A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

A Note on the Displacement Problem of Elastostatics with Singular Boundary Values

“…As far as the two-dimensional case is concerned, the situation is more involved (also for regular data) because of the behavior of the fundamental solution (U(x − y) = O(ln(|x − y|)). As pointed out in [15] (see also [16]), in this case, the search for a solution in the form of a simple layer potential v[ψ] could not lead to existence and uniqueness for degenerate-scale problems. To overcome this difficulty, one may search for the solution in the form of a sum v[ψ] + c, with c constant and ∂Ω ψ = 0 [15].…”

“…As pointed out in [15] (see also [16]), in this case, the search for a solution in the form of a simple layer potential v[ψ] could not lead to existence and uniqueness for degenerate-scale problems. To overcome this difficulty, one may search for the solution in the form of a sum v[ψ] + c, with c constant and ∂Ω ψ = 0 [15]. This could be the starting point for further research into existence and uniqueness with singular data in two-dimensional domains.…”

“…is a solution to (1) which is C ∞ in Ω and satisfies (1) 2 in the sense of (15). Let a j be a regular sequence on ∂Ω which converges to a strongly in W 2−k−1/q,q (∂Ω).…”

“…First of all, we observe that Lemma 1 also holds for exterior domains (Theorem 1 in [11]). Thus, we can apply the Fredholm alternative again, obtaining a solution ψ to (15) and the corresponding solution u = v[ψ] to (21). Then, with the analogous meaning of a j and v[ψ j ], in place of (17) and (18), we get…”

A necessary and sufficient BEM/BIEM for two-dimensional elasticity problems

“…By using the degenerate kernel in terms of the elliptic coordinates [17], the coefficients of the eigenfunction representation of the boundary densities could be obtained as follows [18]: (1) (1) 0 0…”