Zeitschrift f r Analysis und ihre Anwendungen Journal for Analysis and its Applications Volume 18 (1999), No. 4, 953-975 Homogenization of the Poisson Equation in a Thick Periodic Junction T. A. Mel’nyk Abstract. A convergence theorem and asymptotic estimates as C - 0 are proved for a solution to a mixed boundary-value problem for the Poisson equation in a junction Q, of a domain o and a large number N 2 of c-periodically situated thin cylinders with thickness of order e = (*) For this junction, we construct an extension operator and study its properties. Keywords: Homogenization, asymptotic estimates, extension operators AMS subject classification: Primary 35 B 27, secondary 35 B 40, 35 B 25 0. Introduction Let D be a domain in R" which depends on a small parameter c > 0 and, by the limit as c -p 0, is transformed to a submanifold S of dimension in. The number in is called the limit dimension of the domain D. If m < n, then the domain D is called thin. Asymptotic methods for thin domains are well-known. Some years ago several papers appeared that deal with the asymptotic investigation of boundary-value problems in junctions consisting of a finite number of domains with different limit dimensions 11, 5, 11, 12, 22, 23). Boundary-value problems in thick periodic junctions whose number of components increases as c -i 0 have own specific difficulties (see below), and until recently, there were no full asymptotic investigations of these problems. For these junctions we give the following classification: A thick periodic junction fl of type m k : d is a domain in RY’ that is obtained by joining a large number of c-periodically situated thin domains with limit dimensions d to an external part of the boundary (which is the contact zone with limit dimension k m) of a domain Qo (which is the junction’s "body" with limit dimension in n). Here c is a small parameter which depends on the number of the joined thin domains. The junction can have two or more "bodies". These junctions are prototypes of widely adapted engineering constructions such as long bridges on supports, frameworks of houses, industrial installations, spaceship grids as well as other physical systems with very distinct characteristic scales. The objective of studying boundary-value problems in thick periodic junctions is to describe T. A. Mel’nyk: Inst. Math.A Univ., PF 801140, D-70511 Stuttgart ISSN 0232-2064 / $ 2.50 Heldermann Verlag Berlin