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arXiv · 2610.06853

On the Dirichlet problem for the degenerate $2$-Hessian equation

Yasheng Lyu
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Abstract

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 $f\in C^{1,1}(\overlineΩ)$, 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 $σ_{3}[D^{2}u]$ independent of $\inf_Ωf$. For $3\leq k\leq n-1$, we also establish global semiconvexity for $k$-admissible solutions on the unit ball with $C^{3}$ boundary values and $f^{1/(k-1)}\in C^{1,1}(\overline{B}_{1})$, with a bound independent of $\inf_{B_{1}}f$. A counterexample shows that the $C^{3,1}$ boundary value assumption for global $C^{1,1}$ solvability is sharp in the Hölder scale.

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Yasheng Lyu. 2026-10-05. On the Dirichlet problem for the degenerate $2$-Hessian equation. https://arxiv.org/abs/2610.06853

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