On the Second Boundary Value Problem for a Class of Modified-Hessian Equations

TL;DR

This paper introduces a new class of modified-Hessian equations related to optimal transportation, proves the existence of smooth solutions satisfying the second boundary value problem, and extends key a priori estimates without relying on duality.

Contribution

It develops a novel class of modified-Hessian equations and establishes existence results with generalized a priori estimates, advancing the understanding of boundary value problems in this context.

Findings

Existence of globally smooth solutions for the new equations.

Generalization of a priori estimates from Trudinger and Wang.

Global C^2 estimate achieved without duality assumptions.

Abstract

In this paper a new class of modified-Hessian equations, closely related to the Optimal Transportation Equation, will be introduced and studied. In particular, the existence of globally smooth, classical solutions of these equations satisfying the second boundary value problem will be proven. This proof follows a standard method of continuity argument, which subsequently requires various a priori estimates to be made on classical solutions. These estimates are modifications and generalise the corresponding estimates of Trudinger and Wang for the Optimal Transportation Equation. Of particular note is the fact that the global C^2 estimate contained in this paper makes no use of duality in regards to the original equation.