The Infinity-Potential in the Square

TL;DR

This paper constructs an explicit solution for the infinity-Laplace equation in a square with specific boundary conditions, using a hodograph method involving the heat equation and Jacobi's Theta functions, and it disproves a prior conjecture.

Contribution

It provides a novel explicit representation formula for the infinity-potential in a square, employing advanced mathematical techniques like the hodograph method and special functions.

Findings

Explicit formula for the infinity-potential in a square.

Disproves a previously held conjecture about the solution.

Introduces a method combining the heat equation and Theta functions.

Abstract

A representation formula for the solution of the $\infty$-Laplace equation is constructed in a punctured square, the prescribed boundary values being $u=0$ on the sides and $u=1$ at the centre. This so-called $\infty$-potential is obtained with a hodograph method. The heat equation is used and one of Jacobi's Theta functions appears. The formula disproves a conjecture.