PreprintCC BY 4.0arXiv
Logarithmic Hessian estimates for non-uniformly elliptic equations in non-divergence form
Abstract
We establish uniform logarithmic Hessian estimates for solutions of non-uniformly elliptic equations in non-divergence form. The proof is based on a measure estimate from convex analysis without localization or covering arguments. Moreover, we construct a counterexample that rules out uniform power-type Hessian estimates and shows the sharpness of the logarithmic exponent in the plane. Finally, we extend our estimates to strong solutions and apply them to obtain regularity results for Grushin-type equations.
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BibTeX
@misc{kim2026logarithmic,
title = {{Logarithmic Hessian estimates for non-uniformly elliptic equations in non-divergence form}},
author = {Jongmyeong Kim and Se-Chan Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09494},
eprint = {2610.09494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09494},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Logarithmic Hessian estimates for non-uniformly elliptic equations in non-divergence form
AU - Jongmyeong Kim
AU - Se-Chan Lee
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09494
UR - https://arxiv.org/abs/2610.09494
AB - We establish uniform logarithmic Hessian estimates for solutions of non-uniformly elliptic equations in non-divergence form. The proof is based on a measure estimate from convex analysis without localization or covering arguments. Moreover, we construct a counterexample that rules out uniform power-type Hessian estimates and shows the sharpness of the logarithmic exponent in the plane. Finally, we extend our estimates to strong solutions and apply them to obtain regularity results for Grushin-type equations.
ER -