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- W4300777637 abstract "We study global variational properties of the space of solutions to $-varepsilon^2Delta u + W'(u)=0$ on any closed Riemannian manifold $M$. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min-max and have index 1. We show that if $varepsilon$ is not small enough, in terms of the Cheeger constant of $M$, then there are no interesting solutions. However, we show that the number of min-max solutions to the equation above goes to infinity as $varepsilonto 0$ and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed $varepsilon$, as shown recently by G. Smith. We also show that the energy of the min-max solutions accumulate, as $varepsilon to 0$, around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with ${rm Ric}_M>0$, the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet-Rosenberg. Finally, we prove that the min-max energy values are bounded from below by the widths of the area functional as defined by Marques-Neves." @default.
- W4300777637 created "2022-10-04" @default.
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- W4300777637 date "2016-08-23" @default.
- W4300777637 modified "2023-09-26" @default.
- W4300777637 title "The Allen-Cahn equation on closed manifolds" @default.
- W4300777637 doi "https://doi.org/10.48550/arxiv.1608.06575" @default.
- W4300777637 hasPublicationYear "2016" @default.
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