Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

In this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation det ( D 2 u ) = b ( x ) g ( − u ) , u < 0 in Ω and u = 0 on ∂ Ω , $$\mbox{ det}(D^{2} u)=b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and } u=0 \mbox{ on }\partial\Omega, $$ where Ω is a bounded, smooth and strictly convex domain in ℝ N (N ≥ 2), b ∈ C∞(Ω) is positive and may be singular (including critical singular) or vanish on the boundary, g ∈ C 1((0, ∞), (0, ∞)) is decreasing on (0, ∞) with lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $ and g is normalized regularly varying at zero with index −γ(γ>1). Our results reveal the refined influence of the highest and the lowest values of the (N − 1)-th curvature on the second boundary behavior of the unique strictly convex solution to the problem.

Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

In this paper, the equations and systems of Monge-Ampère with parameters are considered. We first show the uniqueness of of nontrivial radial convex solution of Monge-Ampère equations by using sharp estimates. Then we analyze the existence and nonexistence of nontrivial radial convex solutions to Monge-Ampère systems, which includes some new ingredients in the arguments. Furthermore, the asymptotic behavior of nontrivial radial convex solutions for Monge-Ampère systems is also considered. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial convex solutions of the power-type system of Monge-Ampère equations.

Boundary blow-up solutions to the Monge-Ampère equation: Sharp conditions and asymptotic behavior

Consider the boundary blow-up Monge-Ampère problem $$\begin{array}{} \displaystyle M[u]=K(x)f(u) \mbox{ for } x \in {\it\Omega},\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial {\it\Omega})\rightarrow 0. \end{array}$$ Here M[u] = det (uxixj) is the Monge-Ampère operator, and Ω is a smooth, bounded, strictly convex domain in ℝN (N ≥ 2). Under K(x) satisfying appropriate conditions, we first prove that the boundary blow-up Monge-Ampère problem has a strictly convex solution if and only if f satisfies Keller-Osserman type condition. Then the asymptotic behavior of strictly convex solutions to the boundary blow-up Monge-Ampère problem is considered under weaker condition with respect to previous references. Finally, if f does not satisfy Keller-Osserman type condition, we show the existence of strictly convex solutions under different conditions on K(x). The proof combines standard techniques based upon the sub-supersolution method with non-standard arguments, such as the Karamata regular variation theory.

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

This paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation\operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0,where Ω is a strictly convex and bounded smooth domain in{\mathbb{R}^{N}}, with{N\geq 2},{g\in C^{1}((0,\infty),(0,\infty))}is decreasing in{(0,\infty)}and satisfies{\lim_{s\rightarrow 0^{+}}g(s)=\infty}, and{b\in C^{\infty}(\Omega)}is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition ongwhich plays a crucial role in the boundary behavior of such solution.