Contributions to the study of function spaces and PDEs in a class of domains with self-similar fractal boundary.

Summary This thesis is devoted to analytical issues upstream of the modeling of tree-like structures, such as the human lung. In particular, we focus our interest on a class of branched domains of the plane, whose boundary has a self-similar fractal part. We start with a study of function spaces in this class of domains. We first study the Sobolev regularity of the trace on the fractal part of the boundary of functions belonging to Sobolev spaces in the considered domains. We then study the existence of extension operators on the class of branched domains. We finally compare the notion of self-similar trace on the fractal part of the boundary with more classical definitions of trace. Finally, we consider a mixed transmission problem between the branched domain and the outer domain. The interface of the problem is the fractal part of the domain edge. We propose here a numerical approach, by approximating the fractal interface by a prefractal interface. The strategy proposed here is based on the coupling of a self-similar method for solving the inner problem and an integral method for solving the outer problem.