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.

Continuity results for parametric nonlinear singular Dirichlet problems

In this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameter λ > 0 that was considered in [32]. Denoting by Sλ the set of positive solutions of the problem corresponding to the parameter λ, we establish the following essential properties of Sλ: there exists a smallest element $\begin{array}{} u_\lambda^* \end{array}$ in Sλ, and the mapping λ ↦ $\begin{array}{} u_\lambda^* \end{array}$ is (strictly) increasing and left continuous; the set-valued mapping λ ↦ Sλ is sequentially continuous.

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.

Well-posedness of the second-order linear singular Dirichlet problem

Conditions guaranteeing well-posedness of the problem