On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle
Year: 2026
Type: preprint
Abstract: Let $0<κ<1$ and set $p_+=2/(1-κ)$ and $p_-=2/(1+κ)$. We construct a coefficient $κ\leq a\leqκ^{-1}$, smooth outside a compact set of Lebesgue measure zero, for which the $W^{1,2}_{p_+}$ estimate fails for the one-space dimensional nondivergence form parabolic equation. The corresponding solution has second spatial derivative in the weak $L_{p_+}$ space, but not in $L_{p_+}$. By duality, the $W^{1,2}_{p_-}$ a priori estimate also fails. Conversely, we demonstrate that the $W^{1,2}_p$ estimate holds and the equation is uniquely solvable for $|p-2|2-cκ$ and this order is sharp as $κ\to 0$. The corresponding results for divergence form equations are also obtained.
Source: arXiv (Cornell University)
Authors: Dong, Hongjie, Li, Zongyuan
Institutions: Brown University, City University of Hong Kong
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7214068073