We collect here results on the existence and stability of weak solutions of complex Monge-Ampére equation proved by applying pluripotential theory methods and obtained in past three decades. First we set the stage introducing basic concepts and theorems of pluripotential theory. Then the Dirichlet problem for the complex Monge-Ampére equation is studied. The main goal is to give possibly detailed description of the nonnegative Borel measures which on the right hand side of the equation give rise to plurisubharmonic solutions satisfying additional requirements such as continuity, boundedness or some weaker ones. In the last part the methods of pluripotential theory are implemented to prove the existence and stability of weak solutions of the complex Monge-Ampére equation on compact Kähler manifolds. This is a generalization of the Calabi-Yau theorem.

Readership

Graduate students and research mathematicians interested in differential equations.

Table of Contents

• Positive currents and plurisubharmonic functions
• Siciak's extremal function and a related capacity
• The Dirichlet problem for the Monge-Ampère equation with continuous data
• The Dirichlet problem continued
• The Monge-Ampère equation for unbounded functions
• The complex Monge-Ampère equation on a compact Kähler manifold
• Bibliography