Parabolic Monge-Ampere Equation Research Articles

Asymptotic mean value formulas for parabolic nonlinear equations

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Ampere equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.

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 infinity outside some certain bowl-shaped domain in the lower half space for $$n\ge 3$$ , where f is a perturbation of 1 at infinity. We raise this problem for the first time and construct a new subsolution to it. We also use similar method to prove the existence of the entire solutions.

An extension of Jörgens–Calabi–Pogorelov theorem to parabolic Monge–Ampère equation

We extend a theorem of Jorgens, Calabi and Pogorelov on entire solutions of elliptic Monge–Ampere equation to parabolic Monge–Ampere equation, and obtain delicate asymptotic behavior of solutions at infinity. For the dimension $$n\ge 3$$ , the work of Gutierrez and Huang in Indiana Univ. Math. J. 47, 1459–1480 (1998) is an easy consequence of our result. And along the line of approach in this paper, we can treat other parabolic Monge–Ampere equations.

An initial-boundary value problem for parabolic Monge-Ampère equation in mathematical finance

This paper deals with some parabolic Monge-Ampere equation raised from mathematical finance: VsVyy+ryVyVyy−θVy2= 0 (Vyy < 0). The existence and uniqueness of smooth solution to its initial-boundary value problem with some requirement is obtained.

On Harnack’s Inequality for the Linearized Parabolic Monge-Ampère Equation

It is shown that the parabolic Harnack property stands as an intrinsic feature of the Monge-Ampere quasi-metric structure by proving Harnack’s inequality for non-negative solutions to the linearized parabolic Monge-Ampere equation under minimal geometric assumptions.

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-Ampere equation to parabolic Monge-Ampere equation. We use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of parabolic Monge-Ampere equation. Moreover, we prove that the multi-valued solution is continuous in the whole space.

Multi-valued solutions to a class of parabolic Monge-Ampère equations

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 and infinitely valued solutions to the parabolic Monge-Ampere equations. We generalize the results of elliptic Monge-Ampere equations and Hessian equations.

Symmetry of solutions to parabolic Monge-Ampère equations

In this paper, we study the parabolic Monge-Ampere equation -ut det(D 2 u) = f (t, u )i n � × (0, T). Using the method of moving planes, we show that any parabolically convex solution is symmetric with respect to some hyperplane. We also give a counterexample in R n × (0, T) and an example in a cylinder to illustrate the results. MSC: 35K96; 35B06

A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds

We study the asymptotic behavior of the parabolic Monge-Ampere equation in , in , where is a compact complete Riemannian manifold, λ is a positive real parameter, and is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.

BMO spaces associated to generalized parabolic sections

Parabolic sections were introduced by Huang to study the parabolic Monge Ampere equation. In this note, we introduce the generalized parabolic sections P and define BMO(superscript q subscript P) spaces related to these sections. We then establish the John-Nirenberg type inequality and verify that all BMO(superscript q subscript P) are equivalent for q≥1.

Contact Relative Differential Invariants for Non Generic Parabolic Monge-Ampère Equations

We find relative differential invariants of different orders for non generic parabolic Monge-Ampere equations (MAE?s). They are constructed in terms of some tensors associated with the derived flag of the characteristic distribution. The vanishing of such invariants allows one to determine the classes of each non generic parabolic MAE with respect to contact transformations.

CONVERGENCE OF THE KÄHLER-RICCI FLOW ON NONCOMPACT KÄHLER MANIFOLDS

We study the Kahler-Ricci flow on noncompact Kahler manifolds and provide conditions under which the flow has a long time solution converging to a complete negative Kahler-Einstein metric. We also study the complex parabolic Monge-Ampere equation.

Analytic Fourier integral operators, Monge-Amp�re equation and holomorphic factorization

We will show that the factorization condition for the Fourier integral operators \(I_\rho ^\mu (X,Y;\it\Lambda )\) leads to a parametrized parabolic Monge-Ampere equation. For an analytic operator, the fibration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases, by considering a more general continuation problem for the level sets of a holomorphic mapping. The results are applied to obtain Lp-continuity for translation invariant operators in \({\Bbb R}^n\) with \(n\leq 4\) and for arbitrary \({\Bbb R}^n\) with \(d\pi _{X\times Y}|_\Lambda \leq n+2\).

A generalization of a theorem by Calabi to the parabolic Monge-Ampere equation

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 then u is a quadratic polynomial. This statement was extended to all dimensions by Pogorelov [P2] and Cheng and Yau [Ch-Y]. Recently, Caffarelli [Ca3] proved, by using the regularity theory for the Monge-Ampere equation developed in the fundamental papers [Ca1] and [Ca2], that this result holds true for viscosity solutions. The purpose of this article is to investigate the validity of results of the same nature for solutions of the parabolic Monge-Ampere equation −ut detD2u = 1 in Rn × (−∞, 0]. This type of differential operator was first considered by Krylov [K2]. It also appears in connection with the problem of the deformation of a surface by means of its Gauss-Kronecker curvature. Indeed, Tso [T] solved this problem by noting that the support function to the surface that is deforming satisfies an initial value problem involving that parabolic operator. The function u : Rn × (−∞, 0]→ R, u = u(x, t), is called parabolically convex if it is continuous, convex in x and nonincreasing in t. By D2u(x, t) we denote the matrix of second derivatives of u with respect to x and Du denotes the gradient of u with respect to x. We use the standard notation C2k,k(Ω) to denote the class of functions u such that the derivatives Di xD j tu are continuous in Ω for i+ 2j ≤ 2k. We set R − = R n × (−∞, 0]. The main result of this paper is the following theorem.

A complex parabolic type monge-ampère equation

The complex parabolic type Monge-Ampere equation we are dealing with is of the form\((\partial u/\partial t){\text{ det[}}\partial ^2 u/\partial z_i {\text{ }}\partial \overline z _j ] = f\) inB × (0,T),u=ϕ on Γ, whereB is the unit ball in ℂd,d>1, and Γ is the parabolic boundary ofB × (0,T). Solutionu is proved unique in the class\(C(\bar B \times [0,T]) \cap W_{\infty ,loc}^{2,1} (B \times (0,T))\).

Uniqueness of the generalized solution of a parabolic type monge-ampere equation

The parabolic type Monge-Ampere equation treated here is of the form in Dx(0,T)where . A concept of generalized solution is introduced and uniqueness results are obtained for the associated first initial-boundary value problem