On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle
Abstract
Let 0<κ<1 and set p+=2/(1−κ) and p-=2/(1+κ). We construct a coefficient κ≤ a≤κ−1, smooth outside a compact set of Lebesgue measure zero, for which the W1,2p+ estimate fails for the one-space dimensional nondivergence form parabolic equation. The corresponding solution has second spatial derivative in the weak Lp+ space, but not in Lp+. By duality, the W1,2p- a priori estimate also fails. Conversely, we demonstrate that the W1,2p estimate holds and the equation is uniquely solvable for |p−2|<cκ, showing that the size of the solvability interval around $2$ has optimal order κ as κ↓0, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for p>2−cκ and this order is sharp as κ→ 0. The corresponding results for divergence form equations are also obtained.
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BibTeX
@misc{dong2026onespace,
title = {{On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle}},
author = {Hongjie Dong and Zongyuan Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25568},
eprint = {2609.25568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25568},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On one-space dimensional parabolic equations with measurable coefficients: Sobolev estimates and the Alexandrov maximum principle
AU - Hongjie Dong
AU - Zongyuan Li
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25568
UR - https://arxiv.org/abs/2609.25568
AB - Let 0<κ<1 and set p_+=2/(1−κ) and p_-=2/(1+κ). We construct a coefficient κ≤ a≤κ^−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|<cκ, showing that the size of the solvability interval around $2$ has optimal order κ as κ↓0, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for p>2−cκ and this order is sharp as κ→ 0. The corresponding results for divergence form equations are also obtained.
ER -