Neumann problem for Monge-Ampere type equations revisited.

Neil Trudinger,Feida Jiang

doi:10.53733/176

Abstract

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.

Highlights

IntroductionWe revisit the following Neumann problem for Monge-Ampere type equations, considered in [7]: det[D2u − A(x, u, Du)] = B(x, u, Du), in Ω,
In this paper, we revisit the following Neumann problem for Monge-Ampere type equations, considered in [7]: det[D2u − A(x, u, Du)] = B(x, u, Du), in Ω, (1.1)Dνu = φ(x, u), on ∂Ω, (1.2)where Ω ⊂ Rn, u is the unknown scalar function defined on Ω, A is a given n × n symmetric matrix function defined on Ω × R × Rn, B is a nonnegative scalar valued function on Ω × R × Rn, φ is a scalar valued function defined on ∂Ω × R, and ν denotes the unit inner normal vector field on ∂Ω
Where Ω ⊂ Rn, u is the unknown scalar function defined on Ω, A is a given n × n symmetric matrix function defined on Ω × R × Rn, B is a nonnegative scalar valued function on Ω × R × Rn, φ is a scalar valued function defined on ∂Ω × R, and ν denotes the unit inner normal vector field on ∂Ω

Summary

Introduction

We revisit the following Neumann problem for Monge-Ampere type equations, considered in [7]: det[D2u − A(x, u, Du)] = B(x, u, Du), in Ω,. As a consequence we extend the second derivative estimates in the degenerate case in [14] to embrace general regular matrix functions A. We extend the the strictly regular case in [7] to more general degenerate equations. A matrix A, which is twice differentiable with respect to p, is called regular, (strictly regular), 2010 Mathematics Subject Classification 35J66, 35J96. Key words and phrases: Neumann problem, Monge-Ampere type equation, second derivative estimates. A domain Ω is called uniformly A-convex with respect to u ∈ C1(Ω ), if ∂Ω ∈ C2 and n [Diνj(x) − Dpk Aij(x, u, Du)νk]τiτj ≤ −δ0|τ |2 i,j,k=1.

We have

Dξ wj ξ

If we assume just