Transport properties of doubly periodic arrays of circular cylinders and optimal design problems V. Mityushev Abstract We solve an R-linear problem for a multiply connected circular domain in a class of doubly periodic functions in analytic form by a method of functional equations. This problem models transport prop- erties of two-dimensional composite materials made from a collection of disks imbedded in an otherwise uniform host. Institute of Mathematics, Pedagogical College, ul.Arciszewskiego 22 b, 76- 200 Slupsk, Poland MSC subject classification: 30E25, 73V35 Key words. boundary value problem, effective conductivity, functional equation 1 Introduction Transport properties of two-dimensional doubly periodic composite materials made from a collection of non-overlapping circular disks imbedded in an otherwise uniform host are considered. There have been many theoretical approaches to this problem. One of them is to consider specific periodic structure, and to solve approximately or analytically the transport problem. Grigiluk & Filshtinskij [4] applied a method of singular integral equations to boundary value problems for doubly periodic functions. McPhedran et al. [9] (see also papers cited therein) studied the square and triangle arrays of disks. Having based on the classical paper of Lord Rayleigh [14] they reduced the problem to an infinite set of linear algebraic equations. Similar results are also represented in [4] and works cited therein. Kolodziej [6] applied the method of collocations to study a wide class of doubly periodic composite materials. Sangani & Yao [16] developed the method of singular solutions and 1