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Networks and Heterogeneous Media

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

  • 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: 01 October 2009 Revised: 01 May 2010

  • Primary: 35B27; Secondary: 35J60, 49L25.

  • 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.

    Citation: Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems[J]. Networks and Heterogeneous Media, 2011, 6(1): 61-75. doi: 10.3934/nhm.2011.6.61

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  • 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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  • This article has been cited by:

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    2. Daria Ghilli, Claudio Marchi, Rate of convergence for singular perturbations of Hamilton-Jacobi equations in unbounded spaces, 2023, 526, 0022247X, 127225, 10.1016/j.jmaa.2023.127225
    3. Piermarco Cannarsa, Cristian Mendico, Rate of Convergence for First-Order Singular Perturbation Problems: Hamilton–Jacobi–Isaacs Equations and Mean Field Games of Acceleration, 2024, 2153-0785, 10.1007/s13235-024-00594-3
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