A Transmission Problem Across a Fractal Self-Similar Interface

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.

Density and trace results in generalized fractal networks

“…As underlined before, such a trace result has potential applications to inverse problem. Similar questions at a discrete level or on higher-dimensional domains are also considered, let us quote discrete laplacian on infinite networks [11,13,15,18,[20][21][22][23], two-dimensional domains with a fractal boundary [1][2][3][4], pre-fractal domains approximating the Koch snowflake [10]. Let us finally mention some related problems such as the Hamilton-Jacobi equation [5] and the Gauss-Bonnet operator on infinite graphs [6].…”

Gradient estimates for solutions of elliptic systems with measurable coefficients from composite material

“…Finally, we note further that according to Toro and Remark , we consider very rough boundary conditions on the internal boundaries ∂ Ω k for k = 1,…, N and ∂ Ω, including Lipschitz continuous functions or even fractals. These fractal structures are physically very natural and have many applications in multiple fields, such as electrochemistry related to rough electrodes or transfer across irregular membranes, and so on; see Achdou and Deheuvels and related results.…”

“…These problems have been developed in various ways because of their wide applications to many areas such as the conductivity problem through linearly elastic laminates, image smoothing and denoising problem, transmission problems, and so on. We refer to previous studies and references therein for related results.…”

Fractional Equations Via Convergence of Forms

“…A similar example can be given for domains with fractal boundaries as in [22] where the authors have obtained asymptotic results for skew Brownian diffusions across Koch interfaces by using M −convergence results proved in [20], [24], [25]. We recall that M −convergence results have been obtained on fractal structures in order to study several boundary value problems ( [2], [23], [56], [57], [58]), reinforcement problems for variational inequalities ( [26]), dynamical quasi-filling fractal layers, layered fractal fibers and potentials ( [27], [74], [75]).…”