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• W2936236422 abstract "We consider the first eigenvalue $lambda_1(Omega,sigma)$ of the Laplacian with Robin boundary conditions on a compact Riemannian manifold $Omega$ with smooth boundary, $sigmainbf R$ being the Robin boundary parameter. When $sigma>0$ we give a positive, sharp lower bound of $lambda_1(Omega,sigma)$ in terms of an associated one-dimensional problem depending on the geometry through a lower bound of the Ricci curvature of $Omega$, a lower bound of the mean curvature of $partialOmega$ and the inradius. When the boundary parameter is negative, the lower bound becomes an upper bound. In particular, explicit bounds for mean-convex Euclidean domains are obtained, which improve known estimates. Then, we extend a monotonicity result for $lambda_1(Omega,sigma)$ obtained in Euclidean space by Giorgi and Smits to a class of manifolds of revolution which include all space forms of constant sectional curvature. As an application, we prove that $lambda_1(Omega,sigma)$ is uniformly bounded below by $frac{(n-1)^2}4$ for all bounded domains in the hyperbolic space of dimension $n$, provided that the boundary parameter $sigmageqfrac{n-1}{2}$ (McKean-type inequality). Asymptotics for large hyperbolic balls are also discussed" @default.
• W2936236422 created "2019-04-25" @default.
• W2936236422 creator A5057583131 @default.
• W2936236422 date "2019-04-16" @default.
• W2936236422 modified "2023-09-27" @default.
• W2936236422 title "Optimal eigenvalue estimates for the Robin Laplacian on Riemannian manifolds" @default.
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