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Sharp global Alexandrov estimates and entire solutions of Monge-Ampère equations
Tianling Jin, Xushan Tu, Jingang Xiong
This paper continues our work [19] on sharp Alexandrov estimates. We obtain a sharp global uniform distance estimate from a convex function to the class of unimodular convex quadra…
Extremal Alexandrov estimates: singularities, obstacles, and stability
Tianling Jin, Xushan Tu, Jingang Xiong
The classical Alexandrov estimate controls the oscillation of a convex function by the mass of its associated Monge-Ampère measure and yields, for two convex functions of varia…
On the singular set of the free boundary for a Monge-Ampère obstacle problem
Tianling Jin, Xushan Tu, Jingang Xiong
This is a continuation of our earlier work [14] on the Monge-Ampère obstacle problem \[ \det D^2 v = v^q χ_{\{v>0\}}, \quad v \geq 0 \text{ convex} \] with , where we…
Regularity and classification of the free boundary for a Monge-Ampère obstacle problem
Tianling Jin, Xushan Tu, Jingang Xiong
We study convex solutions to the Monge-Ampère obstacle problem \[ \operatorname{det} D^2 v=g v^qχ_{\{v>0\}}, \quad v \geq 0, \] where is a constant and is a bound…
A Liouville theorem for the Neumann problem of the Monge-Ampere equation
Huaiyu Jian, Xushan Tu
In this paper, we study the Neumann problem of Monge-Ampère equations in Semi-space. For two dimensional case, we prove that its viscosity convex solutions must be a quadratic poly…
On A Class of Degenerate And Singular Monge-Ampère Equations
Huaiyu Jian, You Li, Xushan Tu
In this paper we shall prove the existence, uniqueness and global Hlder continuity for the Dirichlet problem of a class of Monge-Ampère type equations which may be degene…