Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach

2009 ◽  
Vol 55 (1-3) ◽  
pp. 269-303 ◽  
Author(s):  
Massimo Lanza de Cristoforis
Keyword(s):  
Asymptotic Behaviour ◽  
Laplace Operator ◽  
Transmission Problem ◽  
Small Hole ◽  
Analytic Approach ◽  
Non Linear ◽  
The Laplace Operator

scholarly journals Neumann to Steklov eigenvalues: asymptotic and monotonicity results

2017 ◽  
Vol 147 (2) ◽  
pp. 429-447 ◽  
Author(s):  
Pier Domenico Lamberti ◽  
Luigi Provenzano
Keyword(s):  
Asymptotic Behaviour ◽  
Laplace Operator ◽  
Mass Concentration ◽  
Explicit Formulae ◽  
Steklov Eigenvalues ◽  
The Laplace Operator ◽  
Neumann Eigenvalues ◽  
Monotonicity Results

We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.

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