scholarly journals On the solvability of a transmission problem for the Laplace operator with a dynamic boundary condition on a nonregular interface

2012 ◽  
Vol 393 (2) ◽  
pp. 651-670 ◽  
Author(s):  
Borys V. Bazaliy ◽  
Nataliya Vasylyeva
Keyword(s):  
Boundary Condition ◽  
Laplace Operator ◽  
Transmission Problem ◽  
Dynamic Boundary Condition ◽  
Dynamic Boundary ◽  
The Laplace Operator

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 $ − ∞ .

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.

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