Hyperbolic Monge-Ampere Equations Research Articles

Smooth Local Solutions to Degenerate Hyperbolic Monge-Ampère Equations

In this paper, we prove the existence of smooth solutions near 0 of the degenerate hyperbolic Monge-Ampere equation $$\det (D^2 u) = K(x)h(x,u,Du)$$ , where $$K(0)=0$$ , $$K\le 0$$ , $$h(0,0,0)>0$$ . We also assume that, the zero set of small perturbation of $$D_n K$$ has a simple structure. For the proof, we first transform the linearized equation into a simple form by a suitable change of variables. Then we proceed to derive a priori estimates for the linearized equation, which is degenerately hyperbolic. Finally we use Nash-Moser iteration to prove the existence of local solutions.

The lifespan of classical solution to the Cauchy problem for the hyperbolic mean curvature flow with a linear forcing term

We investigate the hyperbolic mean curvature flow with a linear forcing term. By means of support function, a hyperbolic Monge-Ampere equation is derived and this equation could be reduced to a system in Riemann invariants. Using the theory of the local solution to Cauchy problem for quasilinear hyperbolic system, we discuss the lifespan (the maximal existence time of unique local classical solutions) of classical solution to the Cauchy problem for the hyperbolic mean curvature flow with a linear forcing term.

Symmetry reduction and exact solutions of a hyperbolic Monge-Ampère equation

By means of the classical symmetry method, a hyperbolic Monge-Ampere equation is investigated. The symmetry group is studied and its corresponding group invariant solutions are constructed. Based on the associated vector of the obtained symmetry, the authors construct the group-invariant optimal system of the hyperbolic Monge-Ampere equation, from which two interesting classes of solutions to the hyperbolic Monge-Ampere equation are obtained successfully.

Hyperbolic mean curvature flow: evolution of plane curves

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves S1 × [0,T) → ℝ2 which satisfies the following evolution equation ∂2F∂t2(u,t)=k(u,t)N→(u,t)-∇ρ(u,t),∀(u,t)∈S1×[0,T) with the initial data F(u,0)=F0(u)and∂F∂t(u,0)=f(u)N→0, where k is the mean curvature and ▪ is the unit inner normal vector of the plane curve F(u,t), f(u) and ▪ are the initial velocity and the unit inner normal vector of the initial convex closed curve Fo, respectively, and Δρ is given by ∇ρ≜(∂2F∂s∂t,∂F∂t)T→, in which ▪ stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution converges to a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time ℝ1,1.

Nondegenerate Monge-Ampere Structures in Dimension 6

We define a nondegenerate Monge–Ampere structure on a six-dimensional manifold as a pair (Ω, ω), such that Ω is a symplectic form and ω is a 3-differential form which satisfies ω∧Ω=0 and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi–Yau structure and we study its integrability from the point of view of Monge–Ampere operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge–Ampere equation in dimension 4. We study from this point of view the example of the Stenzel metric on T*S3.

Bianchi transformation between the real hyperbolic Monge-Amp�re equation and the Born-Infeld equation

By casting the Born-Infeld equation and the real hyperbolic Monge-Ampere equation into the form of equations of hydrodynamic type, we find that there exists an explicit transformation between them. This is Bianchi transformation.