An inverse problem for the double layer potential

We consider an inverse problem for the double layer potential which can be formulated, somewhat loosely, as follows. For which smoothly bounded domains D in Euclidian space does the operator J, which maps a function on the boundary to the boundary values of its double layer potential, admit the eigenvalue 1/2. The question is motivated by Fredholm's solution to the Dirichlet problem by means of the double layer potential.