Consider the Dirichlet boundary value problem of the 2-dimensional Laplace's equation. When the boundary is piecewise smooth, it can be solved by the Green's function for the double layer potential. Consider the case when the boundary is continuous but nowhere differentiable such as a fractal curve. I would like to know the answers or see a rough survey of the answers, or be pointed to the references pertaining, to the following questions.
- What is an analog of the Green's function?
- What are some of the solution methods other than Green's function analog?