Báo cáo sinh học: Integral representations for solutions of some BVPs for the Lame' system in multiply connected domains

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Báo cáo sinh học: " Integral representations for solutions of some BVPs for the Lame system in multiply connected domains"Boundary Value Problems This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Integral representations for solutions of some BVPs for the Lame system in multiply connected domains Boundary Value Problems 2011, 2011:53 doi:10.1186/1687-2770-2011-53 Alberto Cialdea (cialdea@email.it) Vita Leonessa (vita.leonessa@unibas.it) Angelica Malaspina (angelica.malaspina@unibas.it) ISSN 1687-2770 Article type Research Submission date 21 May 2011 Acceptance date 12 December 2011 Publication date 12 December 2011 Article URL http://www.boundaryvalueproblems.com/content/2011/1/53 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Cialdea et al. ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Integral representations for solutions of some BVPs for theLam´ system in multiply connected domains eAlberto Cialdea∗ 1 , Vita Leonessa1 and Angelica Malaspina11 Department of Mathematics and Computer Science, University of Basilicata,V.le dell’Ateneo Lucano, 10, Campus of Macchia Romana, 85100 Potenza, ItalyEmail: Alberto Cialdea∗ - cialdea@email.it; Vita Leonessa - vita.leonessa@unibas.it; Angelica Malaspina -angelica.malaspina@unibas.it;∗ Corresponding authorAbstract The present paper is concerned with an indirect method to solve the Dirichlet and the traction problems forLam´ system in a multiply connected bounded domain of Rn , n ≥ 2. It hinges on the theory of reducible operators eand on the theory of differential forms. Differently from the more usual approach, the solutions are sought in theform 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 and phrases. Lam´ system; boundary integral equations; potential theory; differential forms; emultiply connected domains.1 IntroductionIn this paper we consider the Dirichlet and the traction problems for the linearized n-dimensional elasto-statics. The classical indirect methods for solving them consist in looking for the solution in the form ofa double layer potential and a simple layer potential respectively. It is well-known that, if the boundaryis sufficiently smooth, in both cases we are led to a singular integral system which can be reduced to aFredholm one (see, e.g., [1]). Recently this approach was considered in multiply connected domains for several partial differentialequations (see, e.g., [2–7]). 1 However these are not the only integral representations that are of importance. Another one consistsin looking for the solution of the Dirichlet problem in the form of a simple layer potential. This approachleads to an integral equation of the first kind on the boundary which can be treated in different ways. Forn = 2 and Ω simply connected see [8]. A method hinging on the theory of reducible operators (see [9, 10])and the theory of differential forms (see, e.g., [11, 12]) was introduced in [13] for the n-dimensional Laplaceequation and later extended to the three-dimensional elasticity in [14]. This method can be considered asan extension of the one given by Muskhelishvili [15] in the complex plane. The double layer potential ansatzfor the traction problem can be treated in a similar way, as shown in [16]. In the present paper we are going to consi ...