\(C^{1,1}\) solution of the Dirichlet problem for degenerate \(k\)-Hessian equations (Q2447087)

The paper deals with the Dirichlet problem for degenerate (i.e., \(f\) is non-negative and may vanish somewhere in \(\overline{\Omega}\)) elliptic \(k\)-Hessian equations \(S_k[u]=f\) in a bounded \((k-1)\)- convex domain of \(\mathbb{R}^n\) with \(C^{3,1}\) boundary \(\partial\Omega\). The authors introduce Condition \((H)\) for \(f\geq 0\) which is weaker than condition \(f^\frac{1}{k}\in C^{1,1}(\overline{\Omega})\) usually used by other authors and prove that the Dirichlet problem has a unique \(k\)-admissible solution \(u\in C^{1,1}(\overline{\Omega})\). Uniform a priori estimates for the approximate solutions are established as well.