The Dirichlet problem for Hessian equations on Riemannian manifolds

on a Riemannian manifold (M n , g), where f is a symmetric function of λ ∈ R , κ is a constant, ∇2u denotes the Hessian of a function u on M and, for a (0, 2) tensor h on M , λ(h) = (λ1, · · · , λn ) denotes the eigenvalues of h with respect to the metric g. The Dirichlet problem for equations of type (1.1) in R , with κ = 0, under various hypothesis, is studied by Caffarelli, Nirenberg and Spruck [3], Krylov [15], Trudinger [21] as well as in [16], [22], [24], [4] and [7] etc. Other boundary value problems have also been considered. In [19], Trudinger treated the Dirichlet and Neumann problems in balls for the degenerate case, while Urbas [23] studied nonlinear oblique boundary value problems in two dimensions. Our motivation to study such equations on Riemannian manifolds comes from their close connection with problems in differential geometry, such as the Minkowski-Christoffel problems (see, for example, [5] and [18] for details). In [16], Y. Y. Li treated similar equations on compact manifolds (without boundary) of nonnegative sectional curvature. The Dirichlet problems for Monge-Ampère equations are treated by Guan and Spruck [10] on S and by Atallah and Zuily [1] and Guan and Li [9] (see also [8]) on general Riemannian manifolds. As in [3], we assume f is a smooth function defined in an open, convex, symmetric cone Γ ⊂ R with vertex at the origin, Γ + ⊆ Γ / = R where Γ + ≡ {λ ∈ R : each component λi > 0},