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Global W2,p and C2,α regularity for the sigma-$2$ equation
Abstract
In this paper, we prove that continuous $2$-convex solutions of the σ2 equation with positive continuous density on bounded strictly mean-convex C3 domains with C3 boundary values belong to W2,p for every 1<p<∞ in every dimension. If, in addition, the density is Cα, 0<α<1, the solutions belong to C2,α. Moreover, two counterexamples show that neither the C3 assumption on the domain nor that on the boundary values can in general be weakened to C2,1.
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BibTeX
@misc{lyu2026global,
title = {{Global $W^2,p$ and $C^2,α$ regularity for the sigma-$2$ equation}},
author = {Yasheng Lyu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11909},
eprint = {2609.11909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11909},
}
RIS
TY - UNPB
TI - Global W^2,p and C^2,α regularity for the sigma-$2$ equation
AU - Yasheng Lyu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11909
UR - https://arxiv.org/abs/2609.11909
AB - In this paper, we prove that continuous $2$-convex solutions of the σ_2 equation with positive continuous density on bounded strictly mean-convex C^3 domains with C^3 boundary values belong to W^2,p for every 1<p<∞ in every dimension. If, in addition, the density is C^α, 0<α<1, the solutions belong to C^2,α. Moreover, two counterexamples show that neither the C^3 assumption on the domain nor that on the boundary values can in general be weakened to C^2,1.
ER -