Convergence rate of the variational approximation

Determine the convergence rate of the normalized Evans exponential variational minimizers u_k toward the unique limiting absolute minimizer whenever that limit is unique, and identify assumptions and the optimal function r(k) such that ||u_k − u||∞ ≤ r(k) with r(k) tending to zero.

Background

This problem is conditional on resolving the uniqueness question for the sequence u_k. If all subsequences converge to the same limit u, the paper asks for quantitative control of the uniform approximation error ||u_k − u||_{L∞(𝕋ⁿ)}.

The requested rate is also motivated by numerical approximation of the Aubry set: quantitative convergence of u_k would help determine how rapidly near-contact sets between u_k and the associated backward weak KAM solution approximate the Aubry set.