退化$2$-Hessian方程的Dirichlet问题
On the Dirichlet problem for the degenerate $2$-Hessian equation
- School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文解决了退化2-Hessian方程Dirichlet问题在严格凸域上的全局C^{1,1}可解性（k=2情形），通过证明与inf f无关的σ_3下界揭示半凸结构，并推广到k≥3，反例表明边界正则性假设最优。
AI中文摘要：
本文在有界严格平均凸的$C^{3,1}$域$\Omega$上，对退化$2$-Hessian方程的Dirichlet问题建立了全局$C^{1,1}$可解性，其中边界值取一般的$C^{3,1}$函数，右端项$f\in C^{1,1}(\overline{\Omega})$非负，从而解决了长期悬而未决问题中$k=2$的情形。我们通过证明$\sigma_{3}[D^{2}u]$的下界（该下界与$\inf_{\Omega}f$无关），揭示了一般$2$-容许解的全局半凸结构。对于$3\leq k\leq n-1$，我们还在单位球上对$k$-容许解建立了全局半凸性，其中边界值为$C^{3}$函数，且$f^{1/(k-1)}\in C^{1,1}(\overline{B}_{1})$，所得界与$\inf_{B_{1}}f$无关。一个反例表明，在Hölder尺度下，全局$C^{1,1}$可解性所需的$C^{3,1}$边界值假设是精确的。
英文摘要：
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.
