Monge-Ampère equations on Riemannian manifolds
This article studies the Dirichlet problem: \[ (g^{-1} \text{det} \nabla_i \nabla_j u)^{1/n} =\psi (x,u,\nabla u) \text{ in } \Omega, \quad u= \varphi\text{ on } \partial \Omega, \] where \(\Omega\) is a compact domain with smooth boundary \(\partial\Omega\) in a smooth \(n\)-dimensional Riemannian manifold. Here \(g\) is the determinant of the metric tensor and \(\nabla\) denotes covariant differentiation. Assuming the right hand side \(\psi\) is convex with respect to its last variable and \(\psi_u\geq 0\), the authors give existence and uniqueness theorems for this problem. In the course of their proof, they give an interior estimate for second derivatives (under some convexity assumptions on \(u\) and \(\psi)\): \[ \bigl|\nabla^2 u(x) \bigr|\leq Cd (x)^{-N}, \] where \(C\) and \(N\) are constants and \(d(x)\) is the distance of \(x\in \Omega\) to the boundary \(\partial\Omega\). As a corollary they obtain the following theorem (already known under various additional assumptions): if \(\Omega\) is a strictly convex domain and \(\psi\) depends on \(x\) only, then the above problem admits a unique locally strictly convex solution which is smooth up to the boundary of \(\Omega\).
scientific article; zbMATH DE number 952553