Perturbations of the Spence-Abel equation and deformations of the dilogarithm function

TL;DR

This paper investigates how small changes affect the solutions of the Spence-Abel equation related to the dilogarithm, establishing stability results using cohomological methods.

Contribution

It introduces a new analysis of perturbed Spence-Abel equations and derives a Hyers-Ulam stability result using bounded cohomology techniques.

Findings

Existence and uniqueness of solutions under perturbations

Regularity properties of solutions

Hyers-Ulam stability for the Spence-Abel equation

Abstract

We analyze existence, uniqueness and regularity of solutions for perturbations of the Spence-Abel equation for the Rogers' dilogarithm. As an application we deduce a version of Hyers-Ulam stability for the Spence-Abel equation. Our analysis makes use of a well-known cohomological interpretation of the Spence-Abel equation and is based on our recent results on continuous bounded cohomology of SL(2,R).