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.

scholarly journals Indefinite eigenvalue problem with eigenparameter in the two boundary conditions

1998 ◽  
Vol 21 (4) ◽  
pp. 775-784
Author(s):  
S. F. M. Ibrahim
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Selfadjoint Operator ◽  
Order Differential Equation ◽  
Expansion Theorem ◽  
Compact Resolvent ◽  
Second Order Differential Equation ◽  
Eigenvalue Parameter

The object of this paper is to establish an expansion theorem for a regular indefinite eigenvalue problem of second order differential equation with an eigenvalue parameter,λin the two boundary conditions. We associated with this problem aJ-selfadjoint operator with compact resolvent defined in a suitable Krein space and then we develop an associated eigenfunction expansion theorem.

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.

Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem

2019 ◽  
Vol 22 (1) ◽  
pp. 78-94 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Continuous Functions ◽  
Real Spectrum ◽  
Robin Boundary Conditions ◽  
Schmidt Operator ◽  
Sturm Liouville ◽  
Robin Boundary

Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

scholarly journals ON AN EIGENVALUE PROBLEM INVOLVING THE HARDY POTENTIAL

2010 ◽  
Vol 12 (06) ◽  
pp. 953-975 ◽  
Author(s):  
J. CHABROWSKI ◽  
I. PERAL ◽  
B. RUF
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Principal Eigenvalue ◽  
Robin Boundary Conditions ◽  
Robin Problem ◽  
Hardy Potential ◽  
High Dimensions ◽  
The Neumann Problem ◽  
Robin Boundary ◽  
Second Eigenvalue

In this note we consider the eigenvalue problem for the Laplacian with the Neumann and Robin boundary conditions involving the Hardy potential. We prove the existence of eigenfunctions of the second eigenvalue for the Neumann problem and of the principal eigenvalue for the Robin problem in "high" dimensions.

Sign in / Sign up

Export Citation Format

Share Document