The Dirichlet problem for curvature equations in Riemannian manifolds Flavio CruzJorge de Lira 53A1053C42fully nonlinear elliptic PDEscurvature equations We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allow us to extend some of the existence theorems of Caffarelli, Nirenberg, and Spruck [\textit{Nonlinear second-order elliptic equations. V: The Dirichlet problem for Weingarten hypersurfaces}, Comm. Pure Appl. Math. \textbf{41} (1988), no. 1, 47--70], and Ivochkina, Trudinger, and Lin [N.\:M. Ivochkina, \textit{The Dirichlet problem for the curvature equation of order $m$}, Algebra i Analiz \textbf{2} (1990), no. 3, 192--217 (Russian); English transl., Leningrad Math. J. \textbf{2} (1991), no. 3, 631--654], [N.\:M. Ivochkina, M. Lin, and N.\:S. Trudinger, \textit{The Dirichlet problem for the prescribed curvature quotient equations with general boundary values}, Geometric Analysis and the Calculus of Variations, Int. Press, Cambridge, MA, 1996, pp. 125--141], [M. Lin and N.\:S. Trudinger, \textit{The Dirichlet problem for the prescribed curvature quotient equations}, Topol. Methods Nonlinear Anal. \textbf{3} (1994), no. 2, 307--323] to more general curvature functions under mild conditions on the geometry of the domain. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.4996 10.1512/iumj.2013.62.4996 en Indiana Univ. Math. J. 62 (2013) 815 - 854 state-of-the-art mathematics http://iumj.org/access/