A transmission problem across a fractal self-similar interface

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

Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries

International audienceWe consider a class of ramified bidimensional domains with a self-similar boundary, which is supplied with the self-similar probability measure. Emphasis is put on the case when the domain is not an epsilon-delta domain as defined by Jones and the fractal is not totally disconnected.We compare two notions of trace on the fractal boundary for functions in some Sobolev space, the classical one ( the strict definition ) and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide almost everywhere with respect to the self similar probability measure

JLip versus Sobolev Spaces on a Class of Self-Similar Fractal Foliages

International audienceFor a class of self-similar sets $\Gamma^\infty$ in $\R^2$, supplied with a probability measure $\mu$ called the self-similar measure, we investigate if the $B_s^{q,q}(\Gamma^\infty)$ regularity of a function can be characterized using the coefficients of its expansion in the Haar wavelet basis. Using the the Lipschitz spaces with jumps recently introduced by Jonsson, the question can be rephrased: when does $B_s^{q,q}(\Gamma^\infty)$ coincide with $JLip(s,q,q;0;\Gamma^\infty)$? When $\Gamma^\infty$ is totally disconnected, this question has been positively answered by Jonsson for all $s,q$, $00$, $1\le p,q<\infty$, using possibly higher degree Haar wavelets coefficients). Here, we fully answer the question in the case when $

Trace Results on Domains with Self-Similar Fractal Boundaries

International audienceThis work deals with trace theorems for a family of ramified domains $\Omega$ with a self-similar fractal boundary $\Gamma^\infty$. The fractal boundary $\Gamma^\infty$ is supplied with a probability measure $\mu$ called the self-similar measure. Emphasis is put on the case when the domain is not a $\epsilon-\delta$ domain and the fractal is not post-critically finite, for which classical results cannot be used. It is proven that the trace of a square integrable function belongs to $L^p_\mu$ for all real numbers $p\ge 1$. A counterexample shows that the trace of a function in $H^1(\Omega)$ may not belong to $BMO(\mu)$ (and therefore may not belong to $L^\infty_\mu$). Finally, it is proven that the traces of the functions in $H^1(\Omega)$ belong to $H^s(\Gamma^\infty)$ for all real numbers $s$ such that $0\le s d_H/4$ are supplied. \\ There is an important contrast with the case when $\Gamma^\infty$ is post-critically finite, for which the square integrable functions have their traces in $H^s(\Gamma^\infty)$ for all $s$ such that $0\le

Boundary Value Problems in Some Ramified Domains with a Fractal Boundary: Analysis and Numerical Methods. Part I: Diffusion and Propagation problems.

This paper is devoted to numerical methods for solving boundary value problems in self-similar ramified domains of $\R^2$ with a fractal boundary. Homogeneous Neumann conditions are imposed on the fractal part of the boundary, and Dirichlet conditions are imposed on the remaining part of the boundary. Several partial differential equations are considered. For the Laplace equation, the Dirichlet to Neumann operator is studied. It is shown that it can be computed as the unique fixed point of a rational map. From this observation, a self-similar finite element method is proposed and tested. For the Helmholtz equation, it is shown that the Dirichlet to Neumann operator can also be computed as the limit of an inductive sequence of operators. Here too, a finite element method is designed and tested. It permits to compute numerically the spectrum of the Laplace operator in the irregular domain with Neumann boundary conditions, as well as the eigenmodes. The repartition of the eigenvalues is investigated. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. This permits to solve numerically the wave equation in the self-similar ramified domain

Sobolev extension property for tree-shaped domains with self-contacting fractal boundary

International audienceIn this paper, we investigate the existence of extension operators fromW1,p( ) toW1,p(R2) (1 < p < 1) for a class of tree-shaped domains with a self-similar fractal boundary pre- viously studied by Mandelbrot and Frame. When the fractal boundary has no self-contact, the results of Jones imply that there exist such extension operators for all p 2 [1,1]. In the case when the fractal boundary self-intersects, this result does not hold. Here, we prove however that extension operators exist for p < p? where p? depends only on the dimension of the self-intersection of the boundary. The construction of these operators mainly relies on the self-similar properties of the domains

Well-posed PDE and integral equation formulations for scattering by fractal screens

We consider time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and sound-hard (Neumann) screens embedded in $\R^n$ for $n=2$ or $3$. In contrast to previous studies in which the screen is assumed to be a bounded Lipschitz (or smoother) relatively open subset of the plane, we consider screens occupying arbitrary bounded subsets. Thus our study includes cases where the screen is a relatively open set with a fractal boundary, and cases where the screen is fractal with empty interior. We elucidate for which screen geometries the classical formulations of screen scattering are well-posed, showing that the classical formulation for sound-hard scattering is not well-posed if the screen boundary has Hausdorff dimension greater than $n-2$. Our main contribution is to propose novel well-posed boundary integral equation and boundary value problem formulations, valid for arbitrary bounded screens. In fact, we show that for sufficiently irregular screens there exist whole families of well-posed formulations, with infinitely many distinct solutions, the distinct formulations distinguished by the sense in which the boundary conditions are understood. To select the physically correct solution we propose limiting geometry principles, taking the limit of solutions for a sequence of more regular screens converging to the screen we are interested in, this a natural procedure for those fractal screens for which there exists a standard sequence of prefractal approximations. We present examples exhibiting interesting physical behaviours, including penetration of waves through screens with "holes" in them, where the "holes" have no interior points, so that the screen and its closure scatter differently. Our results depend on subtle and interesting properties of fractional Sobolev spaces on non-Lipschitz sets

Contributions à l'étude d'espaces de fonctions et d'EDP dans une classe de domaines à frontière fractale auto-similaire

Cette thèse est consacrée à des questions d'analyse en amont de la modélisation de structures arborescentes, comme le poumon humain. Plus particulièrement, nous portons notre intérêt sur une classe de domaines ramifiés du plan, dont la frontière comporte une partie fractale auto-similaire. Nous commençons par une étude d'espaces de fonctions dans cette classe de domaines. Nous étudions d'abord la régularité Sobolev de la trace sur la partie fractale de la frontière de fonctions appartenant à des espaces de Sobolev dans les domaines considérés. Nous étudions ensuite l'existence d'opérateurs de prolongement sur la classe de domaines ramifiés. Nous comparons finalement la notion de trace auto-similaire sur la partie fractale du bord à des définitions plus classiques de trace. Nous nous intéressons enfin à un problème de transmission mixte entre le domaine ramifié et le domaine extérieur. L'interface du problème est la partie fractale du bord du domaine. Nous proposons ici une approche numérique, en approchant l'interface fractale par une interface préfractale. La stratégie proposée ici est basée sur le couplage d'une méthode auto-similaire pour la résolution du problème intérieur et d'une méthode intégrale pour la résolution du problème extérieur.We study some questions of analysis in view of the modeling of tree-like structures, such as the human lungs. More particularly, we focus on a class of planar ramified domains whose boundary contains a fractal self-similar part. We start by studying some function spaces defined for this class of domains. We first study the Sobolev regularity of the traces on the fractal part of the boundary of functions in some Sobolev spaces of the ramified domains. Then, we study the existence of Sobolev extension operators for the ramified domains we consider. Finally, we compare the notion of self-similar trace on the fractal part of the boundary with more classical definitions of trace. In the last part, we focus on a mixed transmission problem between the ramified domain and the exterior domain. The fractal part of the boundary is the interface of the problem. We propose a numerical approach where we approximate the self-similar interface by a prefractal interface. The proposed strategy is based on a self-similar method for the resolution of the inner problem coupled with an integral method for the resolution of the outer problem.RENNES1-Bibl. électronique (352382106) / SudocSudocFranceF