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.

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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