Higher-dimensional identification of the Aubry set

Determine whether the identity D_u = 𝒜 holds for subsequential limits u of Evans exponential variational minimizers in dimensions n ≥ 4, and, if it fails, construct a Hamiltonian and a subsequential limit u for which D_u properly contains the projected Aubry set 𝒜.

Background

The main theorem establishes D_u = 𝒜 under the vanishing one-dimensional Hausdorff measure condition for the Mather quotient. A result of Fathi–Figalli–Rifford supplies this condition for smooth Tonelli Hamiltonians in dimensions at most three, but the corresponding geometric information is unavailable in higher dimensions.

The paper identifies the possibility that the equality may fail for n ≥ 4, rather than merely being inaccessible by the current proof. A counterexample would require a Hamiltonian whose exponential-variational subsequential limit has a non-Aubry set of differentiable critical points.