Chang Palais-Smale condition and global inversion

TL;DR

This paper establishes that a locally Lipschitz functional satisfying the Chang Palais-Smale condition implies a local diffeomorphism is a global diffeomorphism, linking variational conditions to global invertibility in Banach spaces.

Contribution

It introduces a new criterion based on the Chang Palais-Smale condition for global invertibility of local diffeomorphisms between Banach spaces.

Findings

Functional F(x)=1/2|f(x)-y|^2 satisfies Chang Palais-Smale condition implies f is a global diffeomorphism.

Extension to weighted Chang Palais-Smale condition.

Relationship to classical global inversion conditions.

Abstract

Let f be local diffeomorphism between real Banach spaces. We prove that if the locally Lipschitz functional F(x)=1/2|f(x)-y|^2 satisfies the Chang Palais-Smale condition for all y in the target space of f, then f is a norm-coercive global diffeomorphism. We also give a version of this fact for a weighted Chang Palais-Smale condition. Finally, we study the relationship of this criterion to some classical global inversion conditions.