On discrete integrable equations with convex variational principles

TL;DR

This paper explores the variational structure of discrete Laplace-type equations linked to integrable quad-equations, establishing conditions for convexity of the action functional to ensure solution existence and uniqueness.

Contribution

It provides new sufficient conditions for convexity of discrete action functionals and connects these to integrable quad-equations and circle patterns.

Findings

Derived conditions for convexity of discrete action functionals.

Linked convex functionals to integrable quad-equations and circle patterns.

Ensured existence and uniqueness of solutions to boundary value problems.

Abstract

We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we derive sufficient conditions (that are often necessary) on the labeling of the edges under which the corresponding generalized discrete action functional is convex. Convexity is an essential tool to discuss existence and uniqueness of solutions to Dirichlet boundary value problems. Furthermore, we study which combinatorial data allow convex action functionals of discrete Laplace-type equations that are actually induced by discrete integrable quad-equations, and we present how the equations and functionals corresponding to (Q3) are related to circle patterns.