Neumann Problems for Elliptic and Parabolic Sum Hessian Equations

This paper studies the Neumann boundary value problem for sum Hessian equations. We first derive a priori $C^2$ estimates for $(k-1)$-admissible solutions in almost convex and uniformly $(k-1)$-convex domains, and prove the existence of admissible solutions via the method of continuity. Furthermore, we obtain existence results for the classical Neumann problem in uniformly convex domains. Finally, we extend these results to the corresponding parabolic problems.