The Obstacle Problem for Monge - Amp Ere Equation

Analytic Fourier Integral Operators, Monge-amp Ere Equation and Holomorphic Factorization

We will show that the factorization condition for the Fourier integral operators I (X; Y ;) leads to a parametrized parabolic Monge-Amp ere equation. In case of an analytic operator the bration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases by considering more general continuation problem for the level sets of a holomorphic mapping. The results are ...

Exterior Monge- Amp`ere Solutions

We discuss the Siciak-Zaharjuta extremal function of a real convex body in C n , a solution of the homogeneous complex Monge-Ampère equation on the exterior of the convex body. We determine several conditions under which a foliation by holomorphic curves can be found in the complement of the convex body along which the extremal function is harmonic. We study a variational problem for holomorphi...

Global Bifurcation for Monge - Amp ere

Adaptive Control of a Diffusion to a Goaland

We study the following adaptive stochastic control problem: to maximize the probability PX(T) = 1] of reaching the \goal" x = 1 during the nite time-horizon 0; T], over \control" processes () which are adapted to the natural ltration of the \observation" process Y (t) = W(t) + Bt; 0 t T and satisfy almost surely R T 0 2 (t)dt < 1 and 0 X(t) = x + R t 0 (s)dY (s) 1; 80 t T. Here W() is standard ...

Special Classes of Three Dimensional Affine Hyperspheres Characterized by Properties of Their Cubic Form

It is well known that locally strongly convex a ne hyperspheres can be determined as solutions of di erential equations of Monge-Amp ere type. The global properties of those solutions are well understood. However, due to the nature of the Monge-Amp ere equation, not much is known about local solutions, particularly if the dimension of the hypersurface is greater then 2. By the fundamental theor...