Robin Green Function Estimates and a Model of Mammalian Lungs

TL;DR

This paper investigates Robin Green functions, providing precise bounds and revealing a phase transition in flow behavior, which models the efficiency of mammalian lungs and bridges Dirichlet and Neumann boundary conditions.

Contribution

It establishes detailed properties of Robin Green functions, including bounds on harmonic measure and confirms a phase transition in flow behavior relevant to lung physiology.

Findings

Sharp bounds on harmonic measure for Robin Green functions

Identification of a phase transition in flow behavior

Connection between mathematical properties and lung efficiency

Abstract

The present paper establishes delicate properties of the Green function with Robin boundary conditions, in particular, elucidating the nature of the passage between the Dirichlet-like and Neumann-like behavior. This yields sharp quantifiable bounds on the corresponding harmonic measure and proves the phase transition in the behavior of the total flow earlier conjectured in physics literature in concert with the efficacy of mammalian lungs.