A transmission problem across a fractal self-similar interface

TL;DR

This paper studies a transmission problem across a fractal interface with infinitely ramified structures, proving convergence of solutions from prefractal approximations to the fractal limit using Mosco convergence and extension properties.

Contribution

It establishes the convergence of energy forms and solutions for transmission problems on fractal interfaces, including cases with and without self-contact, using Mosco convergence techniques.

Findings

Convergence of solutions from prefractal to fractal interface problems.

Extension property for ramified domains with no self-contact.

Mosco convergence of energy forms in fractal transmission problems.

Abstract

We consider a transmission problem in which the interior domain has infinitely ramified structures. Transmission between the interior and exterior domains occurs only at the fractal component of the interface between the interior and exterior domains. We also consider the sequence of the transmission problems in which the interior domain is obtained by stopping the self-similar construction after a finite number of steps; the transmission condition is then posed on a prefractal approximation of the fractal interface. We prove the convergence in the sense of Mosco of the energy forms associated with these problems to the energy form of the limit problem. In particular, this implies the convergence of the solutions of the approximated problems to the solution of the problem with fractal interface. The proof relies in particular on an extension property. Emphasis is put on the geometry of the ramified domain. The convergence result is obtained when the fractal interface has no self-contact, and in a particular geometry with self-contacts, for which an extension result is proved.