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Báo cáo toán học: " Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains | Cialdea et al. Boundary Value Problems 2011 2011 53 http www.boundaryvalueproblems.eom content 2011 1 53 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Integral representations for solutions of some BVPs for the Lamé system in multiply connected domains Alberto Cialdea Vita Leonessa and Angelica Malaspina Correspondence cialdea@email.it Department of Mathematics and Computer Science University of Basilicata V.le dell Ateneo Lucano 10 Campus of Macchia Romana 85100 Potenza Italy Springer Abstract The present paper is concerned with an indirect method to solve the Dirichlet and the traction problems for Lamé system in a multiply connected bounded domain of Rn n 2. It hinges on the theory of reducible operators and on the theory of differential forms. Differently from the more usual approach the solutions are sought in the form of a simple layer potential for the Dirichlet problem and a double layer potential for the traction problem. 2000 Mathematics Subject Classification. 74B05 35C15 31A10 31B10 35J57. Keywords Lam é system boundary integral equations potential theory differential forms multiply connected domains 1 Introduction In this paper we consider the Dirichlet and the traction problems for the linearized n-dimensional elastostatics. The classical indirect methods for solving them consist in looking for the solution in the form of a double layer potential and a simple layer potential respectively. It is well-known that if the boundary is sufficiently smooth in both cases we are led to a singular integral system which can be reduced to a Fredholm one see e.g. 1 . Recently this approach was considered in multiply connected domains for several partial differential equations see e.g. 2-7 . However these are not the only integral representations that are of importance. Another one consists in looking for the solution of the Dirichlet problem in the form of a simple layer potential. This approach leads to an integral equation of the first .