Free interpolation and Dirichlet problem

A compact subset $K$ of the closed upper half-plane $\mathbb C_+\cup\mathbb R$ is called a Dirichlet set if every function continuous on $K$ is the restriction of a function continuous in $\mathbb C_+\cup\mathbb R$ and harmonic in $\mathbb C_+$. A description of Dirichlet sets is given.