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Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three

For each nonnegative integer mm, we construct smooth symmetric 3×33\times3 coefficient matrices AmA_m satisfying the fixed ellipticity boundI≤Am≤281I I\le A_m\le 2^{81}I for which the smooth solutions of uniformly elliptic equations in nondivergence formAm(x):D2um=0in B2⊂R3 A_m(x):D^2u_m=0\qquad\text{in }B_2\subset\mathbb R^3 have common Dirichlet data, satisfy ∥um∥L∞(B2)≤1\|u_m\|_{L^\infty(B_2)}\le1, butlim⁡m→∞∥Dum∥L1(B1)=∞. \lim_{m\to\infty}\|Du_m\|_{L^1(B_1)}=\infty. Thus, there is no interior W1,1W^{1,1} estimate depending only on ellipticity in dimension three, and consequently no such W1,pW^{1,p} estimate for any p≥1p\ge1.

Result
Disproved(see note)
Status
Resolved
AI contribution
AI co-developed
Method
Argument
Field
Partial differential equations
Posed by
Nikolai Nadirashvili; Vladimir Tkachev; Sergei Vlăduţ
Year posed
2014
Years open
12y
Solved
2026-08-13
Model
GPT-5.6 Sol
Vendor
OpenAI
Collaborators
Nam Q. Le, Qi Sun, Hung V. Tran
Verification
Unreviewed
Publication
Preprint
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

The authors construct smooth AmA_m and smooth solutions umu_m in B2⊂R3B_2\subset\mathbb R^3 with one fixed ellipticity boundI≤Am≤281I, I\leq A_m\leq2^{81}I, common boundary data and ∥um∥L∞(B2)≤1\|u_m\|_{L^\infty(B_2)}\leq1, but∥Dum∥L1(B1)→∞. \|Du_m\|_{L^1(B_1)}\to\infty. Thus no interior W1,1W^{1,1} estimate can depend only on dimension and ellipticity. Consequently, no such W1,pW^{1,p} estimate exists for any p≥1p\geq1.

They further obtain a uniformly convergent limit uu with measurable uniformly elliptic coefficient matrix AA, whereu∉BVloc(B1). u\notin BV_{\rm loc}(B_1). The construction even rules out coefficient-independent weak-L1L^1 gradient estimates.

What the AI did

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.

Verification

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

Submitted by VibeGene on

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