Existence of Weak Solutions for a Nonlinear Elliptic System

Abstract

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 in ; in ; , and on .

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 to the local tangential shear stress

(11)

where indicates the amount of slip. When , (1.1) reduces to the no-slip boundary condition. A nonzero implies partial slip. As , the solid surface tends to full slip.

There is a full description of the injection molding process in [3] and in our paper [7]. The formulation of this process as an elliptic system is given here in after.

Problem 1.

Find functions and defined in such that

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(13)

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Here we assume that is a bounded domain in with a boundary. We assume also that , , , , and are given functions, while is a given positive constant related to the power law index; is the pressure of the flow, and is the temperature. The leading order term of the PDE (1.3) 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, equation ()] .

The mathematical model for this system was established in [7]. Some related papers, both rigorous and formal, are [3, 11–13]. In [11, 13], existence results in no-slip surface, , are obtained, while in [3, 7], Navier's slip conditions, and , 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 dimension.

In Section 2, we introduce some notations and lemmas needed in later sections. In Section 3, we investigate the existence, uniqueness, stability, and continuity of solution to the nonlinear equation (1.3). In Section 4, we study the existence of weak solutions to Problem 1.

Using Rothe's method of time discretization and an existence result for Problem 1, one can establish existence of week solutions to the following time-dependent problem.

Problem 2.

Find functions and defined in such that

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The proof is only a slight modification of the proofs given in [11, 13] and is omitted here.

2. Notations and Preliminaries

2.1. Notations

In this paper, for let and denote the usual Sobolev space equipped with the standard norm. Let

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where . The conjugate exponent of is

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

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We further assume that there exist positive numbers and such that

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Finally, we assume that for , a.e. in indicates

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For the convenience of exposition, we assume that

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Next, we recall some previous results which will be needed in the rest of the paper.

2.2. Preliminaries

An important inequality (e.g., see [11, page 550] ) in the study of -Laplacian is as follows:

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where and are certain constants.

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

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where , , and .

Using the Sobolev Embedding Theorem and Hölder's Inequality, we can derive the following results (for more details, see [11, Lemma  3.4] and [13, Lemma  4.2]).

Lemma 2.1.

The following statements hold

() For any positive numbers and , if and then

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moreover,

() If and , then moreover,

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() If and , then , where

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and denotes the conjugate of r, namely, for moreover,

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() If and , then

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where and . Moreover

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The existence proof will use the following general result of monotone operators [14, Corollary  III.1.8, page 87] and [15, Proposition  17.2].

Proposition 2.2.

Let be a closed convex set (), and let be monotone, coercive, and weakly continuous on . Then there exists

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The uniqueness proof is based on a supersolution argument (similar definition can be found in [15, Chapter  3]).

Definition 2.3.

A function is a weak supersolution of the equation

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in if

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whenever is nonnegative.

3. A Dirichlet Boundary Value Problem

We study the following Dirichlet boundary value problem:

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Definition 3.1.

We say that is a weak solution to (3.1) if

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for all and a given .

Theorem 3.2.

Assume that conditions (2.1)–(2.6) are satisfied. Then there exists a unique weak solution to the Dirichlet boundary value problem (3.1) in the sense of Definition 3.1. In addition, the solution satisfies the following properties.

we have

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where is a constant independent of and ;

if a.e. in , then

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The idea behind the existence proof is related to [15, 16]. We will first consider the following Obstacle Problem.

Problem 3.

Find a function in such that

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for all . Here

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Lemma 3.3.

If is nonempty, then there is a unique solution p to the Problem 3 in .

Proof of Lemma 3.3.

Our proof will use Proposition 2.2.

Let and write

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It follows from the proof in [15, Proposition  17.2] that is a closed convex set.

Next we define a mapping by

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By Hölder's inequality,

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Here we used Assumption (2.6), that is, . Therefore we have whenever . Moreover, it follows from inequality (2.7) that is monotone.

To show that is coercive on , fix . Then

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Inequality (2.8) is used to arrive at the last step. This implies that is coercive on .

Finally, we show that is weakly continuous on . Let be a sequence that converges to an element in . Select a subsequence {} such that a.e. in . Then it follows that

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a.e. in . Moreover,

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Thus we have that

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weakly in . Since the weak limit is independent of the choice of the subsequence, it follows that

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weakly in . Hence is weakly continuous on . We may apply Proposition 2.2 to obtain the existence of .

Our uniqueness proof is inspired by [15, Lemmas , , and Theorem ]. Since does not satisfy condition (3.4) of operator in [15], we need to prove the following lemma, which is equivalent to [15, Lemma  3.11]. Then uniqueness can follow immediately from [15, Lemma  3.22].

Lemma 3.4.

If is a supersolution of (2.16) in , then

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for all nonnegative .

Proof.

Let and choose nonnegative sequence such that in . Equation (2.6) and Hölder inequality imply that

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Because , we obtain

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and the lemma follows.

Similar to [15, Corollary  17.3, page 335], one can also obtain the following Corollary.

Corollary 3.5.

Let be bounded and . There is a weak solution to (3.1) in the sense of Definition 3.1.

Proof of Theorem 3.2.

The existence result is given in Corollary 3.5, and we now turn to proof of uniqueness. For a given , assume that there exists another solution Then we have that

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for all . If we take in above equation, from inequality (2.7), we have the following.

() when ,

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where is a positive constant;

() when ,

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Here the Hölder inequality for , namely,

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is applied to the last inequality.

Poincaré's inequality implies that a.e. We complete the uniqueness proof.

Next we prove (3.3). Taking in (3.2), we have

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From (2.4), and the Hölder inequality, we obtain

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Young's inequality with implies

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and (3.3) follows immediately from (2.3) and (2.6).

Finally, we prove (3.4). From weak solution definition (3.2), we know that

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Setting and subtracting from both sides, we obtain that

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

() when ,

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() when ,

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Egorov's Theorem implies that for all , there is a closed subset of such that and uniformly on . Application of the absolute continuity of the Lebesgue Integral implies

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Theorem 3.2 is proved.

4. Nonlinear Elliptic Dirichlet System

Definition 4.1.

We say that is a weak solution to Problem 1 if

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and for all

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and for all

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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, , on the right-hand side of (4.2).

Lemma 4.3.

Suppose that and satisfy

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and (4.3). Then, under the conditions of Theorem 4.2, for all

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

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Proof.

We first show (4.5). Letting in (4.3), we obtain

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After some straightforward computations this yields exactly (4.5).

We now show (4.6). We denote the four terms on the right-hand side of equation (4.5) by I, II, III, and IV, respectively. Under the conditions of Lemma 4.3, we have

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Part (iii) of Lemma 2.1 and Sobolev's imbedding theorems indicate

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where satisfies .

According to Sobolev's imbedding theorems, the integrability of depends on . We estimate II in three different cases.

Case 1 ().

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Case 2 ().

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Case 3 ().

is a bounded continuous function, so

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We next estimate III:

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The estimate of the first term used Hölder inequality and Sobolev's imbedding theorems. The argument of the second estimate is similar to that of I.

Recall that . Similar to II, we estimate IV in three different cases.

Case 1 ().

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Since , we have

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Case 2 ().

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Since , we have

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Case 3 ().

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These estimates lead to

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for some polynomial .

Proof of Theorem 4.2.

Using Theorem 3.2, let then for (3.2) there exists a unique solution satisfying

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Moreover, if a.e. in , then

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Next, using Lemma 4.3, we can define a linear functional determined by

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for all By virtue of (4.6), is well defined, and there exists a constant independent of such that

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We notice that (4.2) is the same as [11,  equation ()]. Therefore, arguments after [11,  equation ()] can be used to complete the proof of Theorem 4.2.