Optimal ellipticity ratio for failure of the interior W^{1,1} estimate

Determine the smallest possible ellipticity ratio for which an ellipticity-only interior W^{1,1} estimate fails for uniformly elliptic equations in nondivergence form in dimension three.

Background

The paper constructs smooth uniformly elliptic coefficient matrices in dimension three with ellipticity ratio bounded by 2{81} and corresponding bounded smooth solutions whose gradients have unbounded L1 norm on an interior ball. This proves that an interior W{1,1} estimate cannot hold with constants depending only on ellipticity for that ratio.

The authors explicitly note that their constant 2{81} is not sharp. Consequently, determining the threshold—or smallest ellipticity ratio—at which an ellipticity-only interior W{1,1} estimate can fail remains unresolved.

References

The explicit constant $2{81}$ is not intended to be sharp. The exact value of the smallest possible ellipticity ratio for which an ellipticity-only $W{1,1}$ estimate can fail remains a separate question.

— Unbounded variation solutions for uniformly elliptic equations in nondivergence form in dimension three  (2608.13380 - Le et al., 13 Aug 2026) in Remark following the remark on the measurable limit