Perturbation and Numerical Methods for Computing the Minimal Average Energy

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 $\text{ep}$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.