Yamabe systems and optimal partitions on manifolds with symmetries

Mónica Clapp; Angela Pistoia; Mónica Clapp; Angela Pistoia

doi:10.3934/era.2021088

Electronic Research Archive

2021, Volume 29, Issue 6: 4327-4338. doi: 10.3934/era.2021088

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Yamabe systems and optimal partitions on manifolds with symmetries

Mónica Clapp ^{1
,
,},
Angela Pistoia ^{2
,}

1.
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, Mexico
2.
Dipartimento SBAI, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italy

Received: 01 July 2021 Revised: 01 September 2021 Published: 26 October 2021
Primary: 35B38, 35J20, 35J47, 35J60; Secondary: 35R35, 49K20, 49Q10, 58J05

We prove the existence of regular optimal $ G $-invariant partitions, with an arbitrary number $ \ell\geq 2 $ of components, for the Yamabe equation on a closed Riemannian manifold $ (M,g) $ when $ G $ is a compact group of isometries of $ M $ with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of $ \ell $ equations, related to the Yamabe equation. We show that this system has a least energy $ G $-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to $ -\infty $, giving rise to an optimal partition. For $ \ell = 2 $ the optimal partition obtained yields a least energy sign-changing $ G $-invariant solution to the Yamabe equation with precisely two nodal domains.
- Competitive elliptic system,
- Riemannian manifold,
- critical nonlinearity,
- optimal partition,
- free boundary problem,
- regularity,
- Yamabe equation,
- sign-changing solution
Citation: Mónica Clapp, Angela Pistoia. Yamabe systems and optimal partitions on manifolds with symmetries[J]. Electronic Research Archive, 2021, 29(6): 4327-4338. doi: 10.3934/era.2021088

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Abstract

We prove the existence of regular optimal $ G $-invariant partitions, with an arbitrary number $ \ell\geq 2 $ of components, for the Yamabe equation on a closed Riemannian manifold $ (M,g) $ when $ G $ is a compact group of isometries of $ M $ with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of $ \ell $ equations, related to the Yamabe equation. We show that this system has a least energy $ G $-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to $ -\infty $, giving rise to an optimal partition. For $ \ell = 2 $ the optimal partition obtained yields a least energy sign-changing $ G $-invariant solution to the Yamabe equation with precisely two nodal domains.

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