Perturbation and Numerical Methods for Computing the Minimal Average Energy

Details Perturbation and Numerical Methods for Computing the Minimal Average Energy Perturbation and Numerical Methods for Computing the Minimal Average Energy

2011-10-08

arxiv.org/abs/1110.1775v1

Mathematics

Analysis of PDEs

Scientific paper

We investigate the differentiability of minimal average energy associated to the functionals $S_\ep (u) = \int_{\mathbb{R}^d} (1/2)|\nabla u|^2 + \ep V(x,u)\, dx$, 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 $\ep$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.

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