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

Partial regularity for elliptic equations

In this paper we study partial and anisotropic Schauder estimates for linear and nonlinear elliptic equations. We prove that if the inhomogeneous term f is Hölder continuous in the Xn-direction, then the mixed derivatives uxxn are Hölder continuous; if f satisfies an anisotropic Hölder continuity condition, then the second derivatives D²v satisfy related anisotropic Hölder continuity estimates.The first author was supported by NSFC Grant 10871199. The second author was supported by the Australian Research Council

Moser-Trudinger type inequalities for the Hessian equation

The k-Hessian equation for k≥2 is a class of fully nonlinear partial differential equation of divergence form. A Sobolev type inequality for the k-Hessian equation was proved by the second author in 1994. In this paper, we prove the Moser-Trudinger typ

A priori estimates for fully nonlinear parabolic equations

In this paper, we prove the Hölder continuity of the second derivatives for fully nonlinear, uniformly parabolic equations. We assume the inhomogeneous term f is Hölder continuous with respect to the spatial variables x and bounded and measurable with

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