Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

• Published in: Calculus of variations and partial differential equations Vol. 51; no. 1-2; pp. 217 - 242
• Main Authors: Fall, Mouhamed Moustapha, Weth, Tobias
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2014; Springer Nature B.V
• Subjects: Analysis; Asymptotic properties; Calculus of variations; Calculus of Variations and Optimal Control; Optimization; Control; Curvature; Dirichlet problem; Eigenvalues; Manifolds; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Scalars; Systems Theory; Theoretical; Topological manifolds; 15A42; 47A11; 35R45; 35B05; 52A40
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-013-0672-y

We study the local Szegö–Weinberger profile in a geodesic ball B g ( y 0 , r 0 ) centered at a point y 0 in a Riemannian manifold ( M , g ) . This profile is obtained by maximizing the first nontrivial Neumann eigenvalue μ 2 of the Laplace–Beltrami Operator Δ g on M among subdomains of B g ( y 0 , r 0 ) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of M at y 0 . As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of Δ g , but additional difficulties arise due to the fact that μ 2 is degenerate in the unit ball in R N and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.