The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds

Abstract

We study the Dirichlet problem for a broad class of fully nonlinear elliptic equations in Euclidean space as well as on general Riemannian manifolds. Under a set of fundamental structure conditions which have become standard since the pioneering work of Cafferalli, Nirenberg and Spruck, we prove that the Dirichlet problem admits a (unique) smooth solution provided that there exists a subsolution. The conditions are essentially optimal, especially with no geometric restrictions to the boundary of the underlying manifold, which is important in applications. We search new ideas and techniques to make use of the concavity condition and subsolution in order to overcome difficulties in deriving key a priori estimates. Along the way we discover some interesting properties of concave functions which should be useful in other fields. Our methods can be adopted to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds. We shall also solve some new equations, which were not covered by previous results even in Rn, with interesting properties.