The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of certain kinds of critical points, in particular saddle points. The Palais-Smale condition is a condition on the functional that one is trying to extremize.
In finite-dimensional spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically for proper maps: functions which do not take unbounded sets into bounded sets. In the calculus of variations, where one is typically interested in infinite-dimensional function spaces, the condition is necessary because some extra notion of compactness beyond simple boundedness is needed. See, for example, the proof of the mountain pass theorem in section 8.5 of Evans.
Strong formulation
A continuously Fréchet differentiable functional
from a Hilbert space H to the reals satisfies the Palais–Smale condition if every sequence
such that:
is bounded, and
in H
has a convergent subsequence in H.
Weak formulation
Let X be a Banach space and
be a Gateaux differentiable functional. The functional
is said to satisfy the weak Palais–Smale condition if for each sequence
such that
,
in
,
for all
,
there exists a critical point
of
with
![{\displaystyle \liminf \Phi (x_{n})\leq \Phi ({\overline {x}})\leq \limsup \Phi (x_{n}).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
References
- Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2.
- Mawhin, Jean; Willem, Michel (2010). "Origin and Evolution of the Palais–Smale Condition in Critical Point Theory". Journal of Fixed Point Theory and Applications. 7 (2): 265–290. doi:10.1007/s11784-010-0019-7. S2CID 122094186.
- Palais, R. S.; Smale, S. (1964). "A generalized Morse theory". Bulletin of the American Mathematical Society. 70: 165–172. doi:10.1090/S0002-9904-1964-11062-4.