Local Estimates for Some Fully Nonlinear Elliptic Equations

We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from C0 estimates. Also, the method is flexible and can be applied to a large class of equations. Let (M, g) be a smooth Riemannian manifold of dimension n ≥ 2. We are interested in a priori estimates for solutions of some classes of fully nonlinear elliptic equations on (M, g). These kinds of equations arise naturally from geometry and other fields of analysis and share structures similar to those of the Monge-Ampere equations. Regularity problems are studied by people in different fields separately. One would like to ask if it is possible to give a unified proof and to generalize further to a large class of equations. The answer is affirmative provided the equations satisfy some algebraic structures which can induce the cancellation phenomenon. We will see how this phenomenon helps us to get the Hessian bound directly. One of the interesting cases is the Schouten tensor equation arising from conformal geometry: σ 1 k k (g −1(∇2u+ du⊗ du− 1 2 |∇u|g + Ag)) = f(x) e−2u where σk is the kth elementary symmetric function. Local C 2 estimates are proved for this equation by Chang, Gursky, and Yang [2] (k = 2, n = 4) and by P.Guan and G.Wang [7] for all k ≤ n. The same results with specific dependence on the radius of the domain are established by Gursky and Viaclovsky [10]. P.Guan and G.Wang [8] also prove the local estimates for quotients of the elementary symmetric functions. Other related works in this direction include [12], [11] and [14]. Another interesting case is the following equation in optics geometry: det(g−1 c (∇v − |∇v|2 2v gc + v 2 gc)) = ( |∇v|2 + v 2v )n ν(x)φ(S(x, v,∇v)).