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

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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 remov ed by combining the integral equations with a suitable combination of one or more supplementary condition(s). We conjecture how the results obtained can, or can not, be generalized to other boundary curves through the concept logarithmic capacity. A few published general results about uniqueness are compared with our findings.

Keywords: Boundary integral equation, fundamental biharmonic problem, uniqueness of solution, logarithmic capacity.

Note: The investigation in this report (which has been accepted for publication in Journal of Computational and Applied Mathematics) will be continued in two further reports.

Technical report 01/98

For further information, please contact, Finn Kuno Christensen, IMM, Bldg. 321, DTU
Phone: (+45) 4588 1433. Fax: (+45) 4588 2673, E-mail: fkc@imm.dtu.dk