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

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 $\Omega$, with general $C^{3,1}$ boundary values and nonnegative right-hand sides $f\in C^{1,1}(\overline{\Omega})$, 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 $\inf_{\Omega}f$. 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 H\"older scale.