On the Martin boundary of a plane domain whose complement is situated on a straight line (Q913994)

Let E be a closed set in \({\mathbb{R}}\) and suppose that all the points of E are regular boundary points of \(D_ E={\mathbb{C}}-E\). It is known that the cone of positive harmonic functions on \(D_ E\) with zero boundary values on E has its dimension equal either to 1 or to 2. Some necessary and sufficient conditions on E which guarantee that the dimension of that cone is equal to 2 are given.