Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds - Fall, Mouhamed Moustapha; Weth, Tobias (Calculus of variations and partial differential equations, 2014)
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.