Pogorelov estimates for semi-convex solutions of 𝑘-curvature equations

In this paper, we consider k k -curvature equations σ k (κ M u) = f (x, u, ∇ u) ₖ (Mᵤ) =f (x, u, u) subject to (k + 1) (k+1) -convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang Duke Math. J. 123 (2004), pp. 235–264. By using the crucial concavity inequality for Hessian operator of Lu Calc. Var. Partial Differential Equations 62 (2023), p. 23, we derive Pogorelov estimates of semi-convex admissible solutions for these k k -curvature equations.