Contour Integrals and Vector Calculus on Fractal Curves and Interfaces

This article develops the definition of contour integrals over fractal curves in the plane by introducing the notion of oriented Iterated Function Systems and directional pseudo-measures. An expression for the contour integral of continuous functions over fractal interfaces is obtained through renormalization. As a result, a vector calculus on fractal interfaces which are boundaries of regular two-dimensional domains is developed by extending Greens theorem in the plane, also to fractal curves. The use of moment analysis makes it possible to obtain recursive relations and closed-form expressions for contour integrals of algebraic functions. Several physical applications are analyzed, including the properties of double-layer potentials and connections with the solution of the Dirichlet problem on bounded two-dimensional domains possessing fractal boundaries.