Poisson-type problems with transmission conditions at boundaries of infinite metric trees

TL;DR

This paper studies a Poisson problem on a mixed structure combining a Euclidean domain with an infinite metric tree, establishing boundary conditions, existence, uniqueness, and approximation methods for solutions.

Contribution

It introduces a rigorous framework for transmission conditions at the boundary of a Euclidean domain and an infinite metric tree, utilizing embedded trace maps and Green-type formulas.

Findings

Established Fredholm properties of the combined Dirichlet-to-Neumann operators.

Proved existence and uniqueness of solutions for the Poisson-type problem.

Demonstrated that finite tree sections can approximate solutions efficiently.

Abstract

The paper introduces a Poisson-type problem on a mixed-dimensional structure combining a Euclidean domain and a lower-dimensional self-similar component touching a compact surface (interface). The lower-dimensional piece is a so-called infinite metric tree (one-dimensional branching structure), and the key ingredient of the study is a rigorous definition of the gluing conditions between the two components. These constructions are based on the recent concept of embedded trace maps and some abstract machineries derived from a suitable Green-type formula. The problem is then reduced to the study of Fredholm properties of a linear combination of Dirichlet-to-Neumann maps for the tree and the Euclidean domain, which yields desired existence and uniqueness results. One also shows that finite sections of tree can be used for an efficient approximation of the solutions.