The Obstacle Problem for Monge - Amp Ere Equation
نویسنده
چکیده
We consider the obstacle problem for the degenerated Monge-Amp ere equation. We prove the existence of the greatest viscosity sub-solution,and the C 1;1-regularity. Then the solution satisses the concave uniformly elliptic equation. We use the author's previous work to show the C 1;;-regularity of the free boundary. Finally, we discuss the stability of the free boundary. In this paper, we consider the greatest viscosity sub-solution of the following equation. > 0 on @ (x o) < 0 for some x 0 2 @ is smooth and convex In the author's paper 1], the obstacle problem is considered for the fully nonlinear uniformly elliptic operators. C 1;;-regularity of the free boundary is established. Then one of the natural question is wheather it still holds for degenerated elliptic operators.We consider the degenerated Monge-Amp ere equation whose Dirichlet problem was solved in 2]. This is one of the papers in this trend. The Monge-Amp ere equation is one of the typical concave and also degenerated nonlinear operator. On the other hand, if the C 1;1-estimate of the solution is gotten, it becomes uniformly elliptic equations 2]. Moreover the diierence between the obstacle and the solution satisses the convex uniformly elliptic equation since the obstacle is above the
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