Memoirs of the American Mathematical Society 2005; 64 pp; softcover Volume: 178 ISBN10: 082183763X ISBN13: 9780821837634 List Price: US$57 Individual Members: US$34.20 Institutional Members: US$45.60 Order Code: MEMO/178/840
 We collect here results on the existence and stability of weak solutions of complex MongeAmpére 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 MongeAmpére 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 MongeAmpére equation on compact Kähler manifolds. This is a generalization of the CalabiYau 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 MongeAmpère equation with continuous data
 The Dirichlet problem continued
 The MongeAmpère equation for unbounded functions
 The complex MongeAmpère equation on a compact Kähler manifold
 Bibliography
