Existence and convexity of local solutions to degenerate hessian equations

In this work, we prove the existence of local convex solution to the degenerate Hessian equationComment: corrections some typos in this versio

C∞C^\infty Local solutions of elliptical 2−2-Hessian equation in R3\mathbb{R}^3

International audienc

Local solvability of the k-Hessian equations

International audienceWe give a classification of second-order polynomial solutions for the homogeneous k-Hessian equation σ k [u] = 0. There are only two classes of polynomial solutions: One is convex polynomial; another one must not be (k + 1)-convex, and in the second case, the k-Hessian equations are uniformly elliptic with respect to that solution. Base on this classification, we obtain the existence of C ∞ local solution for nonhomogeneous term f without sign assumptions