Invertibility of the biharmonic single layer potential operator

Martin Costabel and Monique Dauge

Integral Equations and Operator Theory 24 (1996) 46-67

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The 2X2 system of integral equations corresponding to the biharmonic single layer potential in the plane 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 4X4 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.