[Paper Review] Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation

Mónica Clapp, Angela Pistoia+1 more|arXiv (Cornell University)|Jun 1, 2021

Nonlinear Partial Differential Equations48 references4 citations

TL;DR

This paper establishes the existence of regular optimal partitions with ℓ ≥ 2 components for the Yamabe equation on closed Riemannian manifolds of dimension m ≥ 10, under non-local conformal flatness and additional geometric conditions when m = 10. It proves that a weakly coupled competitive elliptic system admits a least energy solution with nontrivial components, whose components phase-separate as competition intensifies, yielding a regular free boundary partition that exhausts the manifold and gives rise to a least energy sign-changing solution for ℓ = 2.

ABSTRACT

We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$. 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 solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For $\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Motivation & Objective

• To establish conditions under which optimal ℓ-partitions exist for the Yamabe equation on closed Riemannian manifolds.
• To show the existence of least energy solutions with nontrivial components for a weakly coupled competitive elliptic system related to the Yamabe equation.
• To analyze the asymptotic behavior of solutions as the competition parameter λ → −∞, leading to phase separation and optimal partition formation.
• To prove the regularity of the interfaces (free boundaries) and the limit profiles in the phase-separated regime.
• To construct a least energy sign-changing solution to the Yamabe equation with exactly two nodal domains for ℓ = 2.

Proposed method

• Study a weakly coupled competitive elliptic system of ℓ equations involving the conformal Laplacian and critical nonlinearity with negative interaction terms.
• Use variational methods to prove existence of a least energy solution with nontrivial components under coercivity and geometric assumptions on the manifold.
• Analyze the asymptotic limit as the competition parameter λ → −∞ using Almgren's monotonicity formula and blow-up analysis.
• Establish uniform Hölder and Lipschitz bounds on solutions via domain variation and compactness arguments.
• Prove regularity of the free boundary and the limit profiles using a generalized Almgren-type monotonicity formula with variable coefficients.
• Derive a free boundary condition at the interface between nodal domains from the limiting system and Pohozaev-type identities.

Experimental results

Research questions

• RQ1 Under what geometric conditions on a closed Riemannian manifold (M,g) does an optimal ℓ-partition for the Yamabe equation exist for ℓ ≥ 2?
• RQ2 Does a weakly coupled competitive elliptic system with critical nonlinearity and negative coupling admit a least energy solution with nontrivial components when dim M ≥ 10?
• RQ3 How do the components of the solution to the competitive system behave as the competition parameter λ → −∞?
• RQ4 Are the interfaces between nodal domains of the limiting solution regular, and do they satisfy a free boundary condition?
• RQ5 Can the phase-separated limit of the system yield a least energy sign-changing solution to the Yamabe equation for ℓ = 2?

Key findings

• For dim M ≥ 10, if (M,g) is not locally conformally flat and satisfies an additional geometric condition when dim M = 10, the competitive system admits a least energy solution with nontrivial components.
• As λ → −∞, the components of the solution phase-separate spatially, forming an optimal ℓ-partition that exhausts the entire manifold M.
• The interfaces between nodal domains are C^1,α regular, and the limit profiles are Lipschitz continuous with a free boundary condition at the interface.
• The limiting profiles satisfy a generalized Pohozaev identity and the Almgren monotonicity formula holds uniformly in the blow-up center x₀.
• For ℓ = 2, the optimal partition yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.
• The existence of such a solution is guaranteed under the same geometric assumptions as for general ℓ ≥ 2.