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 Eigenstructure of the equilateral triangle, Part II: The Neumann problem

2002 ◽  
Vol 8 (6) ◽  
pp. 517-539 ◽  
Author(s):  
Brian J. McCartin
Keyword(s):  
Boundary Conditions ◽  
Neumann Problem ◽  
Equilateral Triangle ◽  
Orthonormal System ◽  
Neumann Boundary ◽  
Complete Orthonormal System ◽  
Eigenvalues And Eigenfunctions ◽  
Nodal Lines ◽  
Mathematical Techniques ◽  
The Neumann Problem

Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.

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.

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 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.

REINFORCEMENT OF THE POISSON EQUATION BY A THIN LAYER

2011 ◽  
Vol 21 (05) ◽  
pp. 1153-1192 ◽  
Author(s):  
JINGYU LI ◽  
KAIJUN ZHANG
Keyword(s):  
Boundary Condition ◽  
Shear Modulus ◽  
Poisson Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Robin Boundary Condition ◽  
Oscillation Period ◽  
The Poisson Equation ◽  
Robin Boundary

We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

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.

scholarly journals Solutions for parametric double phase Robin problems

2021 ◽  
Vol 121 (2) ◽  
pp. 159-170 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Boundary Condition ◽  
Existence Theorems ◽  
Robin Boundary Condition ◽  
Phase Problem ◽  
Double Phase ◽  
Robin Boundary ◽  
Double Phase Problem

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

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