Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three
resolvedconfidence 70%
VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Precise statement
For each nonnegative integer $m$, we construct smooth symmetric $3\times3$ coefficient matrices $A_m$ satisfying the fixed ellipticity bound
$$
I\le A_m\le 2^{81}I
$$
for which the smooth solutions of uniformly elliptic equations in nondivergence form
$$
A_m(x):D^2u_m=0\qquad\text{in }B_2\subset\mathbb R^3
$$
have common Dirichlet data, satisfy $\|u_m\|_{L^\infty(B_2)}\le1$, but
$$
\lim_{m\to\infty}\|Du_m\|_{L^1(B_1)}=\infty.
$$
Thus, 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 $p\ge1$.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement
fidelity, correctness, priority, or novelty.
What AI did
GPT-5.6 Sol
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.
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.
The authors construct smooth $A_m$ and smooth solutions $u_m$ in $B_2\subset\mathbb R^3$ with one fixed ellipticity bound
$$
I\leq A_m\leq2^{81}I,
$$
common boundary data and $\|u_m\|_{L^\infty(B_2)}\leq1$, but
$$
\|Du_m\|_{L^1(B_1)}\to\infty.
$$
Thus no interior $W^{1,1}$ estimate can depend only on dimension and ellipticity. Consequently, no such $W^{1,p}$ estimate exists for any $p\geq1$.
They further obtain a uniformly convergent limit $u$ with measurable uniformly elliptic coefficient matrix $A$, where
$$
u\notin BV_{\rm loc}(B_1).
$$
The construction even rules out coefficient-independent weak-$L^1$ gradient estimates.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.