A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

Abstract

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.

A minimisation problem in L∞ with PDE and unilateral constraints

“…Interesting results regarding L ∞ variational problems can be found e.g. in [10,14,15,16,17,18,19,28,43,46,47,48].…”

Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

“…The symmetry analysis has been widely studied in the literature. The simplicity on the steps of the theory and the unexpectedly number of new results which were found the last decades on nonlinear systems, in all areas of applied mathematics [4][5][6][7][8][9][10][11][12][13][14][15][16][17], established the Lie symmetry analysis as one of the most important methods for the study of nonlinear differential equations. Indeed, there are many important results in real world problems which followed by Lie symmetry analysis.…”

Soliton solutions to the time-dependent coupled KdV–Burgers’ equation

“…It is based on the study of the invariance under one-parameter Lie group of point transformations [13][14][15][16]. A few but important contributions are in [17][18][19][20][21][22]. Thus, with the applications of Lie's method, one can reduce nonlinear PDEs to the system of ordinary differential equations (ODEs) or at least reduce the number of independent variables.…”

Vectorial Variational Principles in $$L^\infty $$ and Their Characterisation Through PDE Systems

“…The idea of Theorem 5 is inspired by the paper [25] of Evans and Yu, wherein a particular case of the divergence system is derived (in the special scalar case N = 1 for the ∞-Laplacian and only for Ω = O), as well as by new developments on higher order Calculus of variations in L ∞ in [36,38,40].…”

Vectorial variational principles in $L^\infty$ and their characterisation through PDE systems

“…(III) For any ψ ∈ C 1 0 (O; R N ), there exists a non-empty compact set (1.10) The idea of Theorem 5 is inspired by the paper [25] of Evans and Yu, wherein a particular case of the divergence system is derived (in the special scalar case N = 1 for the ∞-Laplacian and only for Ω = O), as well as by new developments on higher order Calculus of variations in L ∞ in [36,38,40].…”