The Dirichlet problem on two-dimensional stratified sets

We consider the Dirichlet problem for harmonic functions on two-dimensional stratified sets, which are assumed for simplicity to be complexes. We show that under certain conditions this problem is Fredholm in the Hölder space and in weighted Hölder spaces of functions satisfying the Hölder condition outside any neighbourhood of the vertex set of the complex and admitting power singularities. We also study the power-logarithmic asymptotics of solutions at these vertices.