Elliptic solutions to nonsymmetric Monge-Amp\`{e}re type equations I. The $d$-concavity and the comparison principle

TL;DR

This paper introduces the concept of $d$-concavity for nonsymmetric Monge-Ampère type functions, proving their concavity and establishing a comparison principle for $ ext{delta}$-elliptic equations, advancing understanding of their solution behavior.

Contribution

It introduces $d$-concavity and proves the concavity of nonsymmetric Monge-Ampère functions, along with a comparison principle for $ ext{delta}$-elliptic equations, providing new tools for analysis.

Findings

Proved concavity of nonsymmetric Monge-Ampère functions in certain sets.

Established a comparison principle for $ ext{delta}$-elliptic equations.

Introduced the concept of $d$-concavity for matrix functions.

Abstract

We introduce the so-called $d$-concavity, $d \geq 0,$ and prove that the nonsymmetric Monge-Amp\`{e}re type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for nonsymmetric Monge-Amp\`{e}re type equations in the case when they are so-called $\delta$-elliptic with respect to compared functions with $0 \leq \delta <1$.