Group invariant solutions of certain partial differential equations

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 \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla u\right\Vert \right) }{\left\Vert \nabla u\right\Vert }\nabla u\right) & =0\text{ in }Ω\\ u|\partialΩ& =φ\end{align*} where $\Omega$ is a $G-$invariant domain of $C^{2,\alpha }$ class in $M$ and $\phi \in C^0\left(\partial \overline{\Omega }\right)$ a $G-$invariant function. Two classical PDE's are included in this family: the $p-$Laplacian $(a(s)=s^{p-1},$ $p>1)$ and the minimal surface equation $(a(s)=s/\sqrt{1+s^2}).$ Our motivation is to present a method in studying $G$-invariant solutions for noncompact Lie groups which allows the reduction of the Dirichlet problem on unbounded domains to one on bounded domains.

To appear in Pacific Journal of Mathematics