Higher-dimensional selection of projected Mather measures

Characterize, in higher dimensions, the projected Mather measures selected by the probability densities σ_k associated with Evans’ exponential variational minimizers, determine conditions under which every weak subsequential limit σ satisfies supp(σ) = 𝓜, and identify the subset of the projected Mather set 𝓜 selected when equality fails.

Background

The densities σ_k are proportional to exp(kH(Du_k(x),x)) and, along subsequences, converge weakly to projected Mather measures. In dimension one, under a suitable nondegeneracy condition, the support of the limiting measure coincides with the entire projected Mather set; degeneracy can cause additional selection.

The unresolved higher-dimensional problem concerns both whether the variational approximation recovers the full projected Mather set and how the selected support is determined when it does not. The question is relevant to obtaining a direct numerical recovery procedure for the Mather set from the exponential approximation.

References

What is the corresponding picture in higher dimensions? In particular, under what conditions does \operatorname{spt}(\sigma)=\mathcal M, and what determines the subset of \mathcal M selected when this equality fails?

— $L^\infty$ Variational Approximation of the Aubry Set  (2609.01557 - Tran et al., 1 Sep 2026) in Section 4, Open Problem (Selection of limiting Mather measures and possible recovery of the Mather set), labeled op:mather-measures