I do not think this is a correctly stated Dirichlet problem. I am assuming you want to solve on the region depicted on the left. For each point in that region there exists a smooth bijection with so that . So if we define then we need to solve on the region on the right. However, the Dirichlet problem only tells us the boundary value and not the initial values. The boundary values that you providied do not completely define the boundary and so I do not think your problem is fairly stated to be solved.