#### Vol. 315, No. 1, 2021

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Group invariant solutions of certain partial differential equations

### Jaime Ripoll and Friedrich Tomi

Vol. 315 (2021), No. 1, 235–254
##### 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

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 ($a\left(s\right)=s∕\sqrt{1+{s}^{2}}\phantom{\rule{0.3em}{0ex}}$). 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.

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