  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.
