On the convergence rate in  multiscale homogenization of  fully nonlinear elliptic problems

Fabio Camilli; Claudio Marchi

doi:10.3934/nhm.2011.6.61

Article Contents

2011, Volume 6, Issue 1: 61-75. Doi: 10.3934/nhm.2011.6.61

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On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems

Fabio Camilli^1, and
Claudio Marchi^2,

1.
Dip. di Matematica Pura e Applicata, Univ. dell'Aquila, loc. Monteluco di Roio, 67040 l'Aquila
2.
Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova

Received: September 30, 2009

Revised: April 30, 2010

Published: February 2011

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Abstract

This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form $u^\epsilon+H^\epsilon (x,\frac{x}{\epsilon},\ldots,\frac{x}{epsilon^k},Du^\epsilon,D^2u^\epsilon)=0$. The operators $H^\epsilon$ are a regular perturbations of some uniformly elliptic, convex operator $H$. As $\epsilon\to 0^+$, the solutions $u^\epsilon$ converge locally uniformly to the solution $u$ of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed.

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Mathematics Subject Classification: Primary: 35B27; Secondary: 35J60, 49L25.

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