{
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  "problem_id": "vibemathed:unbounded-variation-solutions-for-uniformly-elliptic-equations-in-nondivergence-",
  "title": "Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three",
  "canonical_statement": "For each nonnegative integer $m$, we construct smooth symmetric $3\\times3$ coefficient matrices $A_m$ satisfying the fixed ellipticity bound\n$$\nI\\le A_m\\le 2^{81}I\n$$\nfor which the smooth solutions of uniformly elliptic equations in nondivergence form\n$$\nA_m(x):D^2u_m=0\\qquad\\text{in }B_2\\subset\\mathbb R^3\n$$\nhave common Dirichlet data, satisfy $\\|u_m\\|_{L^\\infty(B_2)}\\le1$, but\n$$\n\\lim_{m\\to\\infty}\\|Du_m\\|_{L^1(B_1)}=\\infty.\n$$\nThus, there is no interior $W^{1,1}$ estimate depending only on ellipticity in dimension three, and consequently no such $W^{1,p}$ estimate for any $p\\ge1$.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
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    {
      "id": "vibemathed:unbounded-variation-solutions-for-uniformly-elliptic-equations-in-nondivergence-:event",
      "problem_id": "vibemathed:unbounded-variation-solutions-for-uniformly-elliptic-equations-in-nondivergence-",
      "type": "disproved",
      "status": "resolved",
      "occurred_at": "2026-08-13T00:00:00.000Z",
      "title": "arXiv",
      "summary": "The authors construct smooth $A_m$ and smooth solutions $u_m$ in $B_2\\subset\\mathbb R^3$ with one fixed ellipticity bound\n$$\nI\\leq A_m\\leq2^{81}I,\n$$\ncommon boundary data and $\\|u_m\\|_{L^\\infty(B_2)}\\leq1$, but\n$$\n\\|Du_m\\|_{L^1(B_1)}\\to\\infty.\n$$\nThus no interior $W^{1,1}$ estimate can depend only on dimension and ellipticity. Consequently, no such $W^{1,p}$ estimate exists for any $p\\geq1$.\n\nThey further obtain a uniformly convergent limit $u$ with measurable uniformly elliptic coefficient matrix $A$, where\n$$\nu\\notin BV_{\\rm loc}(B_1).\n$$\nThe construction even rules out coefficient-independent weak-$L^1$ gradient estimates.",
      "ai_contribution": "ai_co_developed",
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      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-08-13T00:00:00.000Z",
      "human_collaborators": [
        "Nam Q. Le",
        "Qi Sun",
        "Hung V. Tran"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The authors state that the main results were obtained through a series of chats with ChatGPT 5.6 Sol and that the key strategies came from ChatGPT. The construction uses repeated localized rank-one Hessian splittings to amplify gradients while maintaining a quantitative saddle condition, allowing all Hessians to be annihilated by coefficient matrices in one fixed ellipticity class. The authors then reworked and rewrote the article entirely, checked and simplified all arguments, and take responsibility for the result.",
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      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.13380",
          "kind": "primary-mathematical-source",
          "license": null
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        {
          "label": "VibeMathed: Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three",
          "url": "https://vibemathed.com/problem/unbounded-variation-solutions-for-uniformly-elliptic-equations-in-nondivergence-",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
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      "independence": "unknown"
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      "verifier": "VibeMathed (source-reported label)",
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      "note": "Unreviewed. arXiv 2608.13380 (version 2, 3 September 2026) read here; the AI-assistance section states that the main results came from chats with ChatGPT 5.6 Sol, that the key strategies were the model's, and that the authors reworked, rewrote, checked and simplified everything and take responsibility. Author-checked, not independently refereed; no formalization.",
      "sources": [
        {
          "label": "VibeMathed: Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three",
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      "url": "https://arxiv.org/abs/2608.13380",
      "kind": "primary-mathematical-source",
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    {
      "label": "VibeMathed: Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three",
      "url": "https://vibemathed.com/problem/unbounded-variation-solutions-for-uniformly-elliptic-equations-in-nondivergence-",
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    "VibeMath has not independently verified the mathematical claim.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}