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.

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES AND EIGENFUNCTIONS OF THE ROBIN PROBLEM WITH LARGE PARAMETER

2017 ◽  
Vol 22 (1) ◽  
pp. 37-51 ◽  
Author(s):  
Alexey V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
Large Parameter ◽  
Eigenvalues And Eigenfunctions ◽  
The Difference ◽  
Robin Boundary ◽  
Outward Unit

We consider the eigenvalue problem with Robin boundary condition ∆u + λu = 0 in Ω, ∂u/∂ν + αu = 0 on ∂Ω, where Ω ⊂ Rn , n ≥ 2 is a bounded domain with a smooth boundary, ν is the outward unit normal, α is a real parameter. We obtain two terms of the asymptotic expansion of simple eigenvalues of this problem for α → +∞. We also prove an estimate to the difference between Robin and Dirichlet eigenfunctions.

scholarly journals Counterexample to Strong Diamagnetism for the Magnetic Robin Laplacian

2020 ◽  
Vol 23 (3) ◽  
Author(s):  
Ayman Kachmar ◽  
Mikael P. Sundqvist
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
Laplace Operator ◽  
Unit Disc ◽  
Uniform Magnetic Field ◽  
Robin Boundary Condition ◽  
Robin Laplacian ◽  
The Laplace Operator ◽  
Robin Boundary ◽  
Lowest Eigenvalue

Abstract We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty $ − ∞ .

Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauchy–Riemann equation

2009 ◽  
Vol 139 (1) ◽  
pp. 157-181 ◽  
Author(s):  
Alip Mohammed ◽  
M. W. Wong
Keyword(s):  
Boundary Condition ◽  
Linear Equations ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
System Of Linear Equations ◽  
Riemann Equation ◽  
Cauchy Riemann ◽  
Robin Boundary ◽  
Poincaré Problem ◽  
Well Posed

The Riemann–Hilbert–Poincaré problem with general coefficient for the inhomogeneous Cauchy–Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coeffcient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.

scholarly journals WEYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS AND DE GENNES' BOUNDARY CONDITION

2008 ◽  
Vol 20 (08) ◽  
pp. 901-932 ◽  
Author(s):  
AYMAN KACHMAR
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Robin Boundary Condition ◽  
The Self ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Weyl Asymptotics ◽  
Robin Boundary

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schrödinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof rely on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [10].

scholarly journals Eigenstructure of the equilateral triangle. Part III. The Robin problem

2004 ◽  
Vol 2004 (16) ◽  
pp. 807-825 ◽  
Author(s):  
Brian J. McCartin
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Equilateral Triangle ◽  
Orthonormal System ◽  
Neumann Boundary ◽  
Robin Boundary Condition ◽  
Complete Orthonormal System ◽  
Robin Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Robin Boundary

Lamé's formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete orthonormal system. Various properties of the spectrum and modal functions are explored.

scholarly journals Multiplicity of solutions for Robin problem involving the p(x)-laplacian

2021 ◽  
Vol 13 (2) ◽  
pp. 321-335
Author(s):  
Hassan Belaouidel ◽  
Anass Ourraoui ◽  
Najib Tsouli
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Robin Boundary Condition ◽  
Robin Problem ◽  
Technical Approach ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity ◽  
Robin Boundary ◽  
Laplacian Equations

Abstract This paper is concerned with the existence and multiplicity of solutions for p(x)-Laplacian equations with Robin boundary condition. Our technical approach is based on variational methods.

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