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

2019 ◽  
Vol 9 (1) ◽  
pp. 729-744 ◽  
Author(s):  
Xuemei Zhang ◽  
Meiqiang Feng
Keyword(s):  
Asymptotic Behavior ◽  
Convex Domain ◽  
Regular Variation ◽  
Blow Up ◽  
Variation Theory ◽  
Type Condition ◽  
Strictly Convex ◽  
Convex Solution ◽  
Monge Ampere ◽  
Standard Techniques

Abstract 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.

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

2021 ◽  
Vol 11 (1) ◽  
pp. 321-356
Author(s):  
Haitao Wan ◽  
Yongxiu Shi ◽  
Wei Liu
Keyword(s):  
Dirichlet Problem ◽  
Convex Domain ◽  
Boundary Behavior ◽  
Strictly Convex ◽  
Convex Solution ◽  
Regularly Varying ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

Abstract 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.

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

2018 ◽  
Vol 18 (2) ◽  
pp. 289-302
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Crucial Role ◽  
Blow Up ◽  
Boundary Behavior ◽  
Structure Condition ◽  
Bounded Smooth Domain ◽  
Convex Solution ◽  
Bounded Smooth ◽  
Monge Ampere ◽  
Singular Dirichlet Problem

AbstractThis 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.

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

2021 ◽  
pp. 1-15
Author(s):  
Li Chen
Keyword(s):  
Type Equation ◽  
Gauss Curvature ◽  
Curvature Flow ◽  
Strictly Convex ◽  
Convex Solution ◽  
Orlicz Minkowski Problem ◽  
Closed Hypersurfaces ◽  
Smooth Measures ◽  
Gauss Curvature Flow ◽  
Monge Ampere

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.

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

2020 ◽  
Vol 10 (1) ◽  
pp. 371-399
Author(s):  
Meiqiang Feng
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Type System ◽  
Power Type ◽  
Sharp Estimates ◽  
Convex Solution ◽  
Existence And Nonexistence ◽  
Monge Ampere

Abstract 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 elliptic problems with nonlinear gradient terms and singular weights

2008 ◽  
Vol 138 (6) ◽  
pp. 1403-1424 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Regular Variation ◽  
Blow Up ◽  
Smooth Boundary ◽  
Elliptic Problems ◽  
Variation Theory ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Comparison Functions ◽  
Positive Index ◽  
Singular Weights

By Karamata regular variation theory, a perturbation method and construction of comparison functions, we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems Δu ± |Δu|q = b(x)g(u), u > 0 in Ω, u|∂Ω = ∞, where Ω is a bounded domain with smooth boundary in ℝN, q > 0, g ∈ C1[0, ∞) is increasing on [0, ∞), g(0) = 0, g′ is regularly varying at infinity with positive index ρ and b is non-negative in Ω and is singular on the boundary.

Existence and asymptotic behavior of ground state solutions of semilinear elliptic system

2017 ◽  
Vol 6 (3) ◽  
pp. 301-315 ◽  
Author(s):  
Habib Mâagli ◽  
Sonia Ben Othman ◽  
Safa Dridi
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic System ◽  
Positive Solutions ◽  
Ground State ◽  
Regular Variation ◽  
Variation Theory ◽  
Content Type ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Alpha Beta

AbstractIn this article, we take up the existence and the asymptotic behavior of entire bounded positive solutions to the following semilinear elliptic system:-Δu = a_{1}(x)u^{\alpha}v^{r}, x\in\mathbb{R}^{n} (n\geq 3), -Δv = a_{2}(x)v^{\beta}u^{s}, x\in\mathbb{R}^{n}, u,v ¿ 0 in \mathbb{R}^{n}, \lim_{|x|\rightarrow+\infty}u(x) = \lim_{|x|\rightarrow+\infty}v(x)=0,where {\alpha,\beta<1}, {r,s\in\mathbb{R}} such that {\nu:=(1-\alpha)(1-\beta)-rs>0}, and the functions a_{1}, a_{2} are nonnegative in {\mathcal{C}^{\gamma}_{\mathrm{loc}}(\mathbb{R}^{n})} (0¡γ¡1) and satisfy some appropriate assumptions related to Karamata regular variation theory.

Asymptotic behavior of solutions of half-linear differential equations and generalized Karamata functions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kusano Takaŝi ◽  
Jelena V. Manojlović
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Continuous Function ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Regular Variation ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Positive Continuous Function ◽  
Linear Differential

AbstractWe study the asymptotic behavior of eventually positive solutions of the second-order half-linear differential equation(p(t)\lvert x^{\prime}\rvert^{\alpha}\operatorname{sgn}x^{\prime})^{\prime}+q(% t)\lvert x\rvert^{\alpha}\operatorname{sgn}x=0,where q is a continuous function which may take both positive and negative values in any neighborhood of infinity and p is a positive continuous function satisfying one of the conditions\int_{a}^{\infty}\frac{ds}{p(s)^{1/\alpha}}=\infty\quad\text{or}\quad\int_{a}^% {\infty}\frac{ds}{p(s)^{1/\alpha}}<\infty.The asymptotic formulas for generalized regularly varying solutions are established using the Karamata theory of regular variation.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

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