Admissible solutions to augmented nonsymmetric $k$-Hessian type equations II. A priori estimates and the Dirichlet problem

TL;DR

This paper establishes regularity estimates and solvability conditions for a class of nonsymmetric $k$-Hessian type equations, extending previous results by removing regularity constraints and analyzing admissible solutions.

Contribution

It proves $C^{2, eta}$ estimates for admissible solutions without regularity assumptions and provides existence and uniqueness conditions using the method of continuity.

Findings

Established $C^{2, eta}$ estimates for solutions

Provided necessary conditions for solution existence

Derived sufficient conditions for unique solvability

Abstract

Using the established $d$-concavity of the $k$-Hessian type functions $F_k(R)=\log(S_k(R)),$ whose variables are nonsymmetric matrices, we prove $ C^{2, \alpha}(\overline{\Omega}) $ estimates for strictly $(\delta, \widetilde{\gamma}_k) $-admissible solutions to the Dirichlet problem without the well-known regularity condition. A necessary condition for the existence of strictly $\delta$-admissible solutions to the equations is given. By the method of continuity, we provide some sufficient conditions for the unique solvability in the class of strictly $(\delta,\widetilde{\gamma}_k)$-admissible solutions to the Dirichlet problem, provided that those skew-symmetric matrices in the equations are sufficiently small in some sense.