A sharp global estimate and an overdetermined problem for Monge-Ampère type equations

Abstract

We consider Monge-Ampère type equations involving the gradient that are elliptic in the framework of convex functions. Through suitable symmetrization we find sharp estimates to solutions of such equations. An overdetermined problem related to our Monge-Ampère type operators is also considered and we show that such a problem may admit a solution only when the underlying domain is a ball.

Keywords

• sharp global estimate
• monge-ampère type equations
• overdetermined problem
• symmetrization
• 35b45
• 35b51
• 35j96
• 35n25
• 49j40