Abstract

A wave propagation or diffusion process inside an object is typically described by a suitable differential or finite difference equation, while the interaction with the surrounding is considered using a boundary condition. In many cases, one looks for solutions of boundary value problems in suitable Sobolev spaces, and the regularity of the trace implies important conclusions on the existence and regularity of the solutions. While there is an established trace theory for Sobolev spaces over smooth domains, many important questions remain open for more complicated geometric objects. In the talk, we consider related questions for a class of special self-similar trees. We discuss the construction of the trace operator defined on Sobolev spaces over a class of infinite radial trees, based on an identification of the abstract boundary with a Euclidean domain. The process of boundary value determination can then be interpreted as an approximation procedure using piecewise constant functions. This requires an analysis of the function spaces on the graph itself, as well as a careful analysis of various characterizations of Besov and Sobolev spaces. In the end, we will formulate a concrete trace theorem which will characterize the Sobolev regularity of the trace functions.