We study the regularity of solutions of interface problems for the Poisson equation in axisynunetric domains. The Fourier decomposition of the 3D-problem into a sequence of 2Dvariational equations end uniform (with respect to the sequence parameter) a prior; estimates of their solutions are derived. Some non-tensorial singular functions describing the behaviour of the solution near interface edges are given and the smoothness of the stress intensity distribution as well as the tangential regularity are characterized in tenns of Sobolev spaces. In a forthcoming part I1 of this paper, the results will be applied to error estimates of the so-called Fourier-finite-element method for solving approximately elliptic interface problems in 3D.In this paper, we study these topics for the Poisson equation -Aii = f under interface conditions involving the piecewise constant coefficient 6 and for f E L2 (6). Furthermore, a the boundary := 6' 6 as well as the interface E* may have edges or E* may meet r at some edge. Interface problems in 3D are considered in [8, 10, 11, 14, 21, 221. For smooth 1991 Maihematics Subjecf Classification. Primary 35 J 25, 35D 10; Secondary 35R05, 42 A 16. K e y w o~d 8 and pkbTases. Elliptic interface problem, axisymmetric interface edges, non-tensorial singular functions, Fourier decomposition.

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