Related papers: Pogorelov type estimates for $(n-1)$-Hessian equat…

We mainly study Pogorelov type $C^2$ estimates for solutions to the Dirichlet problem of Sum Hessian equations. We establish respectively Pogorelov type $C^2$ estimates for $k$-convex solutions and admissible solutions under some…

In this paper, we primarily study the Pogorelov-type $C^2$ estimates for $(k-1)$-convex solutions of the sum Hessian equation under the assumption of semi-convexity, and apply these estimates to obtain a rigidity theorem for global…

In this paper, we derive a Pogorelov type interior $C^2$ estimate for the Hessian quotient equation $\frac{\sigma _n}{\sigma _k}\left( D^2u\right) =f$. As an application, we show that convex viscosity solutions are regular for $k\leq n-3$…

In this paper, We establish Pogorelov type $C^2$ estimates for the admissible solutions with $\sigma_k(D^2u)$ bounded from below of Sum Hessian equations. We also proved the lower bounded condition can be removed when $k = n$.

We establish the Pogorelov type estimates for degenerate prescribed k-curvature equations as well as k-Hessian equations. Furthermore,we investigate the interior C1,1 regularity of the solutions for Dirichlet problems. These techniques also…

We establish an interior $C^2$ estimate for $k+1$ convex solutions to Dirichlet problems of $k$-Hessian equations. We also use such estimate to obtain a rigidity theorem for $k+1$ convex entire solutions of $k$-Hessian equations in…

In this paper, we consider the entire solutions to the parabolic $2$-Hessian equations of the form $-u_t\sigma_2(D^2 u)=1$ in $\mathbb{R}^n\times (-\infty,0]$. We prove some rigidity theorems for the parabolic $2$-Hessian equations in…

In this paper, we exploit the concavity of sums of Hessian operators to derive Pogorelov estimates for corresponding equations under the dynamic semi-convexity assumption, and we further obtain several Liouville-type results. Moreover, when…

We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and…

We consider Pogorelov type estimates and Liouville type theorems to parabolic $k$-Hessian equations of the form $-u_t \sigma_k (D^2u) =1$ in $\mathbb{R}^n\times (-\infty, 0]$. We derive that any \textbf{$k+1$-convex-monotone} solution to…

Let $\Omega$ be a bounded domain (with smooth boundary) on the hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this paper, by using a…

This paper is devoted to the interior $C^2$ estimates for a class of sum Hessian quotient equations. For $0\leq l<k<n$, we establish the interior estimates and the Pogorelov type estimates. In the case $k=n$, we obtain a weaker Pogorelov…

In this paper, we establish the gradient and Pogorelov estimates for $k$-convex-monotone solutions to parabolic $k$-Hessian equations of the form $-u_t\sigma_k(\lambda(D^2u))=\psi(x,t,u)$. We also apply such estimates to obtain a Liouville…

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)…

We are concerned with the Dirichlet problem for a class of Hessian type equations. Applying some new methods we are able to establish the $C^2$ estimates for an approximating problem under essentially optimal structure conditions. Based on…

The curvature estimates of $k$ curvature equations for general right hand side is a longstanding problem. In this paper, we totally solve the $n-1$ case and we also discuss some applications for our estimate.

In this paper, we derive $C^2$ estimates for a class of mixed Hessian type equations with Dirichlet boundary condition, and obtain the existence theorem of admissible solutions for the classical Dirichlet problem of these mixed Hessian type…

In this paper, we consider the Dirichlet problem for a class of prescribed Hessian quotient type curvature equations with homogeneous boundary data in Minkowski space. By establishing the a priori C2 estimates, we obtain the existence…

The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)} \in C^{1,1}$. In…

We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(\lambda)=\log(\sigma_k(\lambda))$ on a subset of the positive cone $\Gamma_n=\{\lambda=(\lambda_{1}, \lambda_{2}, \cdots,\lambda_{n})\in \mathbb{R}^n;…