Uniqueness of the exponential-variational limit

Determine whether the entire sequence of normalized Evans exponential variational minimizers u_k converges uniformly as k tends to infinity, or equivalently whether distinct subsequences can converge to different absolute minimizers.

Background

For each finite k, the normalized minimizer u_k of Evans’ exponential variational functional is uniquely determined, whereas the limiting periodic solutions of the associated Aronsson equation are generally nonunique. The paper proves that, in dimensions n ≤ 3, every subsequential limit u satisfies D_u = 𝒜, where D_u is the set on which the critical equation is attained differentiably and 𝒜 is the projected Aubry set.

The authors emphasize that this characterization does not by itself determine the limiting absolute minimizer uniquely. The unresolved issue is therefore whether the finite-k variational selection mechanism selects one common limit independently of the chosen subsequence.