2021/06/01 by Mónica Clapp, Angela Pistoia, Clapp, Mónica +3 · 4 citations
Computer Science · Mathematics · #35B38 #35J20 #35J47 #35J60 #35R35 #49K20 #49Q10 #58J05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Boundary (topology) #Boundary value problem #Combinatorics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Limit (mathematics) #Manifold (fluid mechanics) #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Partition (number theory) #Riemannian manifold #math.AP #math.DG #msc:35B38 #msc:35J20 #msc:35J47 #msc:35J60 #msc:35R35 #msc:49K20 #msc:49Q10 #msc:58J05
paper · pdf · doi:10.48550/arxiv.2106.00579
published in arXiv (Cornell University) (Cornell University) · 49 pages
arxiv created 2021/06/01 · openalex publication_date 2021/06/01 · arxiv updated 2021/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give conditions for the existence of regular optimal partitions, with an arbitrary number ℓ≥ 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 ℓ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if dim M≥ 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 -∞, 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 ℓ=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.