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.

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 Regular eigenvalue problem with eigenparameter contained in the equation and the boundary conditions

1990 ◽  
Vol 13 (4) ◽  
pp. 651-659 ◽  
Author(s):  
E. M. E. Zayed ◽  
S. F. M. Ibrahim
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Robin Boundary Conditions ◽  
Expansion Theorem ◽  
Eigenvalue Parameter ◽  
The Laplace Operator ◽  
Simply Connected ◽  
Robin Boundary

The purpose of this paper is to establish the expansion theorem for a regular right-definite eigenvalue problem for the Laplace operator inRn,(n≥2)with an eigenvalue parameterλcontained in the equation and the Robin boundary conditions on two “parts” of a smooth boundary of a simply connected bounded domain.

Sublinear eigenvalue problems associated to the Laplace operator revisited

2011 ◽  
Vol 181 (1) ◽  
pp. 317-326 ◽  
Author(s):  
Mihai Mihăilescu ◽  
Vicentţiu Rădulescu
Keyword(s):  
Laplace Operator ◽  
Eigenvalue Problems ◽  
The Laplace Operator

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF THE BOUNDARY VALUE PROBLEM WITH A PARAMETER

2017 ◽  
Vol 21 (6) ◽  
pp. 135-140
Author(s):  
A.V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Laplace Operator ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Simple Eigenvalue ◽  
Robin Problem ◽  
The Laplace Operator ◽  
Robin Boundary

The paper presents the investigation of an eigenvalue problem for the Laplace operator with Robin boundary condition in a bounded domain with smooth boundary. The case of boundary condition containing a real parameter is con- sidered. It is proved that multiplicity of the eigenvalue to the Robin problem for all values of the parameter greater than some number does not exceed the mul- tiplicity of the corresponding eigenvalue to the Dirichlet problem for the Laplace operator. For simple eigenvalue of the Dirichlet problem the convergence of eigen- function of the Robin problem to the eigenfunction of the Dirichlet problem for unlimited increase of the parameter is proved. The formula for derivative on the parameter for eigenvalues of the Robin problem is established. This formula is used to justify the asymptotic expansions of eigenvalues of the Robin problem for large positive values of the parameter.

EIGENVALUE PROBLEMS ASSOCIATED WITH NONHOMOGENEOUS DIFFERENTIAL OPERATORS, IN ORLICZ–SOBOLEV SPACES

2008 ◽  
Vol 06 (01) ◽  
pp. 83-98 ◽  
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Real Number ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Positive Real Number ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
Positive Real

We study the boundary value problem - div ((a1(|∇ u|) + a2(|∇ u|))∇ u) = λ|u|q(x)-2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 3) with smooth boundary, λ is a positive real number, q is a continuous function and a1, a2 are two mappings such that a1(|t|)t, a2(|t|)t are increasing homeomorphisms from ℝ to ℝ. We establish the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any λ ∈ [λ1, ∞) is an eigenvalue, while any λ ∈ (0, λ0) is not an eigenvalue of the above problem.

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