Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach

Analysis ◽  
2008 ◽  
Vol 28 (1) ◽  
Author(s):  
Massimo Lanza de Cristoforis
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Laplace Operator ◽  
Small Hole ◽  
Analytic Approach ◽  
The Laplace Operator

We consider a hypersurface in R

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 A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary

2018 ◽  
Vol 116 ◽  
pp. 211-267 ◽  
Author(s):  
Virginie Bonnaillie-Noël ◽  
Matteo Dalla Riva ◽  
Marc Dambrine ◽  
Paolo Musolino
Keyword(s):  
Dirichlet Problem ◽  
Laplace Operator ◽  
Small Hole ◽  
The Laplace Operator
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