A minimax argument to a stronger version of the Jacobian conjecture

TL;DR

This paper proves the strong real Jacobian conjecture for symmetric cases using a minimax approach, establishing injectivity under specific spectral conditions on the Jacobian and its symmetric part.

Contribution

It introduces a minimax method to prove the strong real Jacobian conjecture for symmetric maps under spectral assumptions, linking it to the classical Jacobian conjecture.

Findings

Proves injectivity of certain $C^1$ maps under spectral conditions.

Establishes a link between the strong real Jacobian conjecture and the classical Jacobian conjecture.

Uses a minimax argument to achieve the main result.

Abstract

The main result of this paper is to prove the strong real Jacobian conjecture under the symmetric assumption and reveals the link between it and the Jacobian conjecture. Precisely, we assume that $F: \mathbb{R}^n \to \mathbb{R}^n$ is of $C^1$ map, $n\geqslant 2$, if for some $\varepsilon >0$, $ 0\notin Spec(F)~~\mbox{and}~ Spec(F+F^T) \subseteq (-\infty,-\varepsilon)~\mbox{or} ~(\varepsilon,+\infty),$ where $Spec (F)$ denotes all eigenvalues of $JF$ and $Spec (F+F^T)$ denotes all eigenvalues of $JF+JF^T$, then we show that $F$ is injective. It is proved by using a minimax argument.