Group invariant solutions of certain partial differential equations

Jaime Ripoll and Friedrich Tomi

Vol. 315 (2021), No. 1, 235–254

DOI: 10.2140/pjm.2021.315.235

Abstract

Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M$. We study the Dirichlet problem

$$\left\{\begin{array}{c}\hfill \\ \hfill \\ \text{ div}\left(\frac{a\left(\parallel \nabla u\parallel \right)}{\parallel \nabla u\parallel }\nabla u\right)=0,\hfill & \text{ in }\Omega ,\hfill \\ u|\partial \Omega =\phi ,\hfill \\ \hfill \\ \hfill \\ \hfill \end{array}\right.$$

where $\Omega$ is a $G$-invariant domain of $C^{2,\alpha }$-class in $M$ and $\phi \in C^{2,\alpha }\left(\partial \overline{\Omega }\right)$ is a $G$-invariant function. Two classical PDEs are included in this family: the $p$-Laplacian ($a\left(s\right)=s^{p-1}$, $p>1$) and the minimal surface equation ($a\left(s\right)=s∕\sqrt{1+s^2}$). Our motivation, by using the concept of Riemannian submersion, is to present a method for studying $G$-invariant solutions for noncompact Lie groups which allows the reduction of the Dirichlet problem on unbounded domains to one on bounded domains.

Keywords

group invariant solutions, Lie groups, Riemannian manifolds, elliptic PDEs

Mathematical Subject Classification

Primary: 53A10, 53C42, 49Q05, 49Q20

Milestones

Received: 9 September 2020

Revised: 7 September 2021

Accepted: 13 September 2021

Published: 13 December 2021