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Homogenization of a class of nonlinear eigenvalue problems

Published online by Cambridge University Press:  12 July 2007

L. Baffico ,
C. Conca  and
M. Rajesh
L. Baffico
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Casilla 170/3 Santiago de Chile (lbaffico@dim.uchile.cl)
C. Conca
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Casilla 170/3 Santiago de Chile (cconca@dim.uchile.cl)
M. Rajesh
Affiliation:
Departamento de Matemática, Facultad de Ciencas Físicas y Matemáticas, Universidad de Concepción, Avda. Esteban Iturra s/n, Barrio Universitario, Concepción, Chile (rmahadevan@udec.cl)
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Abstract

In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form Aε = −div(aε (x, ∇u)), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G-converge to an operator Ahom = −div(ahom(x, ∇u)). We show that any limit point λ of a sequence of eigenvalues λε is an eigenvalue of the limit operator Ahom, where λε is an eigenvalue corresponding to the operator Aε. We also show the convergence of the sequence of first eigenvalues to the corresponding first eigenvalue of the homogenized operator.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 1 , February 2006 , pp. 7 - 22
Copyright
Copyright © Royal Society of Edinburgh 2006

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