Identify the optimal endpoints of the solvability interval around p=2

Identify the exact optimal endpoints of the coefficient-uniform solvability interval around p=2 for one-dimensional parabolic equations with bounded measurable uniformly elliptic coefficients.

Background

The paper proves that the nondivergence- and divergence-form problems are uniquely solvable, with the relevant Sobolev estimates, whenever |p-2|<c\kappa. Its counterexamples show that the size of the largest coefficient-uniform neighborhood of 2 has optimal order \kappa as \kappa\downarrow0.

However, the quantitative result does not determine the exact endpoints of that interval. Thus the precise solvability range for a fixed ellipticity parameter remains unresolved.

References

Thus Theorem \ref{thm:positive} has the optimal order \kappa as \kappa\downarrow0, although we do not identify the optimal endpoints of the solvability interval.

— On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle  (2609.25568 - Dong et al., 22 Sep 2026) in Theorem 2.1 discussion, Section 2

The optimal lower endpoint of p, however, appears to be unknown even in one space dimension.

— On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle  (2609.25568 - Dong et al., 22 Sep 2026) in Introduction, discussion preceding Section 2