A Transmission Problem with a Fractal Interface

Abstract

In this paper we study a transmission problem with a fractal interface $K$, where a second order transmission condition is imposed. We consider the case in which the interface $K$ is the Koch curve and we prove existence and uniqueness of the weak solution of the problem in $V(\Omega ,K)$, a suitable ”energy space”. The link between the variational formulation and the problem is possible once we recover a version of the Gauss-Green formula for fractal boundaries, hence a definition of ”normal derivative”.