ChatGPT 5.6 Sol Helps Rule Out Ellipticity-Only Gradient Estimates in Dimension Three

In a preprint first uploaded on August 13, 2026, Nam Q. Le, Qi Sun, and Hung V. Tran resolve negatively an open regularity question for uniformly elliptic equations in nondivergence form. They construct smooth, uniformly elliptic coefficient matrices and bounded smooth solutions in dimension three whose gradients have unbounded $L^1$ norm, despite common Dirichlet data and a fixed ellipticity ratio.

More precisely, their sequence satisfies $I\leq A_m\leq 2^{81}I$, $A_m:D^2u_m=0$, and $\lVert u_m\rVert_{L^\infty}\leq1$, while $\lVert Du_m\rVert_{L^1}$ tends to infinity. This rules out interior $W^{1,p}$ estimates depending only on ellipticity for every $p\geq1$ in dimensions at least three. The construction also yields a uniformly convergent limit that is not locally of bounded variation for a measurable uniformly elliptic coefficient field.

The authors state that the main results arose through chats with ChatGPT 5.6 Sol and that ChatGPT supplied the key strategies. They report entirely reworking and rewriting the article, checking and simplifying every argument, and accepting responsibility for the result. The paper does not claim that the explicit ellipticity ratio $2^{81}$ is optimal.