Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

✍ Scribed by T. A. Mel'nyk; Iu. A. Nakvasiuk; W. L. Wendland

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 286 KB
• Volume: 34
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1395

✦ Synopsis

We consider a mixed boundary-value problem for the Poisson equation in a thick junction X e which is the union of a domain X 0 and a large number of e-periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as e → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non-uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as e → 0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non-standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in X 0 and an appropriate postprocessing. The equations in X 0 finally are also treated with boundary integral equations.