Timothy Blass, Rafael de la Llave
Perturbation and Numerical Methods for Computing the Minimal Average Energy
(446K, pdf)

ABSTRACT.  We investigate the differentiability of minimal average 
energy associated to the functionals 
$S_\ep (u) = \int_{\mathbb{R}^d} rac{1}{2}|
abla u|^2 + \ep V(x,u)\, dx$, 
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.