A Symmetry Analysis of the $\infty$-Polylaplacian

TL;DR

This paper applies Lie group methods to analyze the symmetry properties of the $ abla$-Polylaplacian PDE, a higher-order nonlinear equation related to variational principles in $L^{ abla}$, and constructs invariant solutions.

Contribution

It is the first to study the Lie symmetries of the $ abla$-Polylaplacian and its reduced form, providing new insights into their structure and solutions.

Findings

Identified Lie symmetries of the $ abla$-Polylaplacian and its reduced equation.

Constructed invariant solutions under one-dimensional symmetry subgroups.

Established a relationship between the reduced equation and the original PDE.

Abstract

In this work we use Lie group theoretic methods and the theory of prolonged group actions to study two fully nonlinear partial differential equations (PDEs). First we consider a third order PDE in two spatial dimensions that arises as the analogue of the Euler-Lagrange equations from a second order variational principle in $L^{\infty}$. The equation, known as the $\infty$-Polylaplacian, is a higher order generalisation of the $\infty$-Laplacian, also known as Aronsson's equation. In studying this problem we consider a reduced equation whose relation to the $\infty$-Polylaplacian can be considered analogous to the relationship of the Eikonal to Aronsson's equation. Solutions of the reduced equation are also solutions of the $\infty$-Polylaplacian. For the first time we study the Lie symmetries admitted by these two problems and use them to characterise and construct invariant solutions under the action of one dimensional symmetry subgroups.