On Rayleigh’s Formula for the First Dirichlet Eigenvalue of a Radial Perturbation of a Ball

2012 ◽  
Vol 23 (3) ◽  
pp. 1427-1440 ◽  
Author(s):  
M. van den Berg
Keyword(s):  
Dirichlet Eigenvalue ◽  
Radial Perturbation ◽  
First Dirichlet Eigenvalue

On two functionals connected to the Laplacian in a class of doubly connected domains

2003 ◽  
Vol 133 (3) ◽  
pp. 617-624 ◽  
Author(s):  
S. Kesavan
Keyword(s):  
Dirichlet Eigenvalue ◽  
First Dirichlet Eigenvalue ◽  
Image Position

Let B1 be a ball of radius R1 in RN with centre at the origin and let B0 be a smaller ball of radius R0 contained inside it. Let u be the solution of the problem −Δu = 1 in B1\B0 vanishing on the boundary. It is shown that is minimal if and only if the balls are concentric. It is also shown that the first (Dirichlet) eigenvalue of the Laplacian in B1\B0 is maximal if and only if the balls are concentric. Generalizations are indicated.

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

2015 ◽  
Vol 160 (2) ◽  
pp. 191-208 ◽  
Author(s):  
SERGEI ARTAMOSHIN
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Negative Curvature ◽  
Connected Space ◽  
Constant Negative Curvature ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
First Dirichlet Eigenvalue ◽  
A New Technique ◽  
Simply Connected

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

Comparison theorems on smooth metric measure spaces with boundary

2016 ◽  
Vol 16 (4) ◽  
Author(s):  
Lin Feng Wang ◽  
Ze Yu Zhang ◽  
Yu Jie Zhou
Keyword(s):  
Mean Curvature ◽  
Comparison Theorems ◽  
Metric Measure Spaces ◽  
Weighted Mean ◽  
Dirichlet Eigenvalue ◽  
Weighted Mean Curvature ◽  
First Dirichlet Eigenvalue ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Weighted Laplacian

AbstractIn this paper we study smooth metric measure spaces with boundary via the Bakry–Émery curvature and the weighted mean curvature of the boundary. We establish the weighted Laplacian comparison theorems and the upper bound estimates of the distance from any point of the manifold to its boundary. As applications, we derive lower bound estimates for the first Dirichlet eigenvalue.

Sign in / Sign up

Export Citation Format

Share Document