Perturbation and numerical methods for computing the minimal average energy

Timothy Blass; Rafael de la Llave; Timothy Blass; Rafael de la Llave

doi:10.3934/nhm.2011.6.241

Networks and Heterogeneous Media

2011, Volume 6, Issue 2: 241-255. doi: 10.3934/nhm.2011.6.241

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Perturbation and numerical methods for computing the minimal average energy

Timothy Blass ^{1
,},
Rafael de la Llave ^{2
,}

1.
Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257
2.
Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257

Received: 01 January 2011 Revised: 01 April 2011
Primary: 35B27, 58E15; Secondary: 41A58, 65M99.

We investigate the differentiability of minimal average energy associated to the functionals $S_\epsilon (u) = \int_{\mathbb{R}^d} \frac{1}{2}|\nabla u|^2 + \epsilon V(x,u)\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter $\epsilon$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.
- Cell problem,
- Plane-like minimizers,
- Minimal average energy,
- Sobolev gradient descent,
- Lindstedt series,
- quasiperiodic solutions of PDE
Citation: Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy[J]. Networks and Heterogeneous Media, 2011, 6(2): 241-255. doi: 10.3934/nhm.2011.6.241

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Abstract

We investigate the differentiability of minimal average energy associated to the functionals $S_\epsilon (u) = \int_{\mathbb{R}^d} \frac{1}{2}|\nabla u|^2 + \epsilon V(x,u)\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter $\epsilon$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.

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