A Transmission Problem across a Fractal Self-similar Interface
Abstract
We consider a transmission problem where the interior domain has infinitely ramified structures. Transmission between the interior and exterior domains only takes place at the fractal component of the boundary of the interior domain. We also consider 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 of the interior domain.
Emphasis is put on the geometry of the ramified domain. The convergence result is proved when the fractal interface has no self-contact, and for a particular geometry with self-contacts for which an extension result is proved.
Domains
Analysis of PDEs [math.AP]
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