Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale Condition

Canadian mathematical bulletin

关键词: $p$-Laplacian;    locally Lipschitz potential;    nonsmooth critical point theory;    principal eigenvalue;    positive solutions;    nonsmooth Mountain Pass Theorem;   

DOI  :  10.4153/CMB-2007-036-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We study two sufficient conditions that imply global injectivityfor a $C^1$ map $Xcolon R^2o R^2$ such that its Jacobian at anypoint of $R^2$ is not zero. One is based on the notion ofhalf-Reeb component and the other on the Palais--Smale condition.We improve the first condition using the notion of inseparableleaves. We provide a new proof of the sufficiency of the secondcondition. We prove that both conditions are not equivalent, moreprecisely we show that the Palais--Smale condition implies thenonexistence of inseparable leaves, but the converse is not true.Finally, we show that the Palais--Smale condition it is not anecessary condition for the global injectivity of the map $X$.

【 授权许可】

Unknown