Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Existence of Weak Solutions for a Nonlinear Elliptic System | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 708389 15 pages doi 2009 708389 Research Article Existence of Weak Solutions for a Nonlinear Elliptic System Ming Fang1 and Robert P. Gilbert2 1 Department of Mathematics Norfolk State University Norfolk VA 23504 USA 2 Department of Mathematical Sciences University of Delaware Newark DE 19716 USA Correspondence should be addressed to Ming Fang mfang@ 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 -A0 k ff Vp r q x in Q -div k 0 Vp r-2 p x Vp r -2 Vp 0 in Q 0 00 and p p0 on dQ. Copyright 2009 M. Fang and R. P. Gilbert. This is an open access article distributed under 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 va to the local tangential shear stress Ta3 Va -pTa3 where p indicates the amount of slip. When p 0 reduces to the no-slip boundary condition. A nonzero p implies partial slip. As p TO 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. 2 Boundary Value Problems Problem 1. Find functions 0 and p defined in Q such that -A0 k 0 Vp r q x
