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

Abstract

In this paper, we consider k k -curvature equations σ k ( κ [ M u ] ) = f ( x , u , ∇ u ) \sigma _k(\kappa [M_u])=f(x,u,\nabla 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.