Group invariant solutions of certain partial differential equations

2020/07/02 by Ripoll, Jaime, Tomi, Friedrich
#53C21 #58J32 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.01040

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 div( (a( \Vert ∇ u\Vert ) )/(\Vert ∇ u\Vert )∇ u) amp; =0 in Ω
u|∂Ωamp; =φ where Ω is a G-invariant domain of C^2,α class in M and φ∈ C⁰( ∂Ω) 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/√ 1+s²). 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.

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