On the existence of smooth plurisubharmonic solutions for certain degenerate monge—ampère equations

Abstract

We construct a special plurisubharmonic defining function for a smoothly bounded, strictly pseudoconvex domain in so that the determinant of the complex Hessian vanishes to high order at the boundary. This construction, coupled with regularity results for solutions of the complex Monge—Ampère equation and a reflection principle, enables us to give a new proof of the Fefferman mapping theorem. Our result, in a sense, completes a program that was initiated by Kerzman, Kohn, and Nirenberg more than twenty years ago.

Keywords:

• Monge—Ampère equation
• Plurisubharmonic function
• Defining function
• Biholomorphic mappings
• Smoothness to the boundary

AMS Subject Classification Numbers::

• 32F07
• 35J25

^∗ [email protected].

^† mail: [email protected].

^∗ [email protected].

^† mail: [email protected].

Notes

^∗ [email protected].

^† mail: [email protected].