In this note we review some results on the transversality conditions for a smooth Fredholm map f : X x (0, T) -> 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 is an element of(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 x (0, T) -> R and C(t) is the set of the critical points of E provides the motivation and the main example of this paper.
Titolo: | On the transversality conditions and their genericity |
Autori: | Agostiniani V; Rossi R; Savaré G |
Rivista: | |
Data di pubblicazione: | 2015 |
Volume: | 64 |
Fascicolo: | 1 |
Pagina iniziale: | 101 |
Pagina finale: | 116 |
Digital Object Identifier (DOI): | http://dx.doi.org/10.1007/s12215-014-0184-4 |
URL: | https://arxiv.org/abs/1310.4846 |
Appare nelle tipologie: | 1.1 Journal article |