On Neumann problem for the degenerate Monge–Ampère type equations

Abstract

In this paper, we study the global $C^{1, 1}$ C 1 , 1 regularity for viscosity solution of the degenerate Monge–Ampère type equation $\det [D^{2}u-A(x, Du)]=B(x, u, Du)$ det [ D 2 u − A ( x , D u ) ] = B ( x , u , D u ) with the Neumann boundary value condition $D_{\nu }u=\varphi (x)$ D ν u = φ ( x ) , where the matrix A is under the regular condition and some structure conditions, and the right-hand term B is nonnegative.

Neumann problem for Monge-Ampere type equations revisited.

“…We can then formulate a global second order derivative estimate, (independent of inf B), for problem (1.1)-(1.2) under the conditions (1.11), (1.12), (1.13), which extends the corresponding result in [14] to general regular matrix functions A satisfying the same hypotheses as in Theorem 1.1. We also include the corresponding estimate for strictly regular matrix functions A, without the barrier conditions (1.8), (1.9) and condition (1.13).…”

“…for any unit tangential vector field τ , where the constant C depends on supΩ |Du| and Ω, see [7,11,14]. We shall deduce the estimate for D νν u on ∂Ω.…”

“…In this paper, we complete the proof of second derivative estimates in [7] through upgrading the auxiliary function used there. As a consequence we extend the second derivative estimates in the degenerate case in [14] to embrace general regular matrix functions A. We also extend the the strictly regular case in [7] to more general degenerate equations.…”

“…Note that in Theorems 1.1 and 1.2, we do not assume monotonicity conditions of A, B and ϕ with respect to u, although these are critical for the construction of barriers Φ in general domains as in [7]. Some historical results for the Neumann problem for equations of Monge-Ampère type can be found in [4,7,8,10,11,13,14]. The results of this paper, along with their application to full second derivative estimates and existence theorems as in [7], may also be extended to parabolic Monge-Ampère type equations, extending [13], as well as the corresponding problems on Riemannian manifolds, extending [4].…”

“…In Section 2, we prove the second order derivative estimates, Theorem 1.1, on uniformly A-convex domains. In Section 3, the second order derivative estimates in Theorem 1.2 are treated by combining the techniques in Section 2 and [7,14].…”