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

 div(a(u) u u) = 0, in Ω, u|Ω = φ,

where Ω is a G-invariant domain of C2,α-class in M and φ C2,α(Ω ¯) is a G-invariant function. Two classical PDEs are included in this family: the p-Laplacian (a(s) = sp1, p > 1) and the minimal surface equation (a(s) = s1 + s2). 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
Authors
Jaime Ripoll
Universidade Federal do R. G. do Sul
Boa Vista
Brazil
Friedrich Tomi
University of Heidelberg
Heidelberg
Germany