A note on the Monge-Amp\`ere type equations with general source terms

TL;DR

This paper studies generalized solutions to Monge-Ampère type equations with arbitrary source terms, establishing comparison principles and boundary behavior, and introduces finite element methods for their numerical approximation.

Contribution

It proves comparison principles and boundary properties for generalized solutions, and develops well-posed finite element methods for these equations.

Findings

Established comparison principle for generalized solutions

Analyzed boundary behavior of solutions

Designed finite element methods for practical computation

Abstract

In this paper we consider the generalised solutions to the Monge-Amp{\`{e}}re type equations with general source terms. We firstly prove the so-called comparison principle and then give some important propositions for the border of generalised solutions. Furthermore, we design well-posed finite element methods for the generalised solutions with the classical and weak Dirichlet boundary conditions respectively.