On the Neumann Problem for Monge-Ampére Type Equations

Abstract

Abstract. In this paper, we study the global regularity for regular Monge-Ampère type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of the Neumann boundary value problem is proved under natural conditions. e techniques build upon the delicate and intricate treatment of the standard Monge-Ampère case by Lions, Trudinger and Urbas in and the recent barrier constructions and second derivative bounds by Jiang, Trudinger and Yang for the Dirichlet problem. We also consider more general oblique boundary value problems in the strictly regular case.

Oblique boundary value problems for augmented Hessian equations I

“…This includes the case when A and G are independent of z as then the interval I becomes irrelevant. As in [16], we will assume that the function 17) where β = G p and ϕ are defined on ∂ × R. If G pp (·, u, Du) ≤ 0 on ∂ for u ∈ C 1 (¯ ) then we say that G is concave in p, with respect to u. Note that we define the obliqueness in (1.3) with respect to the unit inner normal ν, so that our function G keeps the same sign with those in [16] and is the negative of that in [41,45,48].…”

“…Analogously to the situation with uniformly elliptic equations, we obtain gradient estimates in terms of moduli of continuity when the "o" is weakened to "O" in the hypotheses, (1.22) and case (ii), of Theorem 1.3. In particular we will also prove a Hölder estimate for admissible functions in the cones k for k > n/2, when A ≥ O(| p| 2 )I , which extends our gradient estimate in the case k = n in [16], Lemma 4.1. Taking account of this, as well as Theorems 1.2 and 1.3, we have, as an example of our consequent existence results, the following existence theorem for the augmented k-Hessian and Hessian quotient equations.…”

“…Moreover in the Monge-Ampère case, k = n, we recover our definitions of uniform (A, G)-convexity in [16], (called "uniform A-convexity with respect to G" there), which extend the notion of uniform c-convexity with respect to a target domain * in the optimal transportation case, as introduced in [45]. When the interval I = R, we will simply call ∂ uniformly ( , A, G)-convex.…”

“…First we will consider the nontangential estimates and as in [16], the geometric convexity hypotheses on the domain in Theorem 1.2 are crucial for this stage. We assume that the functions G(·, z, p) and ν have been extended to¯ , to be constant along normals to ∂ in some neighbourhood N of ∂ .…”

“…Extensions to the Dirichlet problem for our general class of equations are treated in [14]. Overall this paper provides a comprehensive framework for studying oblique boundary value problems for a large class of fully nonlinear equations, which embraces the Monge-Ampère type case in Section 4 in [16] as a special example.…”