Invertibility of the biharmonic single layer potential operator

The 2 x 2 system of integral equations corresponding to the biharmonic single layer potential in R(2) is known to be strongly elliptic. It is also known to be positive definite on a space of functions orthogonal to polynomials of degree one. We study the question of its unique solvability without this orthogonality condition. To each curve Gamma, we associate a 4 x 4 matrix B-Gamma such that this problem for the family of all curves obtained from Gamma by scale transformations is equivalent to the eigenvalue problem for B-Gamma. We present numerical approximations for this eigenvalue problem for several classes of curves.