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 ($a\left(s\right)=s∕\sqrt{1+s^2}$). 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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