We prove the existence of regular optimal G-invariant partitions, with an arbitrary number l ≥ 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 l 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 −∞, giving rise to an optimal partition. For l = 2 the optimal partition obtained yields a least energy sign-changing G-invariant solution to the Yamabe equation with precisely two nodal domains.

Yamabe systems and optimal partitions on manifolds with symmetries / Clapp, M.; Pistoia, A.. - In: ELECTRONIC RESEARCH ARCHIVE. - ISSN 2688-1594. - 29:6(2021), pp. 4327-4338. [10.3934/era.2021088]

Yamabe systems and optimal partitions on manifolds with symmetries

Pistoia A.
2021

Abstract

We prove the existence of regular optimal G-invariant partitions, with an arbitrary number l ≥ 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 l 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 −∞, giving rise to an optimal partition. For l = 2 the optimal partition obtained yields a least energy sign-changing G-invariant solution to the Yamabe equation with precisely two nodal domains.
2021
Competitive elliptic system; critical nonlinearity; free boundary problem; optimal partition; regularity; Riemannian manifold; sign-changing solution; Yamabe equation
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Yamabe systems and optimal partitions on manifolds with symmetries / Clapp, M.; Pistoia, A.. - In: ELECTRONIC RESEARCH ARCHIVE. - ISSN 2688-1594. - 29:6(2021), pp. 4327-4338. [10.3934/era.2021088]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1680270
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