General interface problems—I

Abstract

Communicated by E. MeisterWe study transmission problems for elliptic operators of order 2m with general boundary and interface conditions, introducing new covering conditions. This allows to prove solvability, regularity and asymptotics of solutions in weighted Sobolev spaces. We give some numerical examples for the location of the singular exponents.

COMPUTATION of generalized stress intensity factors for bonded elastic structures

“…Moreover, the leading terms in the asymptotics (8) have the same structure as in the case of operators with constant coefficients. We extend these ideas to boundary transmission problems (see [24]). Since the principal part of the operator A with coefficients frozen in the corner point P coincides with the original Lamé and boundary operators, we study the boundary transmission problem in the double wedge C 1 (P ) ∪ C 2 (P ) with vanishing right-hand sides…”

“…Let (v 1,j , v 2,j ) and (w 1,j , w 2,j ) be the functions defined by (24) and (26), respectively. Then…”

“…and (w 1,j , w 2,j ) defined by (24) and (26), respectively, satisfy the biorthonormality condition (30). Then the coefficients c j in (25) are given by…”

“…The results [6,20,22,28] about asymptotic expansions for boundary value problems can be immediately extended to boundary transmission problem using ideas from [24].…”

“…Asymptotic expansions of this type were derived by Kondrat'ev [15] for general elliptic boundary value problems using Mellin transformation techniques and were adapted to boundary transmission problems by Nicaise and Sändig [24]. For three-dimensional elastic displacement and harmonic fields near smooth edges or crack fronts the asymptotics can be expressed in cylindrical coordinates (r, ω, z) η u 1 (r, ω, z) u 2 (r, ω, z) ∼ η j c j (z)r αj (z) ϕ 1,j (log r, ω) ϕ 2,j (log r, ω) ,…”