Elliptic Interface Problems in Axisymmetric Domains. Part I: Singular Functions of Non ‐ Tensorial Type

Abstract

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.

Transmission Problems for the Laplace and Elasticity Operators: Regularity and Boundary Integral Formulation

“…51 For domains without vertex singularities, we may cite Grisvard, 20,21 Mikhailov, 43 Roßmann and Sändig, 54 Heinrich, Nicaise and Weber. 22 The vertex singularities are described as in 2-D using spherical coordinates (r, ω) instead of polar coordinates (see the usual references about singularities for details). For a fixed vertex S of Ω common to Ω 1 and Ω 2 , we denote by A S (α) the operator bundles acting on the section G S = C S ∩ S 1 , which is the intersection of the unit sphere with the cone C S of origin S and which coincides with Ω near S (note that C S is split into two cones C S,k corresponding to each piece Ω k ).…”

Boundary-contact problems for domains with edge singularities

“…Now we obtain σ tr (A) (the so-called principal transmission symbol of A) from σ ∂ (A V + ) by applying the push forward (ε −1 ) * to the operators of the second column of (9) from R + to R − , similarly as the relation between operators (7), (8). This gives rise to an operator family…”

“…Problems of this kind have been investigated by several authors, in different context, partly under specific assumptions on the geometry or the involved dimensions, cf. Lemrabet [14], Escauriaza, Fabes, and Verchota [6], Torres and Welland [21], Chkadua [3,4], Li and Vogelius [13], Li and Nirenberg [12], Nicaise and Sändig [16] (numerical method), Heinrich, Nicaise and Weber [8] (the Fourier-finite-element method and singular functions of non-tensorial type), Kapanadze and Schulze [10] (the latter paper studies the case with conical singularities at the interfaces).…”

Fourier series and integral equation method for the exterior Stokes problem

“…Equally, Dauge [13] and Maz'ya [41] work with the Fourier transform in the z-direction of a dihedral which, if truncated by a vertical plane, is a prismatic domain. We mention also the works of Heinrich, Nicaise and Webber [26][27][28]; these authors use the Fourier series in the setting of an axisymmetric domain, which if written in cylindrical co-ordinates is a prism.…”

“…The Fourier series method has proven to be successful in many concrete applications. For instance, this method was again used, not long time ago, for the qualitative and constructive treatment of edge singularities of interface problems on three-dimensional axisymmetric domains (see [26][27][28]). We refer also to the work [51], which includes some contributions on periodic solutions of the Navier-Stokes equations.…”