A Symmetry Analysis of the $\infty$Polylaplacian
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 EulerLagrange 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.06688
 arXiv:
 arXiv:1708.06688
 Bibcode:
 2017arXiv170806688P
 Keywords:

 Mathematical Physics;
 Mathematics  Analysis of PDEs
 EPrint:
 13 pages