The Dirichlet problem for Hessian equations on Riemannian manifolds (Q1283880)

The author studies the Dirichlet problem on subdomains of a Riemannian manifold \((M,g)\) for nonlinear elliptic equations of the form \[ F(\nabla^2 u,u) \equiv f(\lambda(\kappa ug + \nabla^2 u)) = \psi(x,u,\nabla u) \] where \(f\) is a suitable symmetric function defined on an open symmetric convex cone in \({\mathbb R}^n\), \(\kappa\) is a constant, and \(\lambda(\kappa ug + \nabla^2 u)\) denotes the vector of eigenvalues of \(\kappa ug + \nabla^2 u\). Such problems in subdomains of \({\mathbb R}^n\) and with \(\kappa=0\) have been studied by several authors, in particular, \textit{L. Caffarelli, L. Nirenberg} and \textit{J. Spruck} [Acta Math. 155, 261-301 (1985; Zbl 0654.35031)]. As in the author's previous work with \textit{J. Spruck} [Ann. Math. 138, 601-624 (1993; Zbl 0840.53046)] and with \textit{Y. Y. Li} [J. Differ. Equations 132, No. 1, 126-139 (1996; Zbl 0866.58067)] on equations of Monge-Ampère type, the existence of a strict subsolution satisfying the required boundary condition is assumed, which avoids imposing curvature conditions on the boundary of the domain.