Calculus of Variations and Geometric Measure Theory

V. Agostiniani - R. Rossi - G. Savaré

On the transversality conditions and their genericity}

created by rossi on 17 Oct 2013


Submitted Paper

Inserted: 17 oct 2013
Last Updated: 17 oct 2013

Year: 2013


In this note we review some results on the \emph{transversality conditions} for a smooth Fredholm map $f: X \times (0,T) \to Y$ between two Banach spaces $X,Y$. These conditions are well-known in the realm of bifurcation theory and commonly accepted as ``generic''. Here we show that under the transversality assumptions the sections $C(t)=\{x:f(x,t)=0\}$ of the zero set of $f$ are discrete for every $t\in (0,T)$ and we discuss a somehow explicit family of perturbations of $f$ along which transversality holds up to a residual set.

The application of these results to the case when $f$ is the $X$-differential of a time-dependent energy functional $E:X\times (0,T)\to R$ and $C(t)$ is the set of the critical points of $E$ provides the motivation and the main example of this paper.