Hessian estimates for convex solutions to quadratic Hessian equation

THE k-HESSIAN EQUATION

The k-Hessian is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equations. It is elliptic when restricted to k-admissible functions. In this paper we establish the existence and regularity of k-admissible solutions to the Dirichlet problem of the k-Hessian equation. By a g...

C∞ local solutions of elliptical 2−Hessian equation in R

In this work, we study the existence ofC local solutions to 2-Hessian equation in R . We consider the case that the right hand side function f possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function f . The associated linearized operator are uniformly elliptic.

Eigenvalue Estimates And Singularities Of Hessian Fields

Hessian Riemannian Gradient Flows in Convex Programming

In view of solving theoretically constrained minimization problems, we investigate the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions. The first result characterizes Hessian Riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities, giving a new motivation for the ...

Einstein-Hessian barriers on convex cones

On the interior of a regular convex cone K ⊂ R there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the Cheng-Yau metric. The former is associated with a self-concordant logarithmically homogeneous barrier on K with parameter of order O(n), the universal barrier. This barrier is invariant with respect to the unimodular automorphism su...