SOME REMARKS ON THE VARIATIONAL...
Transcript of SOME REMARKS ON THE VARIATIONAL...
![Page 1: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/1.jpg)
FIRENZE 2014
Gabriele Bonanno University of Messina
InternationalWorkshop on
Topological and Variational Methods for ODEs Dedicated to Massimo Furi Professor Emeritus at the University of Florence
Florence, Italy June 3 - 4, 2014
A local minimum theorem and
nonlinear differential problems
![Page 2: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/2.jpg)
Our aim is to present a local minimum theorem for functionals of the type:
A LOCAL MINIMUM THEOREM
- Φ - Ψ
,
![Page 3: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/3.jpg)
A LOCAL MINIMUM THEOREM
(1)
(2)
![Page 4: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/4.jpg)
It is an existence theorem of a critical point for continuously Gâteaux differentiable functions, possibly unbounded from below.
The approach is based on Ekeland’s
Variational Principle applied to a non-smooth variational framework by using also a novel type of Palais-Smale condition which is more general than the classical one.
![Page 5: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/5.jpg)
Let X be a real Banach space, we say that a Gâteaux differentiable functional
I :X → R
verifies the Palais-Smale condition (in short (PS)-condition) if any sequence
{un} such that
PALAIS-SMALE CONDITION
![Page 6: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/6.jpg)
has a convergent subsequence.
Let X be a real Banach space and let Φ : X → R, Ψ : X → R two Gâteaux differentiable functions. Put I = Φ − Ψ.
![Page 7: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/7.jpg)
Fix r1, r2 ∈ [−∞;+∞], with r1 < r2, we say
that the function I verifies the Palais-
Smale condition cut off lower at r1 and
upper at r2 (in short -condition) if any sequence {un} such that
![Page 8: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/8.jpg)
has a convergent subsequence.
Clearly, if r1 = −∞ and r2 = +∞ it coincides with the classical (PS)-condition. Moreover, if r1 = −∞ and r2 ∈ R we denote it by , while if r1 ∈ R and r2 = +∞ we denote it by .
![Page 9: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/9.jpg)
In particular,
![Page 10: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/10.jpg)
![Page 11: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/11.jpg)
To prove the local minimum theorem we use the theory for locally Lipschitz functionals investigated by K.C. Chang, which is based on the Nonsmooth Analysis by F.H. Clarke, and generalizes the study on the variational inequalities as given by A. Szulkin .
![Page 12: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/12.jpg)
This theory is applied to study variational and variational-hemivariational inequalities . In particular, for instance, differential inclusions and equations with discontinuous nonlinearities are investigated.
![Page 13: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/13.jpg)
Here, by using the nonsmooth theory we obtain results for smooth functions.
THE EKELAND VARIATIONAL PRINCIPLE
Arguing in a classical way of the smooth analysis (as, for instance, Ghossoub),
but using the definitions and properties of the non-smooth analysis (as, for
instance, Motreanu-Radulescu, the following consequence of the Ekeland variational
Principle can be obtained.
![Page 14: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/14.jpg)
A CONSEQUENCE OF THE EKELAND VARIATIONAL PRINCIPLE
IN THE NONSMOOTH ANALYSIS FRAMEWORK
![Page 15: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/15.jpg)
Sketch of Proof. Put
Clearly, J is locally Lipschitz and bounded from below. Hence, Lemma and a suitable computation ensure the conclusion.
![Page 16: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/16.jpg)
A LOCAL MINIMUM THEOREM
(1)
(2)
![Page 17: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/17.jpg)
gabriele bonanno
Three consequences
![Page 18: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/18.jpg)
gabriele bonanno
Three consequences
![Page 19: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/19.jpg)
gabriele bonanno
Three consequences
![Page 20: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/20.jpg)
gabriele bonanno
• Consider
Some examples on third consequence of LOCAL MINIMUM THEOREM
![Page 21: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/21.jpg)
gabriele bonanno
![Page 22: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/22.jpg)
gabriele bonanno
![Page 23: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/23.jpg)
gabriele bonanno
![Page 24: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/24.jpg)
gabriele bonanno
We have the same situation for other ordinary nonlinear differential problems
and for elliptic problems involving the p-laplacian with p>n.
However, the same type of low can be obtained also for p=2 .
![Page 25: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/25.jpg)
gabriele bonanno
![Page 26: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/26.jpg)
gabriele bonanno
![Page 27: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/27.jpg)
gabriele bonanno
PHD thesis of Sciammetta:
PHD thesis of Pizzimenti:
![Page 28: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/28.jpg)
gabriele bonanno
![Page 29: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/29.jpg)
gabriele bonanno
![Page 30: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/30.jpg)
gabriele bonanno
AMBROSETTI-BREZIS-CERAMI
![Page 31: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/31.jpg)
gabriele bonanno
AMBROSETTI-BREZIS-CERAMI
![Page 32: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/32.jpg)
gabriele bonanno
• Owing to the local minimum theorem we obtain the first solution directly as a minimum . Moreover, a precise estimate of parameters is given.
![Page 33: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/33.jpg)
gabriele bonanno
![Page 34: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/34.jpg)
gabriele bonanno
![Page 35: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/35.jpg)
gabriele bonanno
![Page 36: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/36.jpg)
gabriele bonanno
![Page 37: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/37.jpg)
gabriele bonanno
![Page 38: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/38.jpg)
gabriele bonanno
![Page 39: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/39.jpg)
gabriele bonanno
![Page 40: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/40.jpg)
gabriele bonanno
![Page 41: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/41.jpg)
WALKING IN THE
MOUNTAINS
![Page 42: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/42.jpg)
Some remarks on the classical Ambrosetti-Rabinowitz theorem are presented. In particular, it is observed that the geometry of the mountain pass, if the function is bounded from below, is equivalent to the existence of at least two local minima, while, when the function is unbounded from below, it is equivalent to the existence of at least one local minimum.
,
![Page 43: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/43.jpg)
So, the Ambrosetti-Rabinowitz theorem actually ensures three or two distinct critical points, according to the function is bounded from below or not.
![Page 44: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/44.jpg)
HISTORICAL NOTES
![Page 45: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/45.jpg)
THE AMBROSETTI-RABINOWITZ THEOREM
![Page 46: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/46.jpg)
THE PUCCI-SERRIN THEOREM
![Page 47: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/47.jpg)
THE GHOUSSOUB-PREISS THEOREM
![Page 48: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/48.jpg)
A REMARK
![Page 49: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/49.jpg)
In fact, in this case the mountain pass geometry implies the existence of two local minima and the Pucci-Serrin theorem ensures the third critical point.
So, the Ambrosetti-Rabinowitz theorem, when the function is bounded from below actually ensures three distinct critical points.
![Page 50: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/50.jpg)
In this case, the following condition is requested:
The function I is bounded from below
on every bounded set of X.
In a similar way it is possible to see that, when the function is unbounded from below, the mountain pass geometry is equivalent to the existence of at least one local minimum.
![Page 51: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/51.jpg)
gabriele bonanno
REMARK
![Page 52: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/52.jpg)
gabriele bonanno
Three consequences
![Page 53: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/53.jpg)
gabriele bonanno
Three consequences
![Page 54: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/54.jpg)
A THREE CRITICAL POINTS THEOREM
From the preceding two variants
of the local minimum theorem, a three critical points theorem is obtained. Here a special case is pointed out.
![Page 55: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/55.jpg)
(3)
A THREE CRITICAL POINTS THEOREM
![Page 56: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/56.jpg)
NONLINEAR DIFFERENTIAL PROBLEMS
Consider the following two point boundary value problem
A TWO-POINT BOUNDARY VALUE PROBLEM
where f : R → R is a continuous function and is λ a positive real parameter.
![Page 57: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/57.jpg)
Moreover, put
for all ξ ∈ R and assume, for clarity, that f
is nonnegative.
![Page 58: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/58.jpg)
(1)
(2)
![Page 59: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/59.jpg)
TWO-POINT BOUNDARY VALUE PROBLEMS
NEUMANN BOUNDARY VALUE PROBLEMS
MIXED BOUNDARY VALUE PROBLEMS
STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS
HAMILTONIAN SYSTEMS
FOURTH-ORDER ELASTIC BEAM EQUATIONS
![Page 60: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/60.jpg)
ELLIPTIC DIRICHLET PROBLEMS INVOLVING THE P-LAPLACIAN WITH P>N
NONLINEAR DIFFERENCE PROBLEMS
BOUNDARY VALUE PROBLEMS ON THE HALF_LINE
ELLIPTIC NEUMANN PROBLEMS INVOLVING THE P-LAPLACIAN WITH P>N
ELLIPTIC SYSTEMS
![Page 61: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/61.jpg)
NONLINEAR EIGENVALUE PROBLEMS IN ORLICS-SOBOLEV SPACES
NONLINEAR ELLIPTIC PROBLEMS ON THE SIERPI NSKI GASKET
GENERALIZED YAMABE EQUATIONS ON RIEMANNIAN MANIFOLDS
ELLIPTIC PROBLEMS INVOLVING THE p(x)-LAPLACIAN
![Page 62: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/62.jpg)
ELLIPTIC PROBLEMS INVOLVING THE p-LAPLACIAN WITH p≤N
(1)
![Page 63: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/63.jpg)
…
![Page 64: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/64.jpg)
…and others
![Page 65: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/65.jpg)
1
A TWO CRITICAL POINTS THEOREM
![Page 66: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/66.jpg)
…and others
2
A TWO CRITICAL POINTS THEOREM
![Page 67: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/67.jpg)
…and others
3
A TWO CRITICAL POINTS THEOREM
![Page 68: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/68.jpg)
![Page 69: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/69.jpg)
(A-R)
![Page 70: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/70.jpg)
(1)
![Page 71: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/71.jpg)
WALKING IN THE
MOUNTAINS
![Page 72: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/72.jpg)
1. Mountain pass geometry 2.Palais-Smale condition Then, there is a critical point.
![Page 73: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/73.jpg)
1. Strong mountain pass geometry 2. Weak Palais-Smale condition Then, there is a critical point near to local minimum.
![Page 74: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/74.jpg)
If we apply two times the first special case of the local minimum theorem and owing to a novel version of the mountain pass theorem where the (PS) cut off upper at r is assumed we can give a variant of the three critical theorem. In the applications it became
A VARIANT OF THREE CRITICAL POINTS THEOREM FOR FUNCTIONS UNBOUNDED FROM BELOW
![Page 75: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/75.jpg)
(1)
(2)
![Page 76: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/76.jpg)
FURTHER APPLICATIONS OF THE LOCAL MINIMUM THEOREM
![Page 77: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/77.jpg)
If we apply iteratively the first special
case of the local minimum theorem in a suitable way, we obtain an infinitely many critical points theorem. As an example of application, here, we present the following result.
INFINITELY MANY CRITICAL POINTS THEOREM
![Page 78: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/78.jpg)
(1)
![Page 79: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/79.jpg)
Previous results can be applied to perturbed problems, as, for instance, the following
PERTURBED PROBLEMS
or in the framework of the non-smooth Analysis. As example, here, the following problem is considered.
![Page 80: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/80.jpg)
VARIATIONAL-HEMIVARIATIONAL INEQUALITIES
![Page 81: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/81.jpg)
![Page 82: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/82.jpg)
A COINCIDENCE POINT THEOREM
Arguing as ARINO-GAUTIER-PENOT and by starting from Ky FAN theorem, the following fixed point theorem for multifunctions with weakly sequentialy closed graph has been established:
![Page 83: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/83.jpg)
Recently, as a consequence, the following coincidence point theorem has been established:
Taking into account that a critical point of Φ-Ψ is
a coincidence point for Φ’ and Ψ’, one has the
following critical point theorem
1
2
![Page 84: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/84.jpg)
(1)
![Page 85: SOME REMARKS ON THE VARIATIONAL METHODSweb.math.unifi.it/convegnofuri2014/slides/SeminarioBonan...THE EKELAND VARIATIONAL PRINCIPLE Arguing in a classical way of the smooth analysis](https://reader034.fdocuments.us/reader034/viewer/2022050401/5f7eb03de17c643b237aa02a/html5/thumbnails/85.jpg)