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CPAA

In this article we investigate a general class of Monge-Ampère equations
in the plane, including the constant Gauss curvature equation. Our first aim
is to prove some maximum and minimum principles for suitable $P$-functions, in the sense of L.E. Payne. Then, these
new principles are employed to solve a general class of overdetermined Monge-Ampère
problems and to investigate two boundary value problems for the constant Gauss curvature
equation. More precisely, when the constant Gauss curvature equation is
subject to the homogeneous Dirichlet boundary condition, we prove several
isoperimetric inequalities, while when it is subject to the contact angle
boundary condition, some necessary conditions of
solvability, involving the curvature of the boundary of the underlying domain and the given
contact angle, are derived.

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