Derivation and analytical investigation of three direct boundary integral equations for the fundamental biharmonic problem

We derive and investigate three families of direct boundary integral equations for the solution of the plane, fundamental biharmonic boundary value problem. These three families are fairly general so that they, as special cases, encompass various known and applied equations as demonstrated by giving many references to the literature. We investigate the families by analytical means for a circular boundary curve where the radius is a parameter. We find for all three combinations of equations (i) that the solution of the equations is non-unique for one or more critical radius/radii, and (ii) that this lack of uniqueness can always be removed by combining the integral equations with a suitable combination of one or more supplementary condition(s). We conjecture how the results obtained can, or cannot, be generalized to other boundary curves through the concept logarithmic capacity. A few published general results about uniqueness are compared with our findings.