From formal graph and density expansions to Mather-measure error bounds

Definitions and mathematical statement

Diogo Gomes's Problem 35 concerns small perturbations of a completely integrable Hamiltonian and a nonresonant unperturbed torus. It reports uniform formal approximations of the corresponding viscosity solution and formal density expansions, and asks how well the Mather measures themselves are approximated. In a usual action-angle model, $H_0=H_0(p)$ and nonresonance of a torus means $k\cdot\nabla H_0(p_0)\ne0$ for all $0\ne k\in{\mathbb{Z}}^n$. The source does not specify a perturbation family, regularity, Diophantine constants, norms, uniqueness of the measure, or a quantitative order to prove.

A Mather measure is an invariant probability minimizing the cohomology-adjusted Lagrangian action; its cotangent version is obtained by the Legendre map. The aid's transfer lemma is purely metric and applies whenever the relevant phase-space probabilities have graph representations. On a compact base $X$ with reference measure ${\,\mathrm{d}} x$, let \[ \mu=(x\mapsto(x,G(x)))_*(m(x)\,{\,\mathrm{d}} x),\qquad \widetilde\mu=(x\mapsto(x,\widetilde G(x)))_*(\widetilde m(x)\,{\,\mathrm{d}} x), \] where $m,\widetilde m\ge0$, $\int m=\int\widetilde m=1$, and both graphs lie in a common phase-space metric set of finite diameter $D$. Assume that the distance between two points over the same base point is bounded by the indicated fiber distance $|G(x)-\widetilde G(x)|$, as in the product metric. Define $W_1$ as the infimum, over probability couplings, of the mean phase-space distance. The aid gives \[ W_1(\mu,\widetilde\mu) \le\int_X\min\{m,\widetilde m\}|G-\widetilde G|\,{\,\mathrm{d}} x +\frac D2\|m-\widetilde m\|_{L^1}. \] The common-density mass is matched over each base point and the remaining mass is half the $L^1$ density difference.

Thus, for a perturbation parameter $\varepsilon$ and order-$N$ approximants defining genuine probabilities in one common bounded phase-space region, bounds \[ \|G_\varepsilon-G_{\varepsilon,N}\|_\infty \le C_G|\varepsilon|^{N+1},\qquad \|m_\varepsilon-m_{\varepsilon,N}\|_1 \le C_m|\varepsilon|^{N+1} \] imply \[ W_1(\mu_\varepsilon,\mu_{\varepsilon,N}) \le(C_G+DC_m/2)|\varepsilon|^{N+1}. \] A formal density truncation that is negative or unnormalized is not automatically a probability; any positivity or normalization correction needs its own error bound. For cotangent graphs $G_\varepsilon=c+D u_\varepsilon$, the required estimate is on gradients, not just potentials. Conversion to velocity graphs requires corresponding control of the Legendre map.

The aid's smooth oscillatory obstruction can be made explicit on $X={\mathbb{R}}/{\mathbb{Z}}$: \[ u_j(x)=\frac{\sin(2\pi jx)}{2\pi j},\quad u_0=0,\quad m_j=m_0=1,\quad G_j=u_j'=\cos(2\pi jx),\quad G_0=0. \] Although $\|u_j-u_0\|_\infty\to0$, in the product metric on $X\times[-1,1]$ the graph probabilities satisfy \[ W_1(\mu_j,\mu_0)=\int_0^1|\cos(2\pi jx)|\,{\,\mathrm{d}} x=2/\pi. \] This demonstrates the failure of a general graph-measure estimate based only on uniform potentials and exact density agreement. It is not asserted that these graphs are Mather measures for the specific near-integrable family in the original problem. The transfer lemma reduces that problem to invariant-graph and density remainder estimates; it does not itself prove those dynamical estimates or their small-divisor assumptions.

Short English statement

To approximate phase-space Mather measures, the aid bounds Wasserstein error by graph displacement and density error. Uniform approximation of the viscosity potential alone is insufficient: an oscillatory example has vanishing potential error but a fixed measure error. The needed invariant-graph and density remainder estimates remain separate dynamical work.

Sources

• Local registry, permanent UnsolvedMath ID $20002491$, exact public code catalogue entry; canonical location aim-pdes-notes.json, notes[102].
• Diogo Gomes, Problem 35, AIM New connections between dynamical systems and PDE's; supplied reference https://aimath.org/WWN/dynpde/dynpde.pdf. The verification field reports that the official workshop page corrects OCR number 5 to 35 and removes the isolated trailing page digit 3. No source was accessed here.
• Supplied research attempt 1: partial_result, run and registry status partially_solved, rigor label proved, low novelty confidence; classification PARTIAL-PROGRESS, statement status reconstructed_unverified, created and updated 14 August 2026. No proposer year, original remarks, literature entry, or primary citation for the coupling lemma is supplied.
• The research aid explicitly says it is machine generated, not peer reviewed, and requires independent verification before citation.