In recent years an alternative method of integral equations has been used for numerical solution of the Dirichlet problem for the Laplace equation or the biharmonic equation in a bounded plane domain. This method is based simply on Green's third identity, and its main advantage is that is leads directly from the prescribed Dirichlet boundary data to the (physically relevant) Neumann data. It has been conjectured recently by S. Christiansen and P. Hougaard that, for certain values of the outer mapping radius of the domain, the integral equation(s) in question has (have) other solutions than that which stems from the solution to the Dirichlet problem. They proved this for a circular domain and gave numerical indications in another case. In the present paper we prove this and related conjectures in the case of a general domain with smooth boundary. We do not discuss the numerical aspects of the method in question.
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Bent Fuglede (1981) studied this question.