I R d tends to infinity if u 1, p , ω tends to infinity.

G ( a ) a = 1 i d F ( u ) , a i e i = F ( u ) , u

and

u 1, p , ω p = 1 i d a i e i 1, p , ω p ( 1 i d | a i | e i 1, p , ω ) p max 1 i d ( e i 1, p , ω p ) ( 1 i d | a i | ) p c a I R p ,

which implies that a I R p tends to infinity if u 1, p , ω tends to infinity.

Now, it suffices to prove that

F ( u ) , u when u 1, p , ω .

Indeed, thanks to the first coercivity condition and the Hölder inequality, we obtain

I = Ω σ ( x , u , D u ) : D u d x λ 2 1 Ω λ 3 ω 0 j α / p | u j | α d x + c 2 1 i , j n , m Ω | D i j u | p ω i j d x

By the Hölder inequality, we have

Ω λ 3 | u j | α ω 0 j α / p d x λ 3 ( p / α ) ( Ω ω 0 j | u j | ( p / α ) α ) α / p c λ 3 ( p / α ) u j 1, p , ω 0 j .

where c is a constant positive. For u 1, p , ω large enough, we can write

| I | λ 2 1 c λ 3 ( p / α ) u j 1, p , ω 0 j α + c 2 1 i , j n , m D u j 1, p , ω i j p λ 2 1 c λ 3 ( p / α ) u 1, p , ω α + c 2 c u 1, p , ω p .

And since

| I | = | v , u | v 1 , p , ω u 1 , p , ω

Finally, it follows from the growth condition ( F 1 ) and G 1 that:

| I | = | Ω f ( x , u , D u ) u d x | ( b 1 p + c 1 D u 1, p , ω + c 2 D u 1, p , ω ) u 1, p , ω c 3 u 1, p , ω

| I | = | Ω g ( x , u ) D u d x | ( b 2 p + u q , γ q p ) D u 1, p , ω c 4 u 1, p , ω ;

with c 4 is a constant. With; 0 < α < p and p > 1 , we get:

I I I c 2 c u 1, p , ω p v 1, p , ω u 1, p , ω c λ 3 ( p / α ) u 1, p , ω α λ 2 1 c 3 . u 1, p , ω (3.1)

Consequently, by using (3.1), we deduce

I I I as u 1, p , ω .

and

I as u 1, p , ω .

F ( u ) , u as u 1, p , ω .

Remark 3.2 The properties of G allow us to construct our Galerkin approximations.

Corollary 3.1 For all k I N , there exists ( u k ) V k such that F ( u k ) , φ = 0 , for all φ V k .

Proof By the proposition 3.1, there exists R > 0 , such that for all a B R ( 0 ) I R d , we have G ( a ) a > 0 . And the usual topological argument see [Zei 86 proposition 2.8] [17] implies that G ( x ) = 0 has a solution x B R ( 0 ) . So, for all k I n , there exists ( u k ) V k , such that

F ( x j e j ) , e j = 0 for all 1 j d , with d = dim V k

Taking u k = ( x k i e i ) , e i V k , so we obtain:

F ( u k ) , φ = 0, for all φ V k .

Proposition 3.2 The Galerkin approximations sequence constructed in corollary (3.1) is uniformly bounded in W 0 1, p ( Ω , ω , I R m ) ; i.e.,

There exists a constant R > 0 , such that u k 1 , p , ω R , for all k I N .

Proof Like in the proof of proposition (3.1), we can see that

F ( u ) , u as u 1, p , ω .

Then, there exists R satisfying F ( u ) , u > 1 when u 1 , p , ω > R . Now, for the sequence of Galerkin approximations ( u k ) V k of corollary (3.1), which satisfying F ( u k ) , u k = 0 , we have the uniform bound u k 1, p , ω R , for all k I N .

Remark 3.3 There exists a subsequence ( u k ) of the sequence ( u k ) V k , such that:

u k u in W 0 1, p ( Ω , ω , I R m )

and

u k u in measure in L r ( Ω , I R m ) ;

with

{ 1 r n p s n ( s + 1 ) p s if p s n ( s + 1 ) r 1 if n ( s + 1 ) p s

The gradient sequence ( D u k ) generates the young measure ϑ x . Since u k u in measure, then ( u k , D u k ) generates the Young measure ( δ u ( x ) ϑ x ) , see [2]. Moreover, for almost x in Ω , we have,

1) ϑ x is the probability measure, i.e., ϑ x m e s = 1 .

2) ϑ x is the W 1, p , ω gradient homogeneous young measure.

3) ϑ x , i d = D u ( x ) , see [18].

Proof. See [2]. (Dolzmann, N. Humgerbuhler S. Muller, Non linear elliptic system …)

4. Passage to the Limit

Now, we are in a position to prove our main result under convenient hypotheses.

Let

I k = ( σ ( x , u k , D u k ) σ ( x , u , D u ) ) : ( D u k D u ) . (4.1)

Lemma 4.1 (Fatou lemma type)(See [2] ) Let: F : Ω × I R m × I M m × n I R be a Carathéodory function, and u k : Ω I R m a measurable sequence, such that ( D u k ) generates the Young measure ϑ x , with ϑ x m e s = 1 , for a.e. x Ω . Then:

lim inf k Ω F ( x , u k , D u k ) d x Ω I M m × n F ( x , u , ζ ) d ϑ x ( ζ ) d x , (4.2)

which provided that the negative part of F ( x , u k , D u k ) is equi-integrable.

Proof.

Lemma 4.2 Let p > 1 and u k be a sequence which is uniformly bounded in W 0 1, p ( Ω , ω , I R m ) . There exists a subsequence of u k (for convenience not relabeled) and a function u W 0 1, p ( Ω , ω , I R m ) such that u k u in W 0 1, p ( Ω , ω , I R m )

And such that u k u in measure on Ω and in L r ( Ω , I R m ) , with:

{ 1 r n p s n ( s + 1 ) p s if p s n ( s + 1 ) r 1 if n ( s + 1 ) p s

Proof. see [10].

Lemma 4.3 The sequence ( I k ) is equi-integrable.

Proof

We have

I k = ( σ ( x , u k , D u k ) σ ( x , u , D u ) ) : ( D u k D u ) = [ σ ( x , u k , D u k ) : D u k ] [ σ ( x , u k , D u k ) : D u ] [ σ ( x , u , D u ) : D u k ] + [ σ ( x , u , D u ) : D u ] = I k 1 + I k 2 + I k 3 + I k 4 (4.3)

We denote ( I k 1 ) = [ σ ( x , u k , D u k ) : D u k ] . Thanks to the coercivity condition (H2), we have

Ω | ( I k 1 ) | d x Ω | λ 2 | + c 2 1 j m ω 0 j α p | λ 3 | | u k j | α + c 1 i , j n , m ω i j | D i j u k | p d x λ 2 1 + Ω ( 1 j m ω 0 j α / p | u k j | α ) p / α λ 3 ( p / α ) + c 2 u k 1, ω , p p (4.4)

with p / α 1 . Therefore,

Ω | ( I k 1 ) | d x λ 2 1 + ( 1 j m ω 0 j | u k j | p ) α / p λ 3 ( p / α ) + c 2 u k 1 , ω , p p λ 2 1 + u k p , ω ¯ 00 α λ 3 ( p / α ) + c 2 u k 1 , ω , p p < ,

for all Ω Ω .

Similarly for | ( I k 4 ) | .

Now, by using the growth condition (H2) and the Hardy inequality (H0), we have

Ω | ( I k 2 ) | d x = Ω | σ ( x , u k , D u k ) : D u k | d x β Ω ω r s 1 / p ( λ 1 + c 1 1 j m γ j 1 / p | u k j | q / p + c 2 1 i , j n , m ω i j 1 / p | D i j u k | p 1 ) D r s u k d x . (4.5)

Thus, by the Hölder inequality, we obtain

Ω | ( I k 2 ) | d x β [ λ 1 p ( Ω | D r s u k | p ω r s d x ) 1 / p + c 1 ( Ω | D r s u k | p ω r s d x ) 1 / p ( Ω ( 1 j m γ j 1 / p | u k j | q / p ) p d x ) 1 / p + c 1 ( 1 j m Ω ( | D i j u k ( x ) | p ( p 1 ) ω i j d x ) 1 / p ) ( Ω | D r s u k | p ω r s d x ) 1 / p ] . (4.6)

So, by combining (4.5) and (4.6), we deduce that

Ω | σ ( x , u k , D u k ) : D u k | d x c β ( λ 1 p u k 1, p , ω + u k 1, p , ω ) < . (4.7)

Similarly to | ( I k 2 ) | , we obtain | ( I k 3 ) | . Finally: I k is equi-integrable.

We choose a sequence φ k such that φ k belongs to the same space V k and φ k φ in W 0 1, p ( Ω , ω , I R m ) , this allows us in particular, to use u k φ k as a test function in (3.1). We have:

Ω | σ ( x , u k , D u k ) : ( D u k D φ k ) | d x = v , u k φ k + Ω f ( x , u k , D u k ) ( u k φ k ) d x Ω g ( x , u k ) : ( D u k D φ k ) d x . (4.8)

The first term on the right in 4.8 converge to zero since ( u k φ k ) 0 in W 0 1, p ( Ω , ω , I R m ) . By the choice of φ k , the sequence φ k uniformly bounded in W 0 1, p ( Ω , ω , I R m ) , and lemma (4.2). Next, for the second term: I I k = Ω f ( x , u k , D u k ) ( u k φ k ) d x in 4.8 it follows from the growth condition F 1 and the Hölder inequality that:

| I I k | ( b 1 p + c 1 D ( u k φ k ) 1, p , ω + c 2 D ( u k φ k ) p p ) u k φ k 1, p , ω ( b 1 p + c D ( u k φ k ) 1, p , ω ) u k φ k 1, p , ω

By the equivalence of the norm in W 0 1, p ( Ω , ω , I R m ) and the sequence u k is uniformly bounded in W 0 1, p ( Ω , ω , I R m ) , u k 1, p , ω is bounded.

Moreover, by the construction of φ k , and lemma (4.2) we have:

u k φ k 1, p , ω u k u 1, p , ω + u φ k 1, p , ω

( u k u 1, p , ω + u φ k 1, p , ω ) 0

We infer that the second term in 4.8 vanishes as k . Finally, for the last term

I I I k = Ω g ( x , u k ) : D ( u k φ k ) d x

in 4.8, we note that

g ( x , u k ) g ( x , u )

Strongly in L p ( Ω , M m × n ) by ( G 0 ) , ( G 1 ) and lemma (4.2).

Indeed we may assure that u k u almost everywhere.

I I I k ( b 2 p + u k φ k q , γ q p ) D ( u k φ k ) 1, p , ω c ( b 2 p + u k φ k q , γ q p ) ( u k φ k ) 1, p , ω c ( b 2 p + u k φ k q , γ q p ) ( u k u 1, p , ω + φ k u 1, p , ω )

φ k u 1, p , ω 0 , u k u 1, p , ω 0 and u k φ k q , γ q p 0

Now, we consider ( I k ) = σ ( x , u k , D u k ) : ( D u k D u ) . We have, I k is equi-integrable because I k it is. So, we define

X = lim inf Ω I k d x = lim inf Ω ( I k ) d x Ω I M m × n ( σ ( x , u , λ ) : ( λ D u ) ) d ϑ x ( λ )

So to prove (??), it suffices to prove that:

X 0. (4.9)

Let ε > 0 , so there exists k 0 I N such that, for all k > k 0 , we have d i s t ( u , V k ) < ε since: lim inf φ k V k u φ k 1 , p , ω < ε , ( u k u )

Or in an equivalent manner d i s t ( u k u , V k ) < ε , k > k 0 then for all v k V k , we have

X = lim inf k Ω ( σ ( x , u k , D u k ) : ( D u k D u ) ) d x = lim inf k [ Ω ( σ ( x , u k , D u k ) : D ( u k u φ k ) ) d x + Ω ( σ ( x , u k , D u k ) : D ( φ k ) ) ]

Combining (H2) and (0.1), we get

X lim inf k Ω β ω r s 1 / p [ λ 1 + c 1 1 j m γ j 1 / p | u k j | q / p + c 1 1 i , j n , m ω i j 1 / p | D i j u k | p 1 ] × | D r s ( u k u φ k ) | d x + v , φ k .

For all ε > 0 , we choose φ k V k such that

u k u φ k 1, p , ω 2 ε , (4.10)

For all k k 0 , which implies that

| v , φ k | | v , φ k + ( u u k ) | + | v , u k u | 2 ε v 1, p , ω + o ( k )

Hence lim k v , u k u = 0 . According to Hölder and Hardy inequalities, and by (4.1) we deduce that

X lim inf k c β ( λ 1 p ( Ω | D r s ( u k u φ k ) | p ω r s d x ) 1 / p + c 1 ( Ω | u k | q γ ) 1 / p ( Ω | D r s ( u k u φ k ) | p ω r s d x ) 1 / p + c 1 ( Ω ω i j | D i j u | p ( p 1 ) ) 1 / p ( Ω ω r s | D r s ( u k u φ k ) | p ) 1 / p ) + | v , φ k | lim inf k c ( λ 1 p u k u φ k 1 , p , ω ) + u k 1 , p , ω q u k u φ k 1 , p , ω + 2 ε v 1 , p , ω + o ( k )

Therefore,

X 2 ε c β ( λ 1 p + u 1, p , ω q + v 1, p , ω * ) .

which proves that X 0 , and finally

Ω I M m × n σ ( x , u , λ ) : λ d ϑ x d x Ω I M m × n σ ( x , u , λ ) : D u d ϑ x ( λ ) d x .

Proof of theorem:

For arbitrary φ in W 0 1, p ( Ω , ω , I R m ) . It follows from the continuity condition ( F 0 ) and ( G 0 ) that

f ( x , u k , D u k ) φ ( x ) f ( x , u , D u ) φ ( x )

and

g ( x , u k ) : D φ ( x ) g ( x , u ) : D φ ( x )

almost everywhere. Since, by the growth conditions ( F 1 ) , ( G 1 ) and the uniform bound of u k , f ( x , u k , D u k ) φ ( x ) and g ( x , u k ) : D φ ( x ) are equi-integrable, it follows that the Vitali’s theorem. This implies that:

lim k Ω f ( x , u k , D u k ) φ ( x ) d x = Ω f ( x , u , D u ) φ ( x ) d x

for all φ k = 1 V k and

lim k Ω g ( x , u k ) : D φ ( x ) d x = Ω g ( x , u ) : D φ ( x ) d x

for all φ k = 1 V k We will start with the easiest case

(d): F σ ( x , u , F ) is strict p-quasi-monotone. (4.11)

Indeed, we assume that ϑ x is not a Dirac mass on the set M with x M of positive Lebesgue measure | M | > 0 . Moreover, by the strict p-quasi-monotonicity of σ ( x , u , ) and ϑ x is an homogeneous W 1, p gradient young measure for a.e. x M . So, for a.e. x M , with λ ¯ = ϑ x , I d = a p D u ( x ) , with a p D u ( x ) is the differentiable approximation in x. We get

I M m × n σ ( x , u , λ ) : ( λ D u ) d ϑ x ( λ ) > I M m × n σ ( x , u , D u ) : ( λ D u ) d ϑ x ( λ ) > σ ( x , u , D u ) : I M m × n λ d ϑ x ( λ ) σ ( x , u , D u ) : D u I M m × n d ϑ x ( λ ) > ( σ ( x , u , D u ) : D u σ ( x , u , D u ) : D u ) = 0 > 0

On the other hand (4.9), integrating over Ω , and using the div-cul inequality we have:

Ω I M m × n σ ( x , u , λ ) : λ d ϑ x ( λ ) d x > Ω I M m × n σ ( x , u , λ ) : D u d ϑ x ( λ ) d x Ω I M m × n σ ( x , u , λ ) : λ d ϑ x ( λ ) d x .

Which is a contradiction with (3.8). Thus ϑ x = δ λ ¯ = δ D u ( x ) for a.e. x Ω . Therefore, D u k D u in measure when k tends to infinity. Then, we get σ ( x , u k , D u k ) σ ( x , u , D u ) for all x Ω . In the other hand, for all φ k I N ϑ k ; σ ( x , u k , D u k ) : D φ σ ( x , u , D u ) : D φ a.e. x Ω . Moveover, for all Ω Ω measurable, it is easy to see that:

Ω σ ( x , u k , D u k ) : D φ d x c β ( λ 1 p + u k 1, p , ω q / p + u k 1, p , ω p / p ) u 1, p , ω < ,

because u k 1, p , ω R . And thanks to Vitali’s theorem, we obtain:

F ( u ) , φ = 0 , for all φ k I N ϑ k .

which proves the theorem in this case.

Remark 4.1 Before treating the cases (a),(b) and (c) of (H3), we note that

Ω I M m × n ( σ ( x , u , λ ) σ ( x , u , D u ) ) : ( λ D u ) d ϑ x ( λ ) d x 0 (4.12)

Since

Ω I M m × n σ ( x , u , λ ) : ( λ D u ) d ϑ x ( λ ) d x = 0,

thanks to the div-Curl inequality in (4.9). On the other hand, the integrand in (4.12) is non negative, by the monotonicity of σ . Consequently, the integrating should be null, a.e., with respect to the product measure d ϑ x d x , which mean

( σ ( x , u , λ ) σ ( x , u , D u ) ) : ( λ D u ) = 0 in s p t ϑ x . (4.13)

Thus,

s p t ϑ x { λ I M m × n / ( σ ( x , u , λ ) σ ( x , u , D u ) ) : ( λ D u ) = 0 } . (4.14)

Case c: We prove that, the map F σ ( x , u , F ) is strictly monotone, for all x Ω and for all u I R m .

Sine σ is strict monotone, and according to (4.14),

s p t ϑ x = { D u } , i .e , ϑ x = δ D u , a .e . in Ω ,

which implies that, D u k D u in measure. For the rest of our prove is similarly to case d.

Case b: We start by showing that for almost all x Ω , the support of ϑ x is contained in the set where W agrees with the supporting hyper-plane.

L = { ( λ , W ( x , u , λ ¯ ) + σ ( x , u , λ ¯ ) : ( λ λ ¯ ) ) } with λ ¯ = D u ( x ) .

So, it suffices to prove that

s p t ϑ x K x = { λ I M m × n / W ( x , u , λ ) = W ( x , u , λ ¯ ) + σ ( x , u , λ ¯ ) : ( λ λ ¯ ) } (4.15)

If λ s p t ϑ x , thanks to (4.14), we have

( 1 t ) ( σ ( x , u , D u ) σ ( x , u , λ ) ) : ( D u λ ) = 0, for all t [ 0,1 ] . (4.16)

On the other hand, since σ is monotone, for all t [ 0,1 ] we have:

( 1 t ) ( σ ( x , u , D u + t ( λ D u ) ) σ ( x , u , λ ) ) : ( D u λ ) 0. (4.17)

By subtracting (4.16) from (4.17), we get

( 1 t ) [ σ ( x , u , λ ¯ + t ( λ λ ¯ ) ) σ ( x , u , λ ¯ ) ] : ( λ ¯ λ ) 0, (4.18)

for all t [ 0,1 ] . Doing the same by the monotonicity in (4.18), we obtain

( 1 t ) [ σ ( x , u , λ ¯ + t ( λ λ ¯ ) ) σ ( x , u , λ ¯ ) ] : ( λ ¯ λ ) 0. (4.19)

Combining (4.18) and (4.19), we conclude that

( 1 t ) [ σ ( x , u , λ ¯ + t ( λ λ ¯ ) ) σ ( x , u , λ ¯ ) ] : ( λ ¯ λ ) = 0, (4.20)

for all t [ 0,1 ] , and for all λ s p t ϑ x .

Now, it follows from (4.19) that

W ( x , u , λ ) = W ( x , u , λ ¯ ) + ( W ( x , u , λ ) W ( x , u , λ ¯ ) ) = W ( x , u , λ ¯ ) + 0 1 [ σ ( x , u , λ ¯ ) + t ( λ λ ¯ ) ] : ( λ λ ¯ ) d t = W ( x , u , λ ¯ ) + σ ( x , u , λ ¯ ) : ( λ λ ¯ )

Witch prove (4.15).

Now, by the coercivity of W, we get

W ( x , u , λ ) W ( x , u , λ ¯ ) + σ ( x , u , λ ¯ ) : ( λ λ ¯ ) ,

for all λ I M m × n . Therefore,

L is a supporting hyper-plane, for all

λ K x . (4.21)

Moveover, the mapping λ W ( x , u , λ ) is continuously differentiable, so we obtain

σ ( x , u , λ ) = σ ( x , u , λ ¯ ) , for all λ K x . (4.22)

Thus,

σ ¯ ( x ) = I M m × n σ ( x , u , λ ) d ϑ x ( λ ) = σ ( x , u , λ ¯ ) . (4.23)

Now, we consider the Carathéodory function

g v ( x , u , ρ ) = | ( σ ( x , u , ρ ) σ ¯ ( x ) ) : D φ | ,

and lets g k ( x ) = g φ ( x , u k , D u k ) is equi-integrable. Thus, thanks to BALL’s theorem, see [6] g k g weakly in L 1 ( Ω ) , and the weakly limit of g is given by

g ¯ φ ( x ) = I R m × I M m × n | σ ( x , η , λ ) σ ¯ ( x ) | d δ u ( x ) ( η ) d ϑ x ( λ ) = s p t ϑ x | σ ( x , u ( x ) , λ ) σ ¯ ( x ) | d ϑ x ( λ ) = 0.

According to (4.22) and (4.23), and since g k 0 , it follow that g k 0 strongly in L 1 ( Ω ) by Fatou lemma, which gives

lim k Ω σ ( x , u k , D u k ) : D φ d x = Ω σ ( x , u , D u ) : D φ d x .

Thus

F ( u ) , φ = 0, φ k I N V k .

This completes the proof of the case (b).

Case (a): In this case, on s p t ϑ x , we affirm that,

σ ( x , u , λ ) : M = σ ( x , u , D u ) : M + ( F σ ( x , u , D u ) : M ) : ( D u λ ) , (4.24)

for all M I M m × n , where F is the derivative with respect to the third variable of σ and λ ¯ = D u ( x ) .

Thanks to the monotonicity of σ , we have

( σ ( x , u , λ ) σ ( x , u , D u + t M ) ) : ( λ D u t M ) 0, for all t I R .

By invoking (4.19), we obtain

σ ( x , u , λ ) : ( t M ) σ ( x , u , D u ) : ( λ D u ) + σ ( x , u , D u + t M ) : ( λ D u t M ) .

On the other hand, F σ ( x , u , F ) is a C 1 function, so

σ ( x , u , D u + t M ) = σ ( x , u , D u ) + F ( x , u , D u ) ( t M ) + o ( t ) .

Thus

σ ( x , u , λ ) : ( t M ) σ ( x , u , D u ) : ( t M ) + F σ ( x , u , D u ) ( t M ) : ( λ D u ) + o ( t ) ,

which gives

σ ( x , u , λ ) : ( t M ) t [ F σ ( x , u , D u ) : ( M ) : ( λ D u ) σ ( x , u , D u ) : ( M ) ] + o ( t ) ,

t is arbitrary in (4.24).

Finally for all φ k I N V k the sequence σ ( x , u k , D u k ) : D φ is equi-integrable. Then, by the BALL’s theorem, see [1] the weak limit is s p t ϑ x σ ( x , u , λ ) : D φ d ϑ x ( λ )

By choosing M = D u in (4.24), we obtain

s p t ϑ x ( D u λ ) ( σ ( x , u , λ ) : D φ ) : D φ d ϑ x ( λ ) = s p t ϑ x σ ( x , u , D u ) : D φ d ϑ x ( λ ) + ( F σ ( x , u , D u ) : D φ ) t s p t ϑ x ( D u λ ) d ϑ x ( λ ) = ( σ ( x , u , D u ) : D φ ) s p t ϑ x d ϑ x ( λ ) = σ ( x , u , D u ) : D φ .

Hence:

σ ( x , u k , D u k ) : D φ σ ( x , u , D u ) : D φ strongly

This proves that

F ( u ) , φ = 0 for all φ V k .

And since V k is dense in W 0 1, P ( Ω , ω , I R m ) , so u is a weak solution of ( Q E S ) f , g , as desired.

Remark 4.2 In case (b) σ ( x , u k , D u k ) : D φ σ ( x , u , D u ) : D φ strongly, but in the case (c) and (d) D u k D u in measure.

Exemple 4.1 We shall suppose that the weight functions satisfy: w i 0 j = 0 , j = 1 , 2 , , m for some i 0 I c ; and ω i j ( x ) = w ( x ) ; x Ω , with I c I = { 0 ; 1 ; 2 ; ; n } , for all i I I c , j = 1 , 2 , , m , and i i 0 with w ( x ) > 0 a.e in Ω then, we can consider the Hardy inequality in the form:

( j = 1 m Ω | u j ( x ) | q γ j ( x ) d x ) 1 q c ( 1 i N , 1 j m Ω | D i j u | p ω i j ) 1 p ,

for every u W 0 1, p ( Ω , ω , I R m ) with a constant c > 0 independent of u and for some q > p . Let us consider the Carathéodory functions: ( ⋆ )

σ i j ( x , η , ξ I ) = ω ( x ) | ξ i j | p 1 s n g ( ξ i j ) , j = 1 , 2 , , m , i I

σ i j ( x , η , ξ I c ) = ω ( x ) | ξ i j | p 1 s n g ( ξ i j ) , j = 1 , 2 , , m , i I c , i i 0

σ i 0 j ( x , η , ξ I c ) = 0 , j = 1 , 2 , , m

f j ( x , η , ξ ) = s i g n ( ξ ) r s ω r s 1 p | ξ | p 1 ω 0 j 1 p

The above functions defined by ( ) satisfies the growth conditions (H2).

In particular, let use the special weight function ω , γ expressed in term of the distance to the boundary Ω denote d ( x ) = d i s t ( x ; Ω ) and ω ( x ) = d λ ( x ) , γ j ( x ) = d μ ( x ) the hardy inequality reads:

( j = 1 m Ω | u j ( x ) | q d μ ( x ) d x ) 1 q c ( 1 i N , 1 j m Ω | D i j u | p d λ ( x ) ) 1 p ,

and the corresponding W 0 1, p ( Ω ; ω ; R m ) L q ( Ω ; γ ; R m ) is compact if:

1) For, 1 < p q <

λ < p 1 ; n q n p + 1 0 ; μ q λ p + n q n p + 1 > 0

2) For, 1 q < p <

λ < p 1 ; n q n p + 1 0 ; μ q λ p + 1 q 1 p + 1 > 0

3) For, q > 1

μ ( q 1 ) < 1 , by the simple modifications of the example in [11]. Moreover, the monotonicity condition are satisfied:

i j ( σ i j ( x , η , ξ I ) σ i j ( x , η , ξ I ) ) ( ξ i j ξ i j ) = ω ( x ) i j ( | ξ i j | p 1 s n g ( ξ i j ) | ξ i j | p 1 s n g ( ξ i j ) ) ( ξ i j ξ i j ) 0

for almost all x Ω and for all, ξ , ξ M n . This last inequality cannot be strict, since for ξ I c ξ I c with ξ i 0 j ξ i 0 j for all j = 1 , 2 , , m . But ξ i j = ξ i j for i I c , i i 0 , j = 1 , 2 , , m the corresponding expression is Zero.

Cite this paper: Barbara, A. , Rami, E. and Azroul, E. (2021) Quasilinear Degenerated Elliptic Systems with Weighted in Divergence Form with Weak Monotonicity with General Data. Applied Mathematics, 12, 500-519. doi: 10.4236/am.2021.126035.
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