PreprintarXiv licencearXiv
Necessity of Ladyzhenskaya \& Ural'tseva's sufficient conditions in nonuniform ellipticity
Abstract
The sufficient conditions of Ladyzhenskaya \& Ural'tseva \cite{lu70} for interior gradient estimates in nonuniformly elliptic problems are found to be also necessary. Specifically, nonparametric minimal surfaces exhibit the maximal nonuniformity rate compatible with the local Lipschitz regularity of solutions, and the classical theory of Finn \cite{fin54} and Bombieri \& De Giorgi \& Miranda \cite{bdm69} doesn't extend beyond Bernstein genre $2$.
Related
- Necessary and sufficient conditions for optimal control of coupled forward-backward stochastic control systems driven by G-Brownian motion
- Sufficient conditions for multiplicity and uniqueness of positive solutions of the Hénon equation
- Necessary and Sufficient Conditions for Adaptive Stabilizability of Continuous-Time Markov Jump Linear Stochastic Systems
- On quantitative sufficient second-order optimality conditions for elliptic optimal control problems
- Sufficient Conditions for Exact Convergence Time in Prescribed-Time Control
BibTeX
@misc{filippis2026necessity,
title = {{Necessity of Ladyzhenskaya \& Ural'tseva's sufficient conditions in nonuniform ellipticity}},
author = {Cristiana De Filippis},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30898},
eprint = {2608.30898},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30898},
}
RIS
TY - UNPB
TI - Necessity of Ladyzhenskaya \& Ural'tseva's sufficient conditions in nonuniform ellipticity
AU - Cristiana De Filippis
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30898
UR - https://arxiv.org/abs/2608.30898
AB - The sufficient conditions of Ladyzhenskaya \& Ural'tseva \cite{lu70} for interior gradient estimates in nonuniformly elliptic problems are found to be also necessary. Specifically, nonparametric minimal surfaces exhibit the maximal nonuniformity rate compatible with the local Lipschitz regularity of solutions, and the classical theory of Finn \cite{fin54} and Bombieri \& De Giorgi \& Miranda \cite{bdm69} doesn't extend beyond Bernstein genre $2$.
ER -