The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds

Abstract

We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and optimal) result in the Euclidean case. We introduce some new ideas and methods in deriving a priori estimates, which can be used to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds.Mathematical Subject Classification (2010): 35J15, 58J05, 35B45. Keywords: Fully nonlinear elliptic equations on Riemannian manifolds, Dirichlet problem, a priori estimates, concavity, subsolutions.

The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

“…, f n when there is no possible confusion. We note that {U ij } and {F ij } can be diagonalized simultaneously and that [7]). We need the following lemma which is Lemma 2.1 in [7].…”

“…As in [7], the existence of u is useful to construct some barrier functions which are crucial to our estimates.…”

“…A generalization of (1.9) was considered in [17]. Guan, Shi and Sui [12] extended the work of [18] using the idea of [7]. As we know, works about the elliptic Hessian equations usually provide useful techniques to deal with the parabolic version.…”

“…As we know, works about the elliptic Hessian equations usually provide useful techniques to deal with the parabolic version. The reader is referred to [21,30,8,9,7,10,11] and their references for examples. The motivation to assume that f is defined in the cone Γ is due to the consideration that many equations are elliptic (or parabolic) with respect to solutions in a cone but they are not in general (see the examples above).…”

“…(1.5) In this work we are interested in the existence of classical solutions to (1.1)- (1.2). Recent research on the Hessian equations of elliptic type (see [9,7]):…”