Necessary conditions of solvability and isoperimetric estimates for some Monge-Ampère problems in the plane

Abstract:

This article is mainly devoted to the solvability of the Monge-Ampère equation $\det \left ( D^{2}u\right ) =1,$ in a $C^{2}$ bounded strictly convex domain $\Omega \subset \mathbb {R}^{2}$, subject to a contact angle boundary condition. A necessary condition for the solvability of this problem, involving the maximal value of the curvature $k\left ( s\right )$ of $\partial \Omega$ and the contact angle, was derived by X.-N. Ma in 1999, making use of a maximum principle for an appropriate P-function. Our main goal here is to prove a complementary result. More precisely, we will derive a new necessary condition of solvability, involving the minimal value of the curvature $k\left ( s\right )$ of $\partial \Omega$ and the contact angle. The main ingredients of our proof are the derivation of a minimum principle for the P-function employed by X.-N. Ma in his proof, respectively, the use of some computations in normal coordinates with respect to the boundary $\partial \Omega$. Finally, a similar minimum principle will be employed to derive some isoperimetric estimates for the classical convex solution of the Monge-Ampère equation, subject to the homogeneous Dirichlet boundary condition.