Gradient and BV regularity of nondivergence elliptic solutions

Do uniformly bounded solutions of smooth uniformly elliptic nondivergence equations in R³ have a uniform L¹ gradient bound? The counterexample target has I≤A_m≤2⁸¹I and common boundary data but diverging gradient mass, with a uniform limit not locally of bounded variation.

Why it matters and what counts as progress

Why it matters: It tests how much first-order regularity follows from uniform ellipticity without additional coefficient control.

What counts as progress: Smooth uniformly elliptic examples with unbounded gradient mass or a non-BV limit, or a regularity theorem identifying sufficient extra coefficient control.

How the problem developed

2014 · Open question [1]
Nikolai Nadirashvili, Vladimir Tkachev, and Serge Vlăduţ Raised the question whether smooth uniformly elliptic nondivergence equations admit an interior W¹,¹ estimate depending only on ellipticity.
3 Sept 2026 · Counterexample [2]
Nam Q. Le, Qi Sun, and Hung V. Tran, with ChatGPT 5.6 Sol supplying key strategies Constructed smooth three-dimensional uniformly elliptic examples with common Dirichlet data and bounded solutions but unbounded interior L¹ gradient norms, ruling out an ellipticity-only W¹,¹ estimate in dimension three.

AI contribution

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

Nam Q. Le, Qi Sun, Hung V. Tran

Le, Sun and Tran construct smooth three-dimensional uniformly elliptic nondivergence equations with I≤Aₘ≤2⁸¹I, common boundary data and bounded solutions whose gradient L¹ norms diverge. They also obtain a uniformly convergent approximation limit that is not locally of bounded variation.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull