Perturbation and Numerical Methods for Computing the Minimal Average Energy

TL;DR

This paper combines numerical Sobolev gradient descent and perturbative Lindstedt series methods to analyze the differentiability of minimal average energy in a variational problem, confirming results through convergence proofs.

Contribution

It introduces a combined numerical and perturbative approach to study minimal average energy and proves convergence of the Lindstedt series for the first time.

Findings

Numerical solutions of the Euler-Lagrange equations for the cell problem.

Representation of solutions as a Lindstedt series in the perturbation parameter.

Proof of convergence for the Lindstedt series.

Abstract

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.