On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems

2010 ◽  
Vol 63 (1) ◽  
pp. 45-74 ◽  
Author(s):  
M. Dambrine ◽  
D. Kateb
Keyword(s):  
Shape Sensitivity ◽  
Two Phase ◽  
Dirichlet Eigenvalue ◽  
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.

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