A class of fully nonlinear equations

Abstract

In this paper, we establish a priori estimates for a class of fully nonlinear equations with Neumann boundary conditions. By the continuity method, we have obtained the existence theorem for the Neumann problem.Mathematical Subject Classification (2010): Primary 35J60, Secondary 35B45.

$$C^{1,1}$$ regularity of geodesics in the space of volume forms

“…We use the same notations as in [3]. For r = (r 0 , r 1 , · · · , r n+1 ), we write Q(r) = r 0 r 1 − n+1 i=2 r 2 i and G(r) = log Q(r).…”

“…Applying ∇ e 1 ∇ e 1 to the equation G(r) = log f (the logarithm of (2.1)) and using the concavity of G (see [7,2,3]), we see that…”

“…Let (M, g) be a Riemannian manifold of real dimension n. We use ∇ to denote the Levi-Civita connection. Recently, Chen-He [3] introduced the following function spacẽ…”

“…For any u 0 , u 1 ∈H, they also introduced the fully nonlinear equation where f is a nonnegative function on M × [0, 1]. In [3], Chen-He solved the equation (1.1) with uniform weak C 2 estimates, which also hold for the degenerate case (see also [9]). When b = 0, a = 1 and f = 0, (1.1) becomes the geodesic equation in the space of volume forms on (M, g).…”