Sharp local estimates for the Szeg "o-Weinberger profile in Riemannian\n manifolds

2011/10/21 by Mouhamed Moustapha Fall, Fall, Mouhamed Moustapha, Tobias Weth +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1110.4770

openalex publication_date 2011/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the local Szeg "o-Weinberger profile in a geodesic ball\nB_g(y₀,r₀) centered at a point y₀ in a Riemannian manifold ( M,g).\nThis profile is obtained by maximizing the first nontrivial Neumann eigenvalue\n\μ₂ of the Laplace-Beltrami Operator \Δ_g on M among subdomains of\nB_g(y₀,r₀) with fixed volume. We derive a sharp asymptotic bounds of this\nprofile in terms of the scalar curvature of M at y₀. As a corollary, we\ndeduce a local comparison principle depending only on the scalar curvature. Our\nstudy is related to previous results on the profile corresponding to the\nminimization of the first Dirichlet eigenvalue of \Δ_g, but additional\ndifficulties arise due to the fact that \μ₂ is degenerate in the unit ball\nin R^N and geodesic balls do not yield the optimal lower bound in the\nasymptotics we obtain.\n

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