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Abstract
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Let
be a complete Riemannian
manifold and
a Lie subgroup
of the isometry group of
acting freely and properly on
.
We study the Dirichlet problem
where
is a
-invariant
domain of
-class
in
and
is a
-invariant
function. Two classical PDEs are included in this family: the
-Laplacian
(,
) and the minimal
surface equation ().
Our motivation, by using the concept of Riemannian submersion, is to present a method for
studying
-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
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Mathematical Subject Classification
Primary: 53A10, 53C42, 49Q05, 49Q20
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Milestones
Received: 9 September 2020
Revised: 7 September 2021
Accepted: 13 September 2021
Published: 13 December 2021
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