Numerical and theoretical results for the real Monge-Ampere equations

The real Monge-Ampere equation$$\left\{\eqalign{&M(u) = det(\nabla\sp2u) = g(x, -u)\quad{\rm in}\ \Omega\cr&u = 0\qquad\qquad\qquad\qquad\qquad\qquad{\rm on}\ \partial\Omega}$$ was studied in this dissertation. This dissertation consists of two different types of numerical solutions for the real Monge-Ampere Equations (one is the mountain pass solution and the other is the minimum solution). The second part is concerned with the monotonicity property of the subsolution and supersolution. The numerical results will be approached by this monotonicity property for $M(u) = det(\nabla\sp2u) = h(x)e\sp{u}$.