On Jörgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge–Ampère equations

  title={On J{\"o}rgens, Calabi, and Pogorelov type theorem and isolated singularities of parabolic Monge–Amp{\`e}re equations},
  author={Jingang Xiong and Jiguang Bao},
  journal={Journal of Differential Equations},
An extension of Jörgens–Calabi–Pogorelov theorem to parabolic Monge–Ampère equation
AbstractWe extend a theorem of Jörgens, Calabi and Pogorelov on entire solutions of elliptic Monge–Ampère equation to parabolic Monge–Ampère equation, and obtain delicate asymptotic behavior of
Symmetry of solutions to parabolic Monge-Ampère equations
AbstractIn this paper, we study the parabolic Monge-Ampère equation −utdet(D2u)=f(t,u)in Ω×(0,T]. Using the method of moving planes, we show that any parabolically convex solution is symmetric with
A Calabi theorem for solutions to the parabolic Monge–Ampère equation with periodic data
Ancient solutions of exterior problem of parabolic Monge–Ampère equations
We use Perron method to prove the existence of ancient solutions of exterior problem for a kind of parabolic Monge–Ampere equation $$-\,u_t\det D^2u=f$$ with prescribed asymptotic behavior at
Multi-valued solutions to a class of parabolic Monge-Ampèreequations
In this paper, we investigate the multi-valued solutions of a class of parabolic Monge-Ampere equation $-u_{t}\det(D^{2}u)=f$. Using the Perron method, we obtain the existence of finitely valued
Entire Solutions of Cauchy Problem for Parabolic Monge–Ampère Equations
Abstract In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation - u t ⁢ det ⁡ D 2 ⁢ u = f ⁢ ( x , t ) -u_{t}\det D^{2}u=f(x,t) and obtain the existence and uniqueness of
Existence of multi-valued solutions with asymptotic behavior of parabolic Monge-Ampère equation
In this paper, we extend the results of multi-valued solutions of elliptic Monge-Ampère equation to parabolic Monge-Ampère equation. We use the Perron method to prove the existence of multi-valued
A Pogorelov estimate and a Liouville-type theorem to parabolic k-Hessian equations
We consider Pogorelov estimates and Liouville-type theorems to parabolic [Formula: see text]-Hessian equations of the form [Formula: see text] in [Formula: see text]. We derive that any [Formula: see


Isolated singularities of Monge-Ampère equations
Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt − s2=E in the punctured disk 0<(x−x0)2+(y−y0)2<ρ2. Assume thatq is
A celebrated result of Calabi [C], that generalizes a two dimensional theorem by Jorgens [J], asserts that if u is a C5 convex solution of the elliptic MongeAmpere equation detD2u = 1 in Rn and n ≤ 5
An extension to a theorem of Jörgens, Calabi, and Pogorelov
(1.1) det(D2u) = 1 in Rn must be a quadratic polynomial. For n = 2, a classical solution is either convex or concave; the result holds without the convexity hypothesis. A simpler and more analytical
W2,p Estimates for the Parabolic¶Monge-Ampère Equation
Abstract When u is a solution to the equation −ut det Dx2u=f with f positive, continuous, and ft satisfying certain growth conditions, we establish estimates in L∞ for ut and show that Dx2u satisfies
Parabolic equations of the second order
from a unified point of view, namely, the extensive use of Green's function. Our main interest is concerned with the first mixed boundary problem (for definition, see ?4) for f linear or nonlinear in
A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity
The purpose of this note is to show a localization property of convex viscosity solutions to the Monge-Ampere inequality 0 1−(2/n)) are strictly convex
Removable singular sets of fully nonlinear elliptic equations
In this paper we consider fully nonlinear elliptic equations, including the Monge-Ampere equation and the Weingarden equation. We assume that F(D 2 u;x)= f( x) x 2 ; u( x)= g( x) x 2 @ has a solution
Maximum principles introduction to the theory of weak solutions Holder estimates existence, uniqueness and regularity of solutions further theory of weak solutions strong solutions fixed point