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Volume 2009, Article ID 708389, 15 pagesdoi:10.1155/2009/708389 Research Article Existence of Weak Solutions for a Nonlinear Elliptic System 1 Department of Mathematics, Norfolk State Un

Volume 2009, Article ID 708389, 15 pages

doi:10.1155/2009/708389

Research Article

Existence of Weak Solutions for

a Nonlinear Elliptic System

1 Department of Mathematics, Norfolk State University, Norfolk, VA 23504, USA

2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Received 3 April 2009; Accepted 31 July 2009

Recommended by Kanishka Perera

We investigate the existence of weak solutions to the following Dirichlet boundary value problem, which occurs when modeling an injection molding process with a partial slip condition on the boundary We have−Δθ  kθ|∇p| r  qx in Ω; −div{kθ|∇p| r−2 βx|∇p| r0 −2∇p}  0 in Ω;

θ  θ0, and p  p0 on ∂Ω.

the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Injection molding is a manufacturing process for producing parts from both thermoplastic and thermosetting plastic materials When the material is in contact with the mold wall surface, one has three choices:i no slip which implies that the material sticks to the surface

ii partial slip, and iii complete slip 1 5 Navier 6 in 1827 first proposed a partial slip

condition for rough surfaces, relating the tangential velocity v αto the local tangential shear

stress τ α3

where β indicates the amount of slip When β  0, 1.1 reduces to the no-slip boundary

condition A nonzero β implies partial slip As β → ∞, the solid surface tends to full slip There is a full description of the injection molding process in3 and in our paper 7 The formulation of this process as an elliptic system is given here in after

Problem 1 Find functions θ and p defined inΩ such that

−Δθ  kθ∇p r

− divk θ∇p r−2 βx∇p r0 −2

Here we assume thatΩ is a bounded domain in RN with a C1 boundary We assume

also that q, θ0, p0, β, and k are given functions, while r is a given positive constant related to the power law index; p is the pressure of the flow, and θ is the temperature The leading order term βx|∇p| r0 −2of the PDE1.3 is derived from a nonlinear slip condition of Navier type Similar derivations based on the Navier slip condition occur elsewhere, for example,8,9 ,

10, equation2.4

The mathematical model for this system was established in7 Some related papers, both rigorous and formal, are3,11–13 In 11,13 , existence results in no-slip surface, β  0,

are obtained, while in 3, 7 , Navier’s slip conditions, β / 0 and r0  0, are investigated, and numerical, existence, uniqueness, and regularity results are given Although the

physical models are two dimensional, we shall carry out our proofs in the case of N

dimension

InSection 2, we introduce some notations and lemmas needed in later sections In

Section 3, we investigate the existence, uniqueness, stability, and continuity of solution p

to the nonlinear equation 1.3 In Section 4, we study the existence of weak solutions to Problem1

Using Rothe’s method of time discretization and an existence result for Problem1, one

can establish existence of week solutions to the following time-dependent problem.

Problem 2 Find functions θ and p defined inΩT such that

θ t − Δθ  kθ∇p r  qx in Ω T ,

− divk θ∇p r−2 βx∇p r0 −2

∇p 0 in ΩT ,

θ  θ0, p  p0 on ∂ Ω × 0, T,

θ  ϕ on Ω × {0}.

1.5

The proof is only a slight modification of the proofs given in11,13 and is omitted here

2 Notations and Preliminaries

2.1 Notations

In this paper, for s > 1, let H 1,s Ω and H 1,s

0 Ω denote the usual Sobolev space equipped with the standard norm Let

σ

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

N

N− 1, if 1 < r < N,

r

r− 1, if r > N,

qN

qN − q  N , if r  N,

2.1

where N < q < ∞ The conjugate exponent of σ is

σ∗

⎧

⎪

⎪

⎪

⎪

N, if 1 < r < N,

r, if r > N, qN

q − N , if r  N.

2.2

We assume that the boundary values θ0 and p0 for Problem 1 can be extended to functions defined onΩ such that

We further assume that there exist positive numbers k2> k1> 0 and β0such that

k1< k θ < k2, ∀θ ∈ R1,

Finally, we assume that for θ m , θ ∈ H 1,σ

0 Ω  θ0, limm→ ∞θ m  θ a.e in Ω indicates

lim

For the convenience of exposition, we assume that

1 < r0< r < τ < ∞. 2.6 Next, we recall some previous results which will be needed in the rest of the paper

2.2 Preliminaries

An important inequalitye.g., see 11, page 550  in the study of p-Laplacian is as follows:



|x| r−2x−yr−2y

x − y≥

⎧

⎪

⎪

ax − yr

, if r ≥ 2,

ax − y2

b  |x| y2−r, if 1 < r < 2, 2.7

where a > 0 and b > 0 are certain constants.

To establish coercivity condition, we will use the following inequality:

where r > 0, a > 0, and b > 0.

Using the Sobolev Embedding Theorem and H ¨older’s Inequality, we can derive the following resultsfor more details, see 11, Lemma 3.4 and 13, Lemma 4.2 

Lemma 2.1 The following statements hold

i For any positive numbers α and ς, if u ∈ L α Ω and v ∈ L ς Ω, then

uv ∈ L γ , where γ  1

α1

ς

−1

moreover, LγΩ LαΩ LςΩ.

ii If p ∈ H 1,r Ω and 1 < r < N, then p|∇p| r−2∇p ∈ L N/ N−1Ω N ; moreover,



p∇p r−2∇p

L N/ N−1Ω≤p

L Nr/ N−rΩ∇p r−1

iii If p ∈ H 1,r Ω and 1 < r < ∞, then |∇p| r−2∇p∇p0∈ L ζ Ω, where

ζ 1

r∗ 1

τ

−1

and r∗denotes the conjugate of r, namely, r∗ r/r − 1 for 1 < r < ∞; moreover,



∇pr−2∇p∇p0

L ζΩ≤∇p r−1

L rΩ∇p0

iv If p ∈ H 1,r Ω and n ≤ r < ∞, then

p ∇p r−2∇p ∈L r∗Ωn , r > n,

p ∇p r−2∇p ∈ L sΩ n , r  n,

2.13

where s  1/r∗ 1/q−1and r < q < ∞ Moreover



p∇p r−2∇p

L r∗Ω≤ C∇p r−1

L rΩ, r > n,



p∇p r−2∇p

L sΩ≤p

L qΩ∇p r−1

L rΩ, r  n.

2.14

The existence proof will use the following general result of monotone operators14, Corollary III.1.8, page 87 and 15, Proposition 17.2

Proposition 2.2 Let K ⊂ X be a closed convex set ( / φ), and let Λ : K → Xbe monotone, coercive, and weakly continuous on K Then there exists

The uniqueness proof is based on a supersolution argumentsimilar definition can be found in15, Chapter 3 

Definition 2.3 A function u ∈ H 1,r

locΩ is a weak supersolution of the equation

− divk θ|∇u| r−2 βx|∇u| r0 −2

inΩ if

 Ω



k θ|∇u| r−2 βx|∇u| r0 −2

whenever ϕ ∈ C∞

0 Ω is nonnegative

3 A Dirichlet Boundary Value Problem

We study the following Dirichlet boundary value problem:

− divk θ∇p r−2 βx∇p r0 −2

∇p 0 in Ω,

Definition 3.1 We say that p θ − p0∈ H 1,r

0 Ω is a weak solution to 3.1 if

 Ω



k θ∇p θr−2 βx∇p θr0 −2

for all ξ ∈ H 1,r

0 Ω and a given θ ∈ H 1,σ

0 Ω  θ0

Theorem 3.2 Assume that conditions 2.1–2.6 are satisfied Then there exists a unique weak solution p θ to the Dirichlet boundary value problem3.1 in the sense of Definition 3.1 In addition, the solution p θ satisfies the following properties.

1 we have

p θ

where C is a constant independent of θ and p θ ;

2 if lim m→ ∞θ m  θ a.e in Ω, then

lim

m→ ∞p θ m  p θ strongly in H 1,r Ω. 3.4

The idea behind the existence proof is related to15,16 We will first consider the following Obstacle Problem

Problem 3 Find a function p in K ψ,p0such that

 Ω



k θ∇p r−2 βx∇p r0 −2

∇p ∇ξ − pdx≥ 0 3.5

for all ξ ∈ K ψ,p0 Here

K ψ,p0Ω p ∈ H 1,r Ω : p ≥ ψ a.e in Ω, p − p0∈ H 1,r

Lemma 3.3 If K ψ,p0is nonempty, then there is a unique solution p to the Problem 3 in K ψ,p0 Proof of Lemma 3.3 Our proof will useProposition 2.2

Let X  L r Ω; R n and write

K∇v : v ∈ K ψ,p0



It follows from the proof in15, Proposition 17.2 that K ⊂ X is a closed convex set

Next we define a mappingΛ : K → Xby

 Ω



k θ|v| r−2 βx|v| r0 −2

vu dx ∀u ∈ X. 3.8

By H ¨older’s inequality,

2 r−1

L rΩ L rΩ β0 r0−1

L r0Ω L r0Ω

≤ C r−1

L rΩ r L0r0−1Ω



L rΩ.

3.9

Here we used Assumption2.6, that is, 1 < r0 < r < τ < ∞ Therefore we have Λv ∈ X

whenever v ∈ K Moreover, it follows from inequality 2.7 that Λ is monotone

To show thatΛ is coercive on K, fix ϕ ∈ K Then





Ω



k θ|u| r−2 β|u| r0 −2

u−k θϕr−2 βϕr0 −2

ϕ

u − ϕdx





Ωk θ|u| r−2u−ϕr−2

ϕ

u − ϕdx



Ωβ

|u| r0 −2u−ϕr0 −2

ϕ

u − ϕdx

≥



Ωk θ|u| r−2u−ϕr−2ϕ

u − ϕdx

≥ k1

ϕr

− k2



r−1ϕ   ϕ r−1 

≥ k12−ru − ϕr − k22r−1ϕu − ϕr−1ϕr−1

− k2ϕr−1ϕ − u  ϕ

3.10

Inequality2.8 is used to arrive at the last step This implies that Λ is coercive on K.

Finally, we show thatΛ is weakly continuous on K Let u i ∈ K be a sequence that converges to an element u ∈ K in L r Ω Select a subsequence u i j such that u i j → u a.e in Ω.

Then it follows that

k θu

i j



r−2u i j  βu

i j



r0−2u i j −→ kθ|u| r−2u  β|u| r0 −2u 3.11 a.e inΩ Moreover,



Ω



kθu i j



r−2u i j  βu

i j



r0−2u i j

r/ r−1 dx ≤ C



Ω u i j



ru

i j



r ×r0−1/r−1



dx

≤ C



Ω



u i j



r dx

Ω



u i j



r dx

r0−1/r−1

≤ C.

3.12 Thus we have that

k θu

i j



r−2u i j  βu

i j



r0−2u i j  k θ|u| r−2u  β|u| r0 −2u 3.13

weakly in L r/ r−1Ω Since the weak limit is independent of the choice of the subsequence,

it follows that

k θ|u i|r−2u i  β|u i|r0 −2u i  k θ|u| r−2u  β|u| r0 −2u 3.14

weakly in L r/ r−1 Ω Hence Λ is weakly continuous on K We may applyProposition 2.2to

obtain the existence of p.

Our uniqueness proof is inspired by15, Lemmas 3.11, 3.22, and Theorem 3.21 Since

kθ|∇u| r−2 βx|∇u| r0 −2∇u does not satisfy condition 3.4 of A operator in 15 , we need

to prove the following lemma, which is equivalent to15, Lemma 3.11 Then uniqueness can follow immediately from15, Lemma 3.22

Lemma 3.4 If u ∈ H 1,r Ω is a supersolution of 2.16 in Ω, then

 Ω



k θ|∇u| r−2 βx|∇u| r0 −2

for all nonnegative ϕ ∈ H 1,r

0 Ω.

Proof Let ϕ ∈ H 1,r

0 Ω and choose nonnegative sequence φ i ∈ C∞

0 Ω such that ϕ i → ϕ in

H 1,rΩ Equation 2.6 and H¨older inequality imply that





Ω



k θ|∇u| r−2 β|∇u| r0 −2

∇u · ∇ϕ dx −

 Ω



k θ|∇u| r−2 β|∇u| r0 −2

∇u · ∇ϕ i dx







Ωk θ|∇u| r−2∇u · ∇ϕ − ϕ i

dx



Ωβ |∇u| r0 −2∇u · ∇ϕ − ϕ i

dx



≤ k2 r−1

L rΩ∇ϕ − ϕ i

L rΩ β0 r0−1

L r0Ω∇ϕ − ϕ i

L r0Ω

≤ C r−1

L rΩ β0 r0−1

L r0Ω

∇ϕ − ϕ

i

L rΩ.

3.16

Because limi→ ∞ i L rΩ 0, we obtain



Ω



k θ|∇u| r−2 β|∇u| r0 −2

∇u · ∇ϕ dx  lim

i→ ∞

 Ω



k θ|∇u| r−2 β|∇u| r0 −2

∇u · ∇ϕ i dx≥ 0

3.17 and the lemma follows

Similar to15, Corollary 17.3, page 335 , one can also obtain the following Corollary

Corollary 3.5 Let Ω be bounded and p0 ∈ H 1,r Ω There is a weak solution p θ ∈ H 1,r

0 Ω  p0to

3.1 in the sense of Definition 3.1

Proof of Theorem 3.2 The existence result is given inCorollary 3.5, and we now turn to proof

of uniqueness For a given θ, assume that there exists another solution p1

θ Then we have that

Δ :

 Ω



k θ ∇p θr−2∇p θ−∇p1

θr−2

∇p1

θ



β ∇p θr0 −2∇p θ−∇p1

θr0 −2

∇p1

θ



· ∇ξ dx  0

3.18

for all ξ ∈ H 1,r

0 Ω If we take ξ  p θ − p1

θin above equation, from inequality2.7, we have the following

i when r ≥ 2,

0 Δ

≥



Ωk θ ∇p θr−2∇p θ−∇p1

θr−2

∇p1

θ



·∇p θ − ∇p1

θ



dx

≥ C

 Ω



∇p θ − ∇p1

θr

dx,

3.19

where C is a positive constant;

ii when 1 < r < 2,

0 Δ

≥



Ωk θ ∇p θr−2∇p θ−∇p1

θr−2

∇p1

θ



·∇p θ − ∇p1

θ



dx

≥ C

 Ω



∇p θ − ∇p1

θ2

b∇p θ ∇p1

θr−2

dx

≥ C

Ω



∇p1

θ − ∇p θr

dx

2/r Ω



b∇p θ ∇p1

θr

dx

r−2/r

.

3.20

Here the H ¨older inequality for 0 < t < 1, namely,



Ωfgdx

 ≥ Ωft dx

1/t Ω

gt∗dx

1/t∗

, t∗ t

is applied to the last inequality

Poincar´e’s inequality implies that p θ  p1

θa.e We complete the uniqueness proof Next we prove3.3 Taking ξ  p θ − p0in3.2, we have



Ωk θ∇p θr

dx≤



Ωk θ∇p θr−2∇p θ ∇p0dx



Ωβ ∇p θr0 −2∇p θ ∇p0dx. 3.22 From2.4, and the H¨older inequality, we obtain

k1



Ω∇p θr

dx ≤ k2

Ω∇p θr

dx

r−1/r

Ω∇p0r

dx

1/r

 β0

Ω∇p θr

dx

r0−1/r

Ω|∇p0|r/ r−r0 1dx

r−r01/r

.

3.23

Young’s inequality with ε implies

k1



Ω∇p θr

dx ≤ ε



Ω∇p θr

dx  C

Ω∇p0r

dx



Ω∇p0r/ r−r0 1dx



3.24

and3.3 follows immediately from 2.3 and 2.6

Finally, we prove3.4 From weak solution definition 3.2, we know that

 Ω



k θ m∇pθ mr−2 β∇p θ mr0 −2

∇p θ m ∇ξ dx



 Ω



k θ∇p θr−2 β∇p θr0 −2

∇p θ ∇ξ dx  0.

3.25

Setting ξ  p θ m − p θand subtracting

Ωkθ m |∇p θ|r−2 β|∇p θ|r0 −2∇p θ ∇ξ dx from both sides,

we obtain that



Ω



k θ m∇p

θ mr−2∇p θ m−∇p θr−2∇p θ



 β∇p

θ mr0 −2p θ m−∇p θr0 −2∇p θ



∇p θ m − p θ

dx





Ωkθ − kθ m∇pθr−2∇p θ∇p θ m − p θ

dx.

3.26

Denote the right-hand side byΔ1 Similar to arguments in the uniqueness proof, we arrive at the folloing:

i when r ≥ 2,

C



Ω∇p θ m − ∇p θr

ii when 1 < r < 2,

C

Ω∇p θ m − ∇p θr

dx

2/r Ω

b∇p θ   ∇p θ mr

dx

r−2/r

≤ Δ1. 3.28

Egorov’s Theorem implies that for all  > 0, there is a closed subsetΩofΩ such that |Ω\Ω | <

 and k θ m  → kθ uniformly on Ω  Application of the absolute continuity of the Lebesgue

Integral implies

Δ1≤



Ω



 Ω\Ω

|kθ m  − kθ|∇p θr−1∇p θ m − p θdx

≤ ε

Ω∇p θr

dx

r−1/r

 2k2

Ω∇p θ m − p θr

dx

1/r

−→ 0 as θ m −→ θ.

3.29

Theorem 3.2is proved

4 Nonlinear Elliptic Dirichlet System

Definition 4.1 We say that {θ, p} is a weak solution to Problem1if

θ − θ0∈ H 1,σ

0 Ω, p − p0∈ H 1,r

and for all v ∈ C∞

0 Ω

−



Ω∇θ · ∇v dx 

 Ω

k θ∇p r  qv dx, 4.2

and for all ξ ∈ H 1,r

0 Ω

 Ω



k θ∇p r−2 βx∇p r0 −2

Theorem 4.2 Assume that 2.1–2.6 hold Then there exists a weak solution to Problem 1 in the sense of Definition 4.1

We shall bound the critical growth,|∇p| r, on the right-hand side of4.2

Lemma 4.3 Suppose that θ and p satisfy

θ − θ0∈ H 1,σ

0 Ω, p − p0∈ H 1,r

and4.3 Then, under the conditions of Theorem 4.2 , for all v ∈ C1Ω



Ωk θ∇p r

vdx



Ωk θ∇p r−2∇p · ∇p0v dx

−



Ωk θ∇p r−2∇pp − p0

· ∇v dx

−



Ωβ ∇p r0 −2∇p · ∇p − p0

v dx

−



Ωβ ∇p r0 −2∇pp − p0

· ∇v dx.

4.5

Moreover, there exists a polynomial F that is independent of θ and p such that



Ωk θ∇p r

v dx ≤ Fp

H 1,rΩ



Proof We first show4.5 Letting ξ  vp − p0 in 4.3, we obtain



Ωk θ∇p r−2∇p ·v∇p − p0

p − p0

∇vdx





Ωβ ∇p r−2∇p ·v∇p − p0

p − p0

∇vdx  0.

4.7

After some straightforward computations this yields exactly4.5

We now show4.6 We denote the four terms on the right-hand side of equation 4.5

by I, II, III, and IV, respectively Under the conditions ofLemma 4.3, we have

∇p r−2∇p ∈ L r∗Ω, ∇p0∈ L τ Ω, r∗ r

Partiii ofLemma 2.1and Sobolev’s imbedding theorems indicate

|I| ≤ k2∇p r−1

L rΩ∇p0

L τΩ L ζ∗Ω

≤ C∇p r−1

L rΩ∇p0

L τΩ H 1,σ∗Ω

≤ C∇p r−1

L rΩ H 1,σ∗Ω,

4.9

where ζ∗ τr/τ − r satisfies r − 1/r  1/τ  1/ζ∗ 1

According to Sobolev’s imbedding theorems, the integrability ofp − p0 depends on

N We estimate II in three different cases

for all ξ ∈ H 1,r

0 Ω If we take ξ  p θ... given θ ∈ H 1,σ

0 Ω  θ0

Theorem... θr−2 βx∇p θr0 −2

for all ξ ∈ H 1,r

0 Ω and a