Extension of the method of auxiliary sources for the eigenvalue problem of the Laplace operator in case of space periodic boundaries

2014 ◽  
Author(s):  
D. Kakulia ◽  
V. Ghoghoberidze ◽  
G. Ghvedashvili
Keyword(s):  
Eigenvalue Problem ◽  
Laplace Operator ◽  
Method Of Auxiliary Sources ◽  
Periodic Boundaries ◽  
The Laplace Operator

scholarly journals On the estimates of eigenvalues of the boundary value problem with large parameter

2015 ◽  
Vol 63 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Alexey V. Filinovskiy
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Boundary Value ◽  
Dirichlet Condition ◽  
Large Parameter ◽  
Robin Condition ◽  
The Difference ◽  
The Laplace Operator

Abstract We consider the eigenvalue problem Δu + λu = 0 in Ω with Robin condition + αu = 0 on ∂Ω , where Ω ⊂ Rn , n ≥ 2 is a bounded domain and α is a real parameter. We obtain the estimates to the difference between λDk - λk(α) eigenvalue of the Laplace operator in with Dirichlet condition and the corresponding Robin eigenvalue for large positive values of for all k = 1,2,… We also show sharpness of these estimates in the power of α.

scholarly journals EIGENVALUE PROBLEMS WITH WEIGHT AND VARIABLE EXPONENT FOR THE LAPLACE OPERATOR

2010 ◽  
Vol 08 (03) ◽  
pp. 235-246
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Variable Exponent ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
The Laplace Operator

This paper deals with an eigenvalue problem for the Laplace operator on a bounded domain with smooth boundary in ℝ N (N ≥ 3). We establish that there exist two positive constants λ* and λ* with λ* ≤ λ* such that any λ ∈ (0, λ*) is not an eigenvalue of the problem while any λ ∈ [λ*, ∞) is an eigenvalue of the problem.

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