5
$\begingroup$

To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is Condition C of Palais and Smale (note that this is automatically satisfied for $M$ compact). It ultimately implies (for instance) the Deformation Lemma and the Mini-Max Principle.

But what if Condition C fails? I'm interested in to what extent I lose control over the function. For instance, I am aware of Uhlenbeck's "perturbation method", where if $f$ doesn't satisfy Condition C then we can look at $f_\varepsilon=f+\varepsilon\cdot g$ (fix function $g$) that satisfies Condition C and try to get critical points of $f$ as limits of those of $f_\varepsilon$.
When Condition C fails, is there a work-around to save the Mini-Max Principle?

Improve this question
$\endgroup$
1
  • $\begingroup$ I don't think you'll get far unless you're at the borderline where Condition C (and compactness) just barely fails. If so, Uhlenbeck and others have shown in certain particular situations (minimal hypersurfaces and self-dual Yang-Mills connections are the most notable examples), there is a global minimum but it lives outside the original Hilbert manifold you started with. I have no idea what could happen at a saddle point. $\endgroup$
    – Deane Yang
    Commented Nov 14, 2013 at 2:27

1 Answer 1

Reset to default
2
$\begingroup$

There are some works on this, but maybe THE expert on this topic is Abbas Bahri. Take a look at his works or google "critical point at infinity".

See, for example: this paper or his book ""Critical Points at Infinity in Some Variational Problem".

Improve this answer
$\endgroup$
3
  • $\begingroup$ I see this is a way to get back the deformation lemma, in particular for studying the flow of a functional. Does this also help us with the critical point structure of the function itself? $\endgroup$
    – Chris Gerig
    Commented Oct 27, 2013 at 23:01
  • $\begingroup$ I think you must make your question more precise: are you trying to find some general condition that will allow you to generalize Palis-Smale (unlikely) or do you have a concrete problem in mind and are wondering what sort of technique or trick will allow you to work around it ? $\endgroup$
    – alvarezpaiva
    Commented Oct 28, 2013 at 13:50
  • $\begingroup$ I think what alvarezpaiva is trying to say is that it's impossible to say much unless you describe a rather specific concrete situation in which you are trying to use the minimax principle. $\endgroup$
    – Deane Yang
    Commented Nov 14, 2013 at 2:23

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .