Related papers: Interior $C^2$ estimate for Hessian quotient equat…

In this paper, we establish an interior $C^2$ estimate for the Hessian quotient equation $\left(\frac{\sigma_3}{\sigma_1}\right)(D^2u)=f$ in dimension three. A crucial ingredient in our proof is a Jacobi inequality.

In this paper, we study the interior $C^{2}$ estimates for Hessian quotient equations $\frac{\sigma_{3}(D^{2}u)}{\sigma_{l}(D^{2}u)}=1$ for $l=1, 2$, in arbitrary dimensions, under the natural ellipticity and semi-convexity conditions. We…

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 the interior Hessian estimates for $2$-convex solutions to $\frac{\sigma_2}{\sigma_1} (D^2 u) = \psi (x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for…

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $\sigma_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$…

We derive a priori interior Hessian estimates and regularity for the sigma-2 Hessian equation $\sigma_{2}(D^2u)=f(x,u,Du)$ with positive $C^{1,1}$ right hand side in dimension 4. In higher dimensions, the same result holds under an…

We prove a priori interior C2 estimate for \sigma_2 = f in R3, which generalizes Warren-Yuan's result.

We derive a priori interior Hessian estimates and interior regularity for the $\sigma_2$ equation in dimension four. Our method provides respectively a new proof for the corresponding three dimensional results and a Hessian estimate for…

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 mainly study the interior $C^2$ estimates for a class of sum Hessian equations. We establish the interior estimates and the Pogorelov type estimates for $0<k<n$. If $k=n$, we derive a weaker Pogorelov type estimates.

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…

We obtain a prior $C^{1,1}$ estimates for some Hessian (quotient) equations with positive Lipschitz right hand sides, through studying a twisted special Lagrangian equation. The results imply the interior $C^{2,\alpha}$ regularity for $C^0$…

In this paper, we consider the Neumann problem for a class of Hessian quotient equations involving a gradient term on the right-hand side in Euclidean space. More precisely, we derive the interior gradient estimates for the $(\Lambda,…

We prove that viscosity solutions to the quadratic Hessian equation $$\sigma_2(D^2u) = 1$$ cannot touch a harmonic function on a minimal surface from below. This can be viewed as a form of strict $2$-convexity. We also prove an a priori…

We study interior curvature estimates for convex graphs which satisfy the quotient equation $\frac{\sigma_{n}}{\sigma_{n-2}}(\lambda)=f(X)>0$ in this paper.

In this note, we present the interior $C^{2,\alpha}$ regularity for viscosity solutions of fully nonlinear uniformly elliptic equations in dimension two.

In this paper, we obtain some important inequalities of Hessian quotient operators, and global $C^2$ estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of…

Let $2\le n\le 5$. We establish an apriori interior H\"older regularity of $C^2$-stable solutions to the semilinear equation $-\Delta u=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex…

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 derive a priori interior Hessian estimates for the special Lagrangian equation $\sigma_{2}=1$ in dimension three.