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Yamabe systems and optimal partitions on manifolds with symmetries

  • 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.

    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

    Related Papers:

  • 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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