Existence of entire solutions of Monge–Ampère equations with prescribed asymptotic behavior

2019 ◽  
Vol 58 (6) ◽  
Author(s):  
Jiguang Bao ◽  
Jingang Xiong ◽  
Ziwei Zhou
Keyword(s):  
Asymptotic Behavior ◽  
Entire Solutions ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

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.

Entire Solutions of Cauchy Problem for Parabolic Monge–Ampère Equations

2020 ◽  
Vol 20 (4) ◽  
pp. 769-781
Author(s):  
Limei Dai ◽  
Jiguang Bao
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Viscosity Solutions ◽  
Existence And Uniqueness ◽  
Entire Solutions ◽  
Perron Method ◽  
The Cauchy Problem ◽  
Monge Ampere

AbstractIn this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation-u_{t}\det D^{2}u=f(x,t)and obtain the existence and uniqueness of viscosity solutions with asymptotic behavior by using the Perron method.

scholarly journals The Dirichlet Problem of Hessian Equation in Exterior Domains

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 666
Author(s):  
Hongfei Li ◽  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Viscosity Solutions ◽  
Exterior Domains ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

In this paper, we will obtain the existence of viscosity solutions to the exterior Dirichlet problem for Hessian equations with prescribed asymptotic behavior at infinity by the Perron’s method. This extends the Ju–Bao results on Monge–Ampère equations det D 2 u = f ( x ) .

scholarly journals Asymptotic Properties of Solutions to Second-Order Difference Equations of Volterra Type

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Janusz Migda ◽  
Małgorzata Migda ◽  
Magdalena Nockowska-Rosiak
Keyword(s):  
Asymptotic Behavior ◽  
Difference Equations ◽  
Existence Of Solutions ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Type Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Volterra Type ◽  
Order Difference ◽  
Prescribed Asymptotic Behavior

We consider the discrete Volterra type equation of the form Δ(rnΔxn)=bn+∑k=1nK(n,k)f(xk). We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we study the asymptotic behavior of solutions. We use o(ns), for given nonpositive real s, as a measure of approximation.

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