Perturbation and numerical methods for computing the minimal average energy

We investigate the differentiability of minimal average energy associated to the functionals S ε(u) = ∫ Rd 1 2 |∇u|2 + εV(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 ε, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.