Existence of Weak Solutions for a Nonlinear Elliptic System
© M. Fang and R.P. Gilbert 2009
Received: 3 April 2009
Accepted: 31 July 2009
Published: 26 August 2009
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 .
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  in 1827 first proposed a partial slip condition for rough surfaces, relating the tangential velocity to the local tangential shear stress
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.
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 . 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.
2. Notations and Preliminaries
In this paper, for let and denote the usual Sobolev space equipped with the standard norm. Let
where . The conjugate exponent of is
We assume that the boundary values and for Problem 1 can be extended to functions defined on such that
We further assume that there exist positive numbers and such that
Finally, we assume that for , a.e. in indicates
For the convenience of exposition, we assume that
Next, we recall some previous results which will be needed in the rest of the paper.
An important inequality (e.g., see [11, page 550] ) in the study of -Laplacian is as follows:
where and are certain constants.
To establish coercivity condition, we will use the following inequality:
where , , and .
The following statements hold
The uniqueness proof is based on a supersolution argument (similar definition can be found in [15, Chapter 3]).
whenever is nonnegative.
3. A Dirichlet Boundary Value Problem
We study the following Dirichlet boundary value problem:
for all and a given .
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.
where is a constant independent of and ;
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.
It follows from the proof in [15, Proposition 17.2] that is a closed convex set.
Here we used Assumption (2.6), that is, . Therefore we have whenever . Moreover, it follows from inequality (2.7) that is monotone.
Inequality (2.8) is used to arrive at the last step. This implies that is coercive on .
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 , 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].
for all nonnegative .
and the lemma follows.
Similar to [15, Corollary 17.3, page 335], one can also obtain the following Corollary.
Let be bounded and . There is a weak solution to (3.1) in the sense of Definition 3.1.
Proof of Theorem 3.2.
for all . If we take in above equation, from inequality (2.7), we have the following.
where is a positive constant;
is applied to the last inequality.
Poincaré's inequality implies that a.e. We complete the uniqueness proof.
and (3.3) follows immediately from (2.3) and (2.6).
Denote the right-hand side by . Similar to arguments in the uniqueness proof, we arrive at the folloing:
Theorem 3.2 is proved.
4. Nonlinear Elliptic Dirichlet System
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).
After some straightforward computations this yields exactly (4.5).
where satisfies .
According to Sobolev's imbedding theorems, the integrability of depends on . We estimate II in three different cases.
Case 1 ( ).
Case 2 ( ).
Case 3 ( ).
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 ( ).
Case 2 ( ).
Case 3 ( ).
for some polynomial .
Proof of Theorem 4.2.
The project is partially supported by NSF/STARS Grant (NSF-0207971) and Research Initiation Awards at the Norfolk State University. The second author's work has been supported in part by NSF Grants OISE-0438765 and DMS-0920850. The project is also partially supported by a grant at Fudan University.
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