PERTURBATION AND NUMERICAL METHODS FOR COMPUTING THE MINIMAL AVERAGE ENERGY

Abstract. 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. 1. Introduction. Let d ∈ N be fixed, and let V: Rd × R → R, be periodic under integer translations. That is V(x + k, y + l) = V(x, y) for all (k, l) ∈ Zd × Z, where (x, y) = (x1,...,xd, y) ∈ Rd × R. Furthermore, assume V is analytic. We consider the formal variational problem Sε(u(x))