Scalar boundary value problems on junctions of thin rods and plates

I. Asymptotic analysis and error estimates

¹ Universitéde Lorraine, Institut Elie Cartan de Lorraine, 7502 UMR, 57045 Metz, France.
renata.bunoiu@univ-lorraine.fr
² University of Sannio − Department of Engineering, Piazza Roma, 21, 84100 Benevento, Italy.
giuseppe.cardone@unisannio.it
³ Mathematics and Mechanics Faculty, St. Petersburg State University 198504, Universitetsky pr., 28, Stary Peterhof, Russia.
srgnazarov@yahoo.co.uk

Received: 13 May 2013
Revised: 7 January 2014

Abstract

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative thickness, appears in the differential equation, too, while the asymptotic structures crucially depend on the contrastness ratio. Asymptotic error estimates are derived in anisotropic weighted Sobolev norms.