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

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 $\mathrm{\Omega }$, with general $C^{3,1}$ boundary values and nonnegative right-hand sides $f\in C^{1,1}(\bar{\mathrm{\Omega }})$, 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 ${\sigma }_3[D^{2}u]$ independent of ${\inf }_{\mathrm{\Omega }}f$. For $3\le k\le n-1$, we also establish global semiconvexity for $k$-admissible solutions on the unit ball with $C^3$ boundary values and $f^{1\mathrm{/}(k-1)}\in C^{1,1}({\bar{B}}_1)$, 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.