eBook ISBN:  9781470404413 
Product Code:  MEMO/178/840.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 
eBook ISBN:  9781470404413 
Product Code:  MEMO/178/840.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 178; 2005; 64 ppMSC: Primary 32; Secondary 53
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.
ReadershipGraduate students and research mathematicians interested in differential equations.

Table of Contents

Chapters

1. Positive currents and plurisubharmonic functions

2. Siciak’s extremal function and a related capacity

3. The Dirichlet problem for the MongeAmpère equation with continuous data

4. The Dirichlet problem continued

5. The MongeAmpère equation for unbounded functions

6. The complex MongeAmpère equation on a compact Kähler manifold


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Requests
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.
Graduate students and research mathematicians interested in differential equations.

Chapters

1. Positive currents and plurisubharmonic functions

2. Siciak’s extremal function and a related capacity

3. The Dirichlet problem for the MongeAmpère equation with continuous data

4. The Dirichlet problem continued

5. The MongeAmpère equation for unbounded functions

6. The complex MongeAmpère equation on a compact Kähler manifold