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On the Dirichlet problem for the degenerate $2$-Hessian equation
Abstract
In this paper, we establish global C1,1 solvability of the Dirichlet problem for the degenerate $2$-Hessian equation on bounded strictly mean convex C3,1 domains Ω, with general C3,1 boundary values and nonnegative right-hand sides f∈ C1,1(Ω̅), resolving the k=2 case of a longstanding open problem. We uncover a global semiconvexity structure for general $2$-admissible solutions by proving a lower bound for σ3[D2u] independent of infΩf. For 3≤ k≤ n−1, we also establish global semiconvexity for k-admissible solutions on the unit ball with C3 boundary values and f1/(k−1)∈ C1,1(B̅1), with a bound independent of infB1f. A counterexample shows that the C3,1 boundary value assumption for global C1,1 solvability is sharp in the Hölder scale.
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BibTeX
@misc{lyu2026dirichlet,
title = {{On the Dirichlet problem for the degenerate $2$-Hessian equation}},
author = {Yasheng Lyu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06853},
eprint = {2610.06853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06853},
}
RIS
TY - UNPB
TI - On the Dirichlet problem for the degenerate $2$-Hessian equation
AU - Yasheng Lyu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06853
UR - https://arxiv.org/abs/2610.06853
AB - In this paper, we establish global C^1,1 solvability of the Dirichlet problem for the degenerate $2$-Hessian equation on bounded strictly mean convex C^3,1 domains Ω, with general C^3,1 boundary values and nonnegative right-hand sides f∈ C^1,1(Ω̅), resolving the k=2 case of a longstanding open problem. We uncover a global semiconvexity structure for general $2$-admissible solutions by proving a lower bound for σ_3[D^2u] independent of inf_Ωf. For 3≤ k≤ n−1, we also establish global semiconvexity for k-admissible solutions on the unit ball with C^3 boundary values and f^1/(k−1)∈ C^1,1(B̅_1), with a bound independent of inf_B_1f. A counterexample shows that the C^3,1 boundary value assumption for global C^1,1 solvability is sharp in the Hölder scale.
ER -