Dipartimento Di Matematica
Corso di Laurea Magistrale in Matematica
Γ-Convergence
and Critical Points
Toward a general theory
Candidato: Pierluigi Puce
Relatore: Prof. Luigi De Pascale Corelatore: Prof. Giuseppe Buttazzo Controrelatore: Prof. Giovanni Alberti
Introduction vii
1 Preliminaries 1
1.1 The Direct Method of the Calculus of Variations . . . 1
1.2 Γ-convergence . . . 6
1.3 Minimax Methods . . . 18
2 Spectral Theory 25 2.1 The p-Laplacian . . . 25
2.2 Index Theory . . . 27
2.3 Building sequences of eigenvalues . . . 36
2.4 Hausdorff distance . . . 43
3 Convergence of Eigenvalues 49 3.1 A result of Γ-Convergence . . . 49
3.2 Asymptotic eigenvalues . . . 52
3.3 Some remarks and open problems . . . 56
Bibliography 59
The purpose of this thesis is to investigate the convergence of critical points of certain sequences of functionals. These critical points are solutions of a class of quasi-linear boundary value problems and are obtained by means of topolog-ical and variational methods of the theory of partial differential equations. In particular, minimax methods and topological index theory are used to define critical points, while the theory of Γ-convergence is employed to obtain their convergence.
The first chapter collects some preliminary results. We present the main ideas regarding the direct method of the calculus of variations, which can be seen as a classical starting point for our work, then we define Γ-convergence and show the fundamental result about the convergence of minima of Γ-converging functionals. The chapter ends with a description of some variational methods and tools, such as Palais-Smale condition, the deformation lemma and mini-max methods.
In the second chapter we start an investigation of the problem of eigenvalues for the p-Laplacian, and we define the tools that we use in the rest of the work: we present index theory and define the Krasnoselskii genus and the Z2
-cohomological index, which are used to define several sequences of eigenvalues for the p-Laplacian as minimax critical values. In the last section, Hausdorff distance on compact sets is investigated.
In the third chapter we develop the main ideas of the work. Given a functional f on a Banach space V we define a new functional in the space of compact subsets of V as F(K) = supKf and we show that under proper hypothesis
if a sequence of functionals fk Γ-converges to f0 then the related sequence
of functionals Fk Γ-converges to F0 with respect to the Hausdorff distance.
As a corollary, we obtain the convergence of sequences of eigenvalues for the p-Laplacian when p → ∞ as a convergence of minima in the framework of variational convergence of functionals.
Variational problems date back to the XVIII century, when Lagrange and Eu-ler, in their study about the tautochrone problem, developed the so called Euler-Lagrange equations, and led to the birth of the mathematical field of the calculus of variations.
A variational problem is a boundary value problem that because of its struc-ture can be studied with the methods of the calculus of variations. To be a bit more specific, given a certain functional equation F (u) = 0 for u belonging to some Banach space V (usually Sobolev spaces), if there exists an energy functional E on V which is Fréchet differentiable with derivative DE and for which F (u) = 0 if and only if DE(u) = 0 we call such equation of variational form. Therefore, solving the equation F (u) = 0 is the same as minimizing the functional E on V .
A quite natural way to deal with this kind of problems is the so called direct method of the calculus of variations, which relies on compactness arguments and originates from the Weierstrass theorem. This method exploits natural conditions on the considered functional such as coerciveness and lower semi-continuity, to ensure the existence of a limit point for a minimizing sequence, i.e. a minimum point. Its development has been the result of the effort of numerous mathematicians of the early XX century, among them B. Levi, R. Courant, S. Zaremba and D. Hilbert.
The version of the direct method that we prove in the first chapter, following the exposition of G. Dal Maso in ([Dal13]), can be summarized as follows: given a topological space X, and F : X → R
F coercive and lower semicontinuous ⇒ F has a minimum point. In many applications it is necessary to study a family of variational problems depending on a certain parameter (imagine a discretization argument, or an approximation process) hence, for instance, one has to deal with a family of problems such
min
u∈V{Fε(u) : ε > 0}
and wants to study its asymptotic behaviour as ε → 0. This because the limit functional may capture the relevant features of the problem and its minimum may be more easily found.
Γ-convergence, since its introduction in the 1970s by E. De Giorgi and T. Franzoni in [DF75] has become a fundamental tool in the investigation of the asymptotic behaviour of minimum problems. A family of functionals (Fε)ε
defined on a metric space V with values in R and ε > 0 Γ-converges to F0 :
V → R as ε → 0 when the following two conditions are fullfilled: 1. For every sequence (xε) ⊂ V such that xε→ x0 ∈ V
lim inf
ε→0 Fε(xε) ≥ F0(x0).
2. For every x0 ∈ V there exists a sequence (xε)ε ⊂ V such that xε → x0
and
lim sup
ε→0
Fε(xε) ≤ F0(x0).
The first request consists in a lower bound for the sequence (Fε)ε, while the
second is a way to sharp such a bound. An important feature of Γ-convergence is that it allows, under additional compactness conditions on the Γ-converging sequence, to obtain converging sequences (or subsequences) of minimizers to a global minimizer of the limit functional. The compactness requested is the equi-coerciveness of the sequence, which is realized when there exists a non-empty compact set K ⊂ V such that for every ε > 0 infV Fε= infKFε. Hence
roughly speaking the convergence of absolute minima can be summerized as Equicoerciveness + Γ-convergence ⇒ Convergence of absolute minima. In this work we use the theory of Γ-convergence to study the convergence of the eigenvalues of the non-linear problem
(
−∆pu = λ|u|p−2u in Ω
u = 0 on ∂Ω (0.0.1)
where Ω is a bounded regular open set of RN (N ≥ 1), p > 2 and ∆
p is the
p-Laplacian operator
∆pu = div(|∇u|p−2∇u), p ∈ (1, ∞),
which arises in several fields of applications, such as turbolent filtration in porous media, glaceology, non-Newtonian fluid flows, plasticity theory.
In a variational framework, eigenvalues of (0.0.1) are the critical values of the C1-functional Fp(u) =
R
Ω|∇u|
p on S = {u ∈ W1,p
0 (Ω) : kukLp = 1}.
Except for the smallest eigenvalue, which is computed by standard methods using the fact that F satisfies a compactness condition on S (Palais-Smale con-dition), other eigenvalues need different tools in order to be found, specifically
Lusternick-Schnirelmann theory and minimax methods.
Denote by A the class of closed symmetric subsets of S and by γ(A) = inf{k ≥ 1 : ∃an odd continuous map A → Sk−1} the genus of A ∈ A, where Sk−1 is the unit sphere of Rk, then
λk(A) = inf A∈A γ(A)≥k sup u∈A Fp(u)
is an increasing and unbounded sequence of eigenvalues.
The genus γ is an index, and other sequences of eigenvalues can be built by means of similar techniques, but, in general, it is not known whether either sequence is a complete list or if they are actually a unique one.
The main result of the work relates the theory of Γ-convergence to the conver-gence of sequences of eigenvalues found as minimax critical values. Following the ideas of T. Champion and L. De Pascale ([CD07]), we define, given Fp as
above, a new functional defined on A as Fp(A) = sup
u∈A
Fp(u),
therefore, minimax critical values become minima for F, and we can use the techniques of Γ-convergence to actually show that for every k as p → ∞ the sequence λk
p converges to
λk∞= inf{sup
u∈G
k∇uk∞ : G ∈ Ks(Ω), G ⊂ W01,∞(Ω) ∩ {kvk∞= 1}, i(G) ≥ K},
where i is a topological index such as the genus.
In particular, in Theorem 3.1 we show that under some additional conditions the Γ-convergence of the functionals Fp(u) implies that also the family Fp
converges in the same sense in the space Ksof compact and symmetric subsets
of S equipped with the Hausdorff distance, and this implies, as a corollary, the convergence of the sequences of eigenvalues.
Background material
When one writes a master thesis, it is necessary to suppose the reader to be already acquainted with some arguments, that will represent the starting point of the work.
Here, the topics that will be assumed as known are definitions and basic facts about partial differential equations, boundary value problems, calculus of vari-ations, Sobolev spaces and smooth manifolds. General references for such topics are [Bre11], [Eva10] and [Lee12].
Preliminaries
1.1
The Direct Method of the Calculus of Variations
In this first section we give a brief introduction to boundary value problems and to the classical Tonelli’s direct method for the existence of minimum points of variational problems.
A problem involving Partial Differential Equations together with some bound-ary conditions (a Boundbound-ary Value Problem, BVP shortly), can be written abstractly as
(
J (u) = 0 in Ω,
u = g on ∂Ω. (1.1.1) where J : X → Y is a (possibly nonlinear) differential operator from a function space X to Y , u is the unknown function and g is given.
If J turns out to be (in some sense we will specify soon) the derivative of a particular “energy functional” E, equation (1.1.1) can be written as
E0(u) = 0.
Therefore, solutions for (1.1.1) are critical points for E. This way of looking at the problem can be useful, because while solution of the standard BVP can be really hard to find, minima for the energy functional may be discovered through the lens of Calculus of Variations and Functional Analysis, and that can be hopefully easier to do.
This kind of BVPs that can be treated with calculus of variations are called variational problems, and one of their natural settings are Sobolev spaces, therefore now we discuss partial differentiability of functionals defined on a generic Banach space V with values in R, with dual V∗ and duality pairing
(·, ·) : V × V∗ → R.
Definition 1.1. A functional E on a Banach space V is Frèchet differentiable at u ∈ V if there exists a bounded linear map DE(u) ∈ V∗, called the
ential of E at u, such that
|E(u + v) − E(u) − (DE(u), v)| ||v||V
→ 0 as ||v||V → 0.
E is of class C1, if the map u → DE(u) is continuous
This way, we can consider variational problems as functional equations of the type DE(u) = 0, for u belonging to some Banach space V , where E is a functional on V , and is Frèchet differentiable with derivative DE.
Definition 1.2. We call a point u ∈ V critical if DE(u) = 0, otherwise, u is called regular. A critical value β ∈ R is the image of a critical point, otherwise β is called regular value.
Given a set M ⊂ V , a point u ∈ M is an absolute minimizer for E on M if for every v ∈ M there holds E(v) ≥ E(u), or analogously
E(u) = inf
v∈ME(v).
A point u ∈ M is a relative minimizer for E on M if for some neighborhood U of u in V it is an absolute minimizer in M ∩ U.
Moreover, we can define saddle points, that is, critical points of E such that any neighborhood U of u in V contains points v, w such that E(v) < E(u) < E(w). In the rest of this section, we will deal with a topological space X, and for every x ∈ X, N (x) will be the set of all open neighborhoods of x in X. Definition 1.3. (Semicontinuity) We say that a function F : X → R is lower semicontinuous (l.s.c) at x ∈ X if and only if
F (x) ≤ sup
U ∈N (x)
inf
y∈UF (y). (1.1.2)
F is said to be upper semicontinuous if −F is l.s.c.
Remark 1.1. Each result regarding lower semicontinuity has a counterpart for upper semicontinuous functions. However, since lower semicontinuity is the natural requirement when dealing with minima problems, we will focus on this concept.
Remark 1.2. Since F (x) ≥ infy∈UF (y) for every U ∈ N (x), F is l.s.c. at x ∈ X if and only if F (x) = supU ∈N (x)infy∈UF (y).
It follows quite immediately from this definition that for a l.s.c. function at x ∈ X holds that
F (x) ≤ lim inf
j→∞ F (xj) (1.1.3)
F (x) = min{lim inf
j→∞ F (xj) : xj → x}. (1.1.4)
The converse is also true under the assumption that X satisfies the first axiom of countability (i.e. the neighborhood system of each point x ∈ X has a countable base).
Remark 1.3. When the first axiom of countability is satisfied, topological notions admit sequential characterizations. For instance, a subset M of a topological space X which satisfies the first axiom of countability is closed when it contains every point of closure (i.e. points x ∈ X such that each U ∈ N (x)contains a point of M), or sequentially speaking when every x ∈ X which admits a sequence in M converging to x is a point of M.
In the following, we will always suppose that our topological space X satisfies the first axiom of countability, hence we will use with no distinction (1.1.2), (1.1.3) or (1.1.4) as definitions for lower semicontinuity.
Definition 1.4. (Level sets) For every function F : X → R and every t ∈ R we define
Ft= {x ∈ X : F (x) ≤ t}.
In the same way, we can define the sets Ft= {x ∈ X : F (x) ≥ t}.
Definition 1.5. (Lower and upper limits) Let F : X → R. We define the lower limit (lim inf) of F at x as
lim inf
y→x F (y) = inf{lim infj→∞ F (xj) : xj ∈ X, xj → x}
= inf{ lim
j→∞F (xj) : xj ∈ X, xj → x, ∃ limj→∞F (xj)},
and the upper limit (lim sup) of f at x as lim sup
y→x
F (y) = sup{lim sup
j→∞
F (xj) : xj ∈ X, xj → x}
= sup{ lim
j→∞F (xj) : xj ∈ X, xj → x, ∃ limj→∞F (xj)}.
The following proposition provides a useful characterization of l.s.c. func-tions.
Proposition 1.1. Let F : X → R be a function. The following properties are equivalent
1. F is lower semicontinuous (sequentially l.s.c.) on X; 2. F (x) = lim infy→xF (y);
Proof. The equivalence of (1) and (2) has already been discussed above, while in order to include also (3) is enough to consider that (1) implies that if F (xj) ≤
t and xj → x then F (x) ≤ t, while if there exists x and xj → x such that
F (x) > t > lim infjF (xj) then (3) would be no more valid.
Example 1.1. By the point (3) of Proposition 1.1 it follows that the charac-teristic function F = 1E of a set E, defined as 1E(x) = 1 if x ∈ E, 1E(x) = 0
otherwise, is l.s.c. if and only if E is open, while the indicator function χE,
defined by χE(x) = 0if x ∈ E and χE(x) = +∞ if x /∈ E, is l.s.c. if and only
if E is closed.
In order to state and prove Tonelli’s theorem we need some other classical definitions.
Definition 1.6. (Coerciveness) A function F : X → R is coercive if for all t ∈ R the set Ft is precompact, i.e. its closure is compact.
Definition 1.7. (Minimizing sequence) Let F : X → R be a functional. A minimizing sequence for F in X is a sequence (xj) in X such that
inf
y∈XF (y) = limj→∞F (xj).
Theorem 1.1. (The Direct Method of the Calculus of Variations) Let F : X → R be coercive and lower semicontinuous. Then
(a) F has a minimum point in X;
(b) if (xj)is a minimizing sequence of F in X and x is the limit of a
subse-quence of (xj), then x is a minimum point of F in X;
(c) if F is not identically +∞ then every minimizing sequence for F has a convergent subsequence.
Proof. If F is identically +∞ the result is trivial, infact every point of X is a minimum point for the functional F .
Otherwise, Let (xj)be a minimizing sequence for F in X. Since F is coercive
and
lim
j→∞F (xj) = infy∈XF (y) < +∞ (1.1.5)
we have that the sequence (xh)has a subsequence which converges to a certain
point x ∈ X, whence (c) is proved.
Since F is lower semicontinuous, by (1.1.3) and (1.1.5) the following inequalities hold:
inf
y∈XF (y) ≤ F (x) ≤ lim supj→∞ F (xj) = infy∈XF (y),
It must be observed that the two conditions we have used to obtain the existence of a minimum are competing at topological level. Indeed the lower semicontinuity needs a fine topology to be easily fulfilled, while the coerciveness condition calls for a coarse topology. In applications, for instance when dealing with integral functionals defined on Sobolev spaces, this issue is treated by choosing a proper topology on the space that can satisfy both requests. In this setting, Theorem 1.1 can be restated as follows
Theorem 1.2. Suppose V is a reflexive Banach space with norm k·k, and let M ⊂ V be a weakly closed subset of V . Suppose E → R is a coercive functional on M with respect to V , that is
E(u) → ∞as kuk → ∞, u ∈ M,
and (sequentially) weakly lower semi-continuous on M with respect to V , that means that for every u ∈ M and any sequence (uj) in M such that uj * u
weakly in V
E(u) ≤ lim inf
j→∞ E(uj)
Then E is bounded from below on M and attains its infimum in M.
We can now present a brief example of application of the direct method. Example 1.2. (p-laplacian equation.) Let Ω be a bounded domain in Rn, p ∈ [2, ∞) with conjugate exponent q, and let f ∈ W−1,q(Ω), the dual of W01,p(Ω). Then there exists a weak solution u ∈ W01,p(Ω)of the BVP
(
−∇ · (|∇u|p−2∇u) = f in Ω
u = 0 on ∂Ω (1.1.6)
which means that u satisfies the equation Z
Ω
(|∇u|p−2∇u∇ϕ − f ϕ)dx = 0, ∀ ϕ ∈ C0∞(Ω). (1.1.7) Infact, the BVP is in variational form, because the left part of (1.1.7) represents the directional derivative of the C1-functional
E(u) = 1 p Z Ω |∇u|pdx − Z Ω f udx on the Banach space V = W1,p
0 (Ω), in the direction ϕ. The bounds on p make
W01,p(Ω)a reflexive space. Moreover, E is coercive because of these inequalities E(u) ≥1 pkuk p H01,p− kf kH−1,qkukH01,p ≥ ≥ 1 p kukp H01,p− c kukH01,p ≥ c−1kukp H01,p− C.
In order to conclude, we need to demonstrate that E is (sequentially) weakly lower semi-continuous: if uj * u weakly in W01,p(Ω), since f belongs to the
dual space of W1,p 0 (Ω)we have Z Ω f ujdx → Z Ω f udx, from the definition of weak convergence. Since 1
p
R
Ω|∇u|pdx is a norm on
W01,p(Ω)we conclude that there exists a minimizer u ∈ W01,p(Ω)of E.
1.2
Γ-convergence
Γ-convergence has been introduced by Ennio De Giorgi and Tullio Franzoni in [DF75]. It is a notion of convergence for sequences of functionals which is the most appropriate for variational problems, since under some additional compactness conditions, it implies the convergence of minima of the sequence considered to the minimum of the limit functional.
In this section we will always consider X as a metric space.
Definition 1.8. (Γ-convergence) Let Fj : X → R be a sequence of functionals. We say that (Fj) Γ-converges in X to F∞: X → R if for all x ∈ X we have
1. liminf inequality: for every sequence (xj)converging to x
F∞(x) ≤ lim inf
j→∞ Fj(xj); (1.2.1)
2. limsup inequality: there exists a sequence (xj)converging to x such that
F∞(x) ≥ lim sup j→∞
Fj(xj). (1.2.2)
The functional F∞ is called the Γ-limit of (Fj) and we will write
F∞= Γ- lim j→∞Fj
Remark 1.4. If (xj) is a sequence converging to x satisfying the limsup in-equality, than by (1.2.1) and (1.2.2) we have
F∞(x) ≤ lim inf
j→∞ Fj(xj) ≤ lim supj→∞ Fj(xj) ≤ F∞(x),
so that for the sequence (xj)actually holds
F∞(x) = lim
j→∞Fj(xj). (1.2.3)
Such a sequence is called recovery sequence and its existence can replace the limsup inequality.
Remark 1.5. We can write the liminf inequality as F∞(x) ≤ inf{lim inf
j→∞ Fj(xj) : xj → x}.
Now, we always have inf{lim inf
j→∞ Fj(xj) : xj → x} ≤ inf{lim supj→∞ Fj(xj) : xj → x},
whence, if (˜xj)is a recovery sequence, from the limsup inequality follows that
inf{lim sup
j→∞
Fj(xj) : xj → x} ≤ lim sup j→∞
Fj(˜xj) ≤ F∞(x).
Therefore, we have obtained the following equalities F∞(x) = min{lim inf
j→∞ Fj(xj) : xj → x} = min{lim supj→∞ Fj(xj) : xj → x},
(1.2.4) and both minima are reached as limits along a recovery sequence.
Actually, (1.2.4) suggests to consider the definition of Γ-convergence as the following equality of infima
F∞(x) = inf{lim inf
j→∞ Fj(xj) : xj → x} = inf{lim supj→∞ Fj(xj) : xj → x}. (1.2.5)
Remark 1.6. Γ-convergence is stable under continuous perturbations. Indeed, if (Fj) Γ-converges to F∞and G : X → R is a continuous function then (Fj+G)
Γ- converges to F∞+ G. Following definition 1.8, from liminf inequality for all
x ∈ X and xj → x we get
F∞(x) + G(x) ≤ lim inf
j→∞ Fj(xj) + limj G(xj) = lim infj→∞ (Fj(xj) + G(xj)),
while from the limsup inequality follows that F∞(x) + G(x) = lim
j→∞Fj(xj) + limj→∞G(xj) = limj→∞(Fj(xj) + G(xj)),
and we have found a recovery sequence for F∞+ G.
It is possible to give a topological definition of Γ-convergence (similar to the one we gave for semicontinuity) and also to prove that the Γ-limit of a se-quence is always a lower-semicontinuous function. Hence, a constant sese-quence (Fj)j∈Nwith Fj = F for every j ∈ N converges to scF , the lower
semincontinu-ous envelope of the function F , i.e. the greatest lower-semicontinusemincontinu-ous function not greater than F .
For a deeper insight on Γ-convergence, the reader can consult the book of Braides [Bra02] and Dal Maso [Dal13].
We conclude this summary with the important theorem about convergence of minima for variational problems via Γ-convergence. First, we need a proposi-tion.
Proposition 1.2. Let Fj, F∞ : X → R be functionals, and suppose that
F∞= Γ-limj→∞Fj. Then
(a) if K ⊂ X is compact we have inf
K F∞≤ lim infj→∞ infK Fj;
(b) if U ⊂ X is open then inf
U F∞≥ lim supj→∞ infU Fj.
Proof. (a) Let (˜xj) be a minimizing sequence for K for which
lim inf
j→∞ infK Fj = lim infj→∞ Fj(˜xj)
Extracting a subsequence, we get (˜xjk)such that
lim
k→∞Fjk(˜xjk) = lim infj→∞ Fj(˜xj) = lim infj→∞ infK Fj,
and xjk → ¯x ∈ K. If xj = ( ˜ xjk if j = jk ¯ x if j 6= jk for every k, we get inf K F∞≤ F∞(¯x) ≤ lim infj→∞ Fj(xj) ≤ ≤ lim inf
k→∞ Fjk(xjk) = limk→∞Fjk(˜xjk) = lim infj→∞ infK Fj (1.2.6)
and we are done with the first part.
(b) Fix a δ > 0, let x ∈ U be a point for which F∞(x) ≤ infUF∞+ δ. If (xj)
is a recovery sequence for x we have inf
U F∞+ δ ≥ F∞(x) ≥ lim supj→∞ Fj(xj) ≥ lim supj→∞ infU Fj, (1.2.7)
and by letting δ go to 0 we obtain the conclusion.
For the result we are going to demonstrate, we need an improvement of coerciveness Definition 1.6.
Definition 1.9. (Equi-coerciveness) A functional F : X → R is mildly coercive if infXF = infKF for some non-empty compact set K ⊂ X. A sequence (Fj)
is equi-mildly coercive if there exists a non-empty compact set K ⊂ X such that infXFj = infKFj for all j.
Obviously, coerciveness implies mild coerciveness, infact, if the functional F is not identically +∞, it is enough to take as K the closure of the sublevel set Ftfor a t for which this is not empty.
We are now in position to state and prove the required convergence result. Theorem 1.3. (Convergence of absolute minima) Let (Fj) be a sequence of
equi-mildly coercive functionals on X, and let F∞= Γ- limjFj; then
∃ min
X F∞= limj→∞infX Fj. (1.2.8)
Moreover, if (xj) is a precompact sequence such that
lim
j→∞Fj(xj) = limj→∞infX Fj,
then converging subsequences of (xj) converge to a minimum point for F∞.
Proof. If ¯x is as in the proof of Proposition 1.2 then by (1.2.6) and (1.2.7) with U = X and by the equi-mild coerciveness
inf
X F∞≤ infK F∞≤ F∞(¯x) ≤ lim infj→∞ infK Fj =
= lim inf
j→∞ infX Fj ≤ lim supj→∞ infX Fj ≤ infX F∞.
Remark 1.7. Γ-convergence does not imply convergence of local minimizer. Consider for instance the sequences of functions
fj(x) = x2+ sin(jx), gj(x) = ex+ sin(jx)
having Γ-limits respectively f∞(x) = x2− 1 and g∞(x) = ex− 1. The only
minima converging are the absolute ones.
The following example from [JS09] shows how equicoercivity may be a necessary condition also for the convergence of critical points.
Example 1.3. Let λ > 0 and f, fε: R2 → R be the functions defined as fε(x) = x1 h tanh(x2) − λ 1 cosh(x1 ε ) i , f (x) = x1tanh(x2),
where (x1, x2) are the coordinates of R2.
Then fε ε→0
−−−→ f uniformly, hence follows that fε−→ fΓ . Indeed, f (x) − fε(x) = λε x1 ε 1 cosh(x1 ε) ,
and such a difference is a family of uniformly Lipschitz functions that converges to 0, so an interpolation inequality implies that fε → f in C0,α as ε → 0 for
any α ∈ (0, 1).
−4 −2 0 2 4 −5 0 5 −5 0 5 x y f (x, y )
Figure 1.1: The Γ-limit function f(x) = x1tanh(x2).
• f has one critical point at (0, 0) (Figure 1.1).
• if λ ≥ 1, fε has no critical points: in this case there is no compactness
condition holding for any ε > 0 (Figure 1.2).
• for 0 < λ < 1, fε has a unique critical point (which is independent of
ε), at x = (0, tanh−1(λ)), and this point can be arbitrarly far from the critical point (0, 0) of f, since tanh−1
(λ) % ∞as λ % 1 (see Figure 1.3).
Example 1.4. Consider Ω ⊂ RN open and bounded with regular boundary. On the space (C0(Ω), k·k∞)we can consider the family of maps defined as the
Lp norms of a function as p varies in [1, +∞], then by the boundedness of Ω such a family is well defined. What we want to show is that
k·kp−→ k·kΓ ∞ as p → +∞.
Consider first the lim inf inequality. Let (fp)p⊂ C0(Ω)be a sequence
converg-ing to a certain f with respect to the ∞-norm, we need to show that lim inf
−4 −2 0 2 4 −5 0 5 −5 0 5 x y f (x, y )
Figure 1.2: The function fε(x) with λ ≥ 1
−4 −2 0 2 4 −5 0 5 −5 0 5 x y f (x, y )
In order to do this, let p ∈ [1, +∞) and consider δ > 0 such that δ < kfpk∞.
Let Cδ= {x ∈ Ω : |fp(x)| ≥ kfpk∞− δ}, we can write
kfpkp = Z Ω |fp(x)|pdx1/p ≥ Z Cδ | kfpk∞− δ|p1/p= (kfpk∞− δ)|Cδ|1/p
where |Cδ|is the Lebesgue measure of the set Cδ. As p → ∞, we get
lim inf
p→∞ kfpkp≥ kf k∞− δ
and the conclusion follows from the arbitrariness of δ.
We turn now on the lim sup inequality, let f be a function in (C0(Ω), k·k∞),
and consider the constant sequence fp ≡ f, then we have for q < p
kf kp = Z Ω |f (x)|pdx 1/p = Z Ω |f (x)|p−q|f (x)|qdx 1/p ≤ kf k p−q p ∞ Z Ω |f (x)|q1/p= kf k p−q p ∞ kf kq/pq , hence kfkp ≤ kf k p−q p ∞ kf kq/pq and lim sup p→∞ kf kp≤ kf k∞.
Holding both the liminf and the limsup inequality, we conclude.
An example from microstructure theory
We conclude the section with a classical example of Γ-convergence of function-als, the Modica-Mortola theorem, following the presentation given by Alberti in [Alb00].
The aim is to study equilibrium states of a system composed by a container filled with two incompressible and immiscible fluids, considering this as an isolated physical system (hence we will not take into account gravity or the interaction of the fluids with the walls).
The mathematical model proposed by Cahn and Hilliard in [CH58] is the following: the container is a bounded open set Ω ⊂ RN, and V such that
0 < V < vol(Ω)is the volume of the second fluid. Each configuration is repre-sented by a measurable function u : Ω → [0, 1] for which RΩu = V holds, and we denote by X the space of all such functions, endowed with the L1 norm.
The meaning of each configuration function is that at every point x ∈ Ω, u(x) is the volume density of the second fluid at that point (hence u(x) = 0 means that there is only the first fluid at the point x, u(x) = 1 the second, while u(x) = 1/2 denotes that each fluid is present at x with the same concentra-tion).
To every configuration u is associated the energy functional Eε(u) := ε2 Z Ω |∇u|2+ Z Ω W (u)
where ε is a small positive parameter and W a continuous positive function which vanishes at 0 and 1. Minimizing Eε, the term R W (u) favors those
configurations which take values close to 0 and 1, which means that the two fluids are separated, while the first term penalizes the inhomogeneity of u. When ε is small enough the second term will prevail, causing the two fluids to arrange themselves separately, with the phase transition occurring in a thin layer.
Intuitively, the stable configuration of the system will occur when the interface between the two fluids will have the smallest measure possible, and that is actually what happens. We can expect infact that in the limit for ε → 0 the system can be represented by functions u : Ω → {0, 1} such that RΩu = V, while the energy of the system is the functional
F (u) := σHN −1(Su)
where σ is a parameter called surface tension between the fluids and HN −1(Su)
is the (N-1)-dimensional Hausdorff measure of the set Su, the singularities of u, which is the interface between the two fluids (if you think of the classical case in which N = 3, Su turns out to be a surface, and everything looks more familiar).
Modica in [Mod87], applying a result proved by Modica and Mortola in [MM77] and previously conjectured by De Giorgi, established the connection between these two models. Modica showed that an opportune rescaling of the func-tionals Eε Γ-converges to F , and as a corollary, that the minimizers of Eε
converge to those of F . We recall that BV (Ω, {0, 1}) is the set of all functions u : Ω → {0, 1} with bounded variation, for which Su is the set of essential singularities.
Theorem 1.4. For every ε > 0, let Fε(u) := 1 εEε(u) = ( εRΩ|∇u|2+1 ε R ΩW (u) if u ∈ W 1,2(Ω) ∩ X, +∞ elsewhere (1.2.9) and F (u) := ( ¯ σHN −1(Su) if u ∈ BV (Ω, {0, 1}) ∩ X, +∞ elsewhere (1.2.10)
where ¯σ is a constant we will define later.
Then the functionals Fε Γ-converge to F in X, and the compactness condition
() let be given sequences (εn) and (un) such that εn → 0 and Fεn(un) is
bounded; then (un) is pre-compact in L1(Ω)and every limit point belongs
is satisfied.
Remark 1.8. In this example, we will call strong convergence either conver-gence in measure or in Lp for p ∈ [1, ∞), which are equivalent because we will
consider sequence of functions having values in [0, 1], while with weak conver-gence we will address every converconver-gence induced by weak Lp topologies for
p ∈ [1, ∞), which are actually the same.
The proof of the 1-dimensional case will be enough for our purposes, we address the reader to the reference for a discussion of the general case, which will be mostly a reduction to what we will describe here.
Let Ω be an interval of the real line R, here the Hausdorff measure will just count points. We recall that in this setting each function u ∈ W1,2(Ω)admits
a continuous representant on Ω.
Before starting the proof, we point out the main features of the problem, and give some lemmas. In what follows, we need to consider functionals Fε not
just as functions of the variable u, but also specifying the integration domain, hence we will write
Fε(u, A) := ε Z A | ˙u|2+1 ε Z A W (u) (1.2.11) and in particular Fε(u) = Fε(u, Ω).
Another useful property that we will use is the scaling identity Fε(u, A) = F1(uε,
1
εA) (1.2.12) where uε(x) = u(εx)and 1
εA = {x : εx ∈ A}. This follows immediately from
computation F1(uε, 1 εA) = Z 1 εA | ˙u(εx)|2ε2dx + Z 1 εA W (u(εx))dx = = ε Z A | ˙u(y)|2dy + 1 ε Z A W (u(y))dy = Fε(u, A)
where in the last equality we changed variable using y = εx.
This scaling property suggests that our problem may be related to the case ε = 1. Consider the minimum problem
¯
σ = inf{F1(u, R) : u : R → [0, 1], lim
x→−∞u(x) = 0, limx→+∞u(x) = 1} (1.2.13)
called optimal profile problem, then the following result holds:
Lemma 1. Let I ⊆ R be an interval, and u : I → [0, 1]. Suppose that there exists a, b ∈ I and δ > 0 such that u(a) ≤ δ and u(b) ≥ 1 − δ. Then for every ε > 0
Proof. Without any loss of generality suppose ε = 1 and I = (a, b) (otherwise, apply the scaling property or work just with (a, b)). In order to compare Fε(u, I)and F1(u, R) we need to extend u to the entire real line, it sufficies to
do it linearly, redefining u(x) := 0 if x ∈ (−∞, a − δ) x + δ − a if x ∈ (a − δ, a) u(x) if x ∈ (a, b) x + (1 − δ) − b if x ∈ (b, b + δ) 1 if x ∈ (b + δ, +∞) Now set K = maxx∈[0,1]W (x), then we get
F1(u, R I) = Z R I | ˙u|2dx + Z R I W (u)dx ≤ ≤ Z a a−δ 1dx + Kδ + Z b+δ b 1dx + Kδ = 2δ + 2Kδ Define O(δ) := 2δ + 2Kδ, then
F1(u, I) = F1(u, R) − F1(u, R I) ≥ ¯σ − O(δ)
concludes the proof
The next preliminary step consists in solving problem (1.2.13) Lemma 2. The minimum in (1.2.13) is attained at ¯u, moreover
¯ σ = 2 Z 1 0 p W (¯u)d¯u, which is the constant of theorem 1.4.
Proof. We apply Cauchy’s inequality 2ab ≤ a2+ b2 with a = ˙u(x) and b =
pW (u), to get F1(u, R) = Z R ˙ u2(x) + W (u(x))dx ≥ ≥ 2 Z R p W (u(x)) ˙u(x)dx = 2 Z 1 0 p W (u)du = σ.
Recall that, in the Cauchy’s inequality, equality holds if a = b. Hence we look for u solution of the differential equation ˙u = pW (u). Note that the constant functions 0 and 1 solve such equation, because W (0) = W (1) = 0, but we
need a growth condition on the solution. Consider then the following Cauchy problem ( ˙ u =pW (u), u(0) = 1/2.
Since√W is continuous we have a global solution, and the fact that W (u) > 0 on (0, 1) implies that such a solution has two asyntotes, the function identically equal to 1 at +∞ and 0 at −∞.
Proof of Theorem 1.4. We start by noticing that the proof of Theorem 1.4 consists in demonstrating the following facts
(a) compactness condition ();
(b) lower bound inequality - if u ∈ BV (Ω, {0, 1}), (uε) ⊂ W1,2(Ω)and uε→
u in L1(Ω)then
lim inf
ε→0 Fε(uε) ≥ ¯σH
N −1(Su); (1.2.15)
(c) upper bound inequality - for every u ∈ BV (Ω, {0, 1}) exists (uε) ⊂
W1,2(Ω)such that uε→ u in L1(Ω), R uε=R ufor every ε and
lim sup
ε→0
Fε(uε) ≤ ¯σHN −1(Su). (1.2.16)
We start by the compactness condition. The idea behind the proof is the following: consider a sequence (uε)such that Fε(uε) is bounded, say Fε(uε) ≤
C < +∞. Then we can say that such a sequence is bounded in W1,2(Ω)and that RΩW (uε) ≤ Cε, hence, recalling that W is a continuous positive function
such that W (0) = W (1) = 0, we can affirm that functions (uε) take values
different from 0 and 1 just in a set with measure of order ε.
If (uε) converges weakly to a function u, then by Lemma 1 we know that the
cost of each oscillation in terms of the energy Fε is of order ¯σ, whence the
boundedness of Fε(uε)implies that a finite number of oscillations are possible,
from which we obtain strong convergence.
To make this argument rigorous, it is possible to use the notion of Young measure generated by a sequence: suppose that (uε)is a sequence of functions,
then we can extract a subsequence ”converging” to a family of probability measures {νx : x ∈ Ω} called the Young measure generated by (uε). In
particular the functions uε converge weakly to a u such that
u(x) := Z
Ω
udνx(u),
and we have strong convergence if and only if νx is a dirac mass for almost
function f ∈ Ω×[0, 1] → R continuous with respect to the second variable and bounded Z Ω f (x, uε(x))dx → Z Ω Z 1 0 f (x, u)dνx(u)dx. (1.2.17)
We are now in position to actually prove compactness condition (). Consider (uε) such that Fε(uε) ≤ C < +∞ and suppose that it generates a Young
measure {νx : x ∈ Ω}. Then RΩW (uε) converges to 0 and by the limit
property (1.2.17) Z Ω Z 1 0 W (u)dνx(u)dx = 0.
So for almost every x ∈ Ω the measure νx is supported on {0, 1} and we can
assume then that it has the form
νx = λ(x)δ0+ (1 − λ(x))δ1
for a proper λ(x) ∈ [0, 1].
Consider now any interval I ⊂ Ω such that λ is not a.e. equal to 0 neither a.e. equal to 1, functions uε must take there values close to 0 and to 1, so because
of lemma 1 Fε(uε, I) ≥ ¯σ − o(1). Therefore Fε(uε) ≤ C implies that we can
find at most C/¯σ such intervals.
Thus λ agrees a.e. with a function u which has a finite number of transitions from values 0 and 1, which takes almost everywhere indeed.
We just proved that u ∈ BV (Ω, {0, 1}) and νx = δ0 where u(x) = 0 and
νx = δ1 where u(x) = 1, this, by what we have said about Young measures,
implies the strong convergence of the sequence (uε)to u.
From the previous argument on the number of oscillations of the limit function uwe get that ¯σH0(Su) ≤ C, whence the lower-bound inequality 1.2.15 follows, because we can take, eventually passing to subsequences, C arbitrarly close to lim infε→0Fε(uε).
To conclude we just need to prove (1.2.16). For every u ∈ BV (Ω, {0, 1}) we need to construct a sequence (uε) such that uε → u and lim supε→0Fε(uε) ≤
¯
σH(Su).
We consider the simple case in which the function u has only a variation at the point x = 0, hence
u(x) = (
1 if x ≥ 0, 0 if x < 0.
Then a proper sequence can be found considering the solution γ of the optimal profile problem (1.2.13), and setting
uε(x) := γ(x/ε)
hence uε(x)ε→0→ u(x)for every x 6= 0 because of the asymptotes of the function
γ. We apply then the scaling property 1.2.12 to find Fε(uε) = F1(γ,
1
Taking the limsup on the left hand of the inequality we conclude. The con-straint R uε =R ucan be easily fulfilled by opportune translations of uε, while
the general case of u ∈ BV (Ω, {0, 1}) is solved repeating the above construction for every variation point.
1.3
Minimax Methods
The purpose of this section is to introduce some minimax methods that we will use from now on. We start by giving an outline of the Palais-Smale compact-ness condition, then we introduce deformation lemmas and we conclude with a description of minimax method itself.
The setting for this section will always be a Banach Space V , together with a norm k·k. Analogously to what we have done in the first section, DE : V → V∗
will denote the Fréchet derivative of a functional E, and (·, ·) the pairing be-tween V and V∗. The norm of the dual space V∗ will always be denoted by
| · |, as the norm on R.
Palais-Smale condition is a compactness condition that will ensure that when we consider a sequence that is a good candidate to reach a critical point, it actually does it (up to subsequences, of course).
A good reference is the book of Struwe [Str08].
Definition 1.10. (Palais-Smale sequence) Let (un) ⊂ V be a sequence, and E : V → R a functional, (un) is a Palais-Smale sequence ((P S) for short) for
E if
1. E(un) is bounded, i.e. there exists C ∈ R such that |E(un)| < C for
every n ∈ N; 2. |DE(un)|
n→∞
−−−→ 0.
A refinement of this condition is the following:
Definition 1.11. Let (un) ⊂ V be a sequence, and E : V → R a functional, (un) is a Palais-Smale sequence at level c ∈ R ((P S)c) if
1. E(un)−−−→ cn→∞ ,
2. |DE(un)| n→∞
−−−→ 0.
Therefore, we will say that the functional E satisfies the Palais-Smale condition when each (PS)-sequence admits a subsequence converging to some ¯
u ∈ V. Obviously, this is just a technical condition: whenever we deal with a variational problem, we have to check the Palais-Smale condition for the func-tional considered.
An improvement of the Palais-Smale condition is the Cerami condition, weaker then the previous one.
Definition 1.12. Let (un) ⊂ V be a sequence, and E : V → R a functional, (un)is a Cerami sequence for E at level c if
1. E(un)−−−→ cn→∞ ,
2. (1 + kunk)|DE(un)|−−−→ 0n→∞ .
A functional then satisfies Cerami condition if each Cerami sequence admits a convergent subsequence.
It is easy to check that if a functional satisfies the Palais-Smale condition then automatically the Cerami condition follows, hence the latter is weaker than (PS).
The next step is to define pseudo-gradients on Banach spaces. Suppose E ∈ C1(V ) is a functional on V , we denote with
˜
V = {u ∈ V : DE(u) 6= 0}
the set of regular points with respect to the Fréchet derivative. Then
Definition 1.13. A pseudo-gradient vector field for E is a locally Lipschitz continuous vector field v : ˜V → V such that the conditions
1. kv(u)k < 2 min{|DE(u)|, 1},
2. (v(u), DE(u)) > min{|DE(u)|, 1}|DE(u)| hold for all u ∈ ˜V.
Proposition 1.3. Each functional E ∈ C1(V )admits a pseudo-gradient vector field.
Proof. Consider u ∈ ˜V, then there exists w = w(u) such that the definition of pseudo-gradient vector field holds:
kwk < 2 min{|DE(u)|, 1},
(w, DE(u)) > min{|DE(u)|, 1}|DE(u)|; (1.3.1) and by continuity, being the inequalities strict, for each u ∈ ˜V there exists a neighborhood W = W (u) such that (1.3.1) holds for w(u) and each ˜u ∈ W . Now, ˜V ⊂ V is metrizable, hence paracompact, and the covering {W (u)}u∈ ˜V admits a locally finite subcovering {Wi}i∈I that we can use to build a Lipschitz
continuous partition of unity {ϕi}such that supp(ϕi) ⊂ Wi and Piϕi(u) ≡ 1
for each u ∈ ˜V.
A construction of such a family is the following: consider ρi(u) = dist(u, V \Wi)
and define
ϕ(u) = Pρi(u)
j∈Iρj(u)
From its definitions, follows that 0 ≤ ϕi ≤ 1, ϕ|V \Wi ≡ 0and Pi∈Iϕi ≡ 1. For
every u ∈ ˜V there exists a neighborhood W of u for which W ∩ Wi6= ∅ for at
most finitely many indices, because {Wi}i∈I is locally finite, and the Lipschitz
continuity of the maps ρi implies the Lipschitz continuity of each ϕi.
Finally, we can let our pseudo-gradient vector field to be v(u) =X
i∈I
ϕi(u)w(ui).
Conditions of pseudo-gradient vector field are satisfied, being (1.3.1) linear in wand the sum a convex combination.
Remark 1.9. Suppose that on the space V we consider the action of the group Z2 = {1, −1}, and that the functional is invariant under such action, that
is, the functional is even and hence invariant under the action of the group of symmetries {id, −id}. Hence we are allowed to consider an odd pseudo-gradient vector field by choosing
˜
v(u) = 1
2(v(u) − v(−u)). This will be of crucial importance in Chapter II.
We define some special subsets of V , which will be useful in the following Kβ = {u ∈ V : E(u) = β, DE(u) = 0}, Nβ,δ = {u ∈ V : |E(u) − β| < δ, |DE(u)| < δ}, Uβ,ρ= [ u∈Kβ {v ∈ V : kv − uk < ρ}.
Remark 1.10. If E ∈ C1(V ) satisfies (PS), then Kβ is compact and the
families {Uβ,ρ}ρ>0 and {Nβ,δ}δ>0 are fundamental systems of neighborhoods
of Kβ. In fact any sequence (um) in Kβ has a convergent subsequence by the
Palais-Smale condition. By the continuity of E and DE every cluster point of (um)lies in Kβ and then Kβ is compact.
To show that {Uβ,ρ}ρ>0 is a fundamental neighborhoods system, observe that
for every ρ > 0, Uβ,ρ is a neighborhood of Kβ. Vice versa let U be an open
neighborhood of Kβ. Suppose that for a sequence ρm → 0 there exists a
sequence wm ∈ Uβ,ρm\ U. By the definition of Uβ,ρm we can find vm ∈ Kβ
such that kwm− vmk ≤ ρm. Since Kβ is compact, vm → v ∈ Kβ up to a
subsequence and this implies that wm → v and wm ∈ U for m big enough,
which proves the statement.
The proof for the family {Nβ,δ}δ>0 is quite the same, hence we will omit it.
Theorem 1.5. Let E ∈ C1(V )satisfying (PS). Let β ∈ R, ¯ε > 0 be given and let N be any neighborhood of Kβ. Then there exist a number ε ∈]0, ¯ε[ and a
continuous family of homeomorphisms Φ(·, t) of V , 0 ≤ t < ∞, such that 1. Φ(u, t) = u, if t = 0, or DE(u) = 0, or |E(u) − β| ≥ ¯ε;
2. for any u ∈ V , E(Φ(u, t)) is non-increasing in t; 3. Φ(Eβ+ε N, 1) ⊂ Eβ−ε, and Φ(Eβ+ε, 1) ⊂ Eβ−ε∪ N.
Moreover, Φ : V × [0, ∞[→ V has the semi-group property, that is, Φ(·, t) ◦ Φ(·, s) = Φ(·, s + t) for all s, t ≥ 0, and may be chosen odd if the functional E is even.
Proof. The proof consists in building a proper vector field which is a sort of modified pseudo-gradient and then in defining Φ as the solution of a proper differential equation.
Since the families of neighborhoods we have defined above form a fundamental neighborhood system, we can consider numbers δ, ρ > 0 such that
N ⊃ Uβ,2ρ⊃ Uβ,ρ⊃ Nβ,δ.
Consider η to be a Lipschitz function on V such that 0 ≤ η ≤ 1, η ≡ 1 outside Nβ,δ, η ≡ 0 in Nβ,δ/2.
Then let ϕ be a Lipschitz function on R such that 0 ≤ ϕ ≤ 1, ϕ(s) ≡ 0 if |β − s| ≥ min{¯ε, δ/4}, ϕ(s) ≡ 1, if |β − s| ≤ min{¯ε/2, δ/8}.
Finally, let v : ˜V → V be a pseudo-gradient vector field for E. Define e(u) =
(
−η(u)ϕ(E(u))v(u), if u ∈ ˜V , 0, otherwise.
By the choice of ϕ and η, the vector field e vanishes identically (and therefore is Lipschitz) near critical points u of E. Hence e is locally Lipschitz throughout V. Moreover, since kvk < 2 uniformly, also kek ≤ 2 is uniformly bounded, and there will be a global solution Φ : V × R → V of the initial value problem
(
Φt(u, t) = e(Φ(u, t))
Φ(u, 0) = u.
Φ has every desired feature: it is continuous in u, differentiable in t and has the semi-group property, in particular, for any t ∈ R the map Φ(·, t) is a homeomorphism of V . Moreover, properties 1 and 2 are trivially satisfied by construction and by the properties of v, and for ε ≤ min{¯ε/2, δ/8} and u ∈ Eβ+ε, if E(Φ(u, 1)) ≥ β − ε it follows from 2 that |E(Φ(u, t)) − β| ≤ ε and
hence that ϕ(E(Φ(u, t))) = 1 for all t ∈ [0, 1].
In order to prove property 3, we differentiate, and by the chain rule obtain E(Φ(u, 1)) = E(u) + Z 1 0 d dtE(Φ(u, t)dt < β + ε − Z 1 0
η(Φ(u, t))(v(Φ(u, t)), DE(Φ(u, t)))dt < β + ε −
Z
{t : Φ(u,t) /∈Nβ,δ}
(v(Φ(u, t)), DE(Φ(u, t)))dt ≤ β + ε − |{t : Φ(u, t) /∈ Nβ,δ}| · δ2.
But if either u /∈ N or Φ(u, 1) /∈ N, by uniform boundedness kek ≤ 2 and since V N and Nβ,δ are separated by the annulus Uβ,2ρ Uβ,ρ of width ρ, we get
|{t : Φ(u, t) /∈ Nβ,δ}| ≥ ρ 2. Hence, if we choose ε ≤ δ2ρ
4 the above estimates give
E(Φ(u, 1)) ≤ β + ε − δ
2ρ
2 ≤ β − ε, and we conclude.
Remark 1.11. Both the existence of pseudo-gradients vector fields and the deformation lemma can be stated and proved on complete smooth manifolds, by an adaptation of the above arguments, and that is actually what we will make use of in the following. However, for the sake of clearness and conciseness we will not prove here such results, simply assuming them as true.
Moreover, in the following, we will always consider smooth manifolds.
Minimax methods are devoted to the search of saddle points of functionals defined on manifolds.
The trick used to prove that a certain value is critical is the following: it is necessary to find a family of subsets which is invariant with respect to a family of homeomorphisms, then by means of the deformation lemma the infimum among all these invariant sets of the supremum of the functional on each set turns out to be a critical value.
Definition 1.14. Let Φ : M × [0, ∞[→ M be a semi-flow on a manifold M. A family F of subsets of M is called (positively) Φ-invariant if and only if Φ(F, t) ∈ F for all F ∈ F, t ≥ 0.
Theorem 1.6. Suppose M is a complete smooth manifold, and E ∈ C1(M ) satisfies (PS). Let F ⊂ P(M) be a collection of sets which is invariant with
respect to any continuous semi-flow Φ : M × [0, ∞[→ M such that Φ(·, 0) = id|M, Φ(·, t) is a homeomorphism of M for any t ≥ 0 and E(Φ(u, t)) is non-increasing in t for any u ∈ M. Then, if
β = inf
F ∈Fsupu∈FE(u)
is finite, β is a critical value of E.
Proof. By reductio ad absurdum, suppose β ∈ R is a regular value of E. With the notation of Theorem 1.5, consider ¯ε = 1, N = ∅ and ε > 0, let Φ : M × [0, ∞[→ M be the 1-parameter family of homeomorphisms. By definition of β there exists F ∈ F for which
sup
u∈F
E(u) < β + ε;
or, in other words, F ⊂ Eβ+ε. But we know by property 3 of Φ in Theorem
1.5 and by invariance of F, that F1 = Φ(F, 1) is in F and F1 ⊂ Eβ−ε, or in
other words
sup
u∈F1
E(u) < β − ε, which is a contradiction, hence we conclude.
Spectral Theory
2.1
The p-Laplacian
The p-Laplacian operator
∆pu = div(|∇u|p−2∇u), p ∈ (1, ∞) (2.1.1)
arises in many areas of application, and it has been studied intensively in the last fifty years. We give here some results about sequences of eigenvalues that can be defined for such operator. A good survey about the spectrum of the p-Laplacian is [Lin11].
Consider the nonlinear eigenvalue problem (
−∆pu = λ|u|p−2u in Ω
u = 0 on ∂Ω (2.1.2)
where Ω is a bounded and regular domain in Rn, n ≥ 1. λ 6= 0 for which
there exists some u 6= 0 that solve the boundary value problem (2.1.2) is called eigenvalue, while the function u is said eigenfunction associated to it.
Solutions should be intended in the following weak sense Z Ω |∇u|p−2∇u · ∇ηdx = λ Z Ω |u|p−2uηdx ∀ η ∈ C∞ 0 (Ω) (2.1.3)
The set of the eigenvalues for the problem is called spectrum and will be de-noted by σ(−∆p).
Some facts which are well known are that the first eigenvalue λ1 is positive,
simple, and is isolated in the spectrum σ(−∆p). It has an associated
eigen-function that is positive on Ω. In the ODE case n = 1, considering Ω = (0, 1), (2.1.2) reduces to the problem
(
−(|u0|p−2u0)0 = λ|u|p−2u in (0, 1),
u(0) = u(1) = 0. (2.1.4) 25
and it is possible to explicitly compute the eigenvalues and the eigenfunctions, obtaining that the spectrum consists of a sequence of simple eigenvalues λk%
∞ and the eigenfunction associated to λk has exactly k − 1 interior zeroes. Indeed multiplying the differential equation in (2.1.4) by u0 on both sides we
obtain the following equations.
(|u0|p−2u0)0u0+ λ|u|p−2uu0 = 0 ⇔ ((p − 2)|u0|p−3 u 0 |u0|u 00 u0+ |u|p−2u00)u0+ λ|u|p−2uu0 = 0 ⇔ ((p − 2)|u0|p−2u0u00+ |u0|p−2u0u00+ λ|u|p−2uu0 = 0 ⇔ (p − 1)|u0|p−1 u 0 |u0|u 00+ λ|u|p−1 u |u|u 0 = 0 ⇔ 1 p d dt((p − 1)|u 0|p+ λ|u|p) = 0,
suppose that u0(0) 6= 0, if λ is an eigenvalue, consider an eigenfunction u
associated to λ such that u0(0) = 11, then we have
(p − 1)|u0|p+ λ|u|p = p − 1 (2.1.5)
and we can write
|u0| = (1 − λ |u|
p
p − 1)
1
p. (2.1.6)
Assume now that u does not vanish in (0, 1). By uniqueness u(t) = u(1 − t) for every t ∈ (0, 1) hence u0(1/2) = 0and t = 1/2 is the unique value where u0
is 0. Using then (2.1.5) we obtain u(1/2) =p−1 λ
1p
. Then from (2.1.6) after integration over (0, 1/2) we get
1/2 = Z (p−1 λ ) 1/p 0 du 1 − λp−1|u|p 1/p = λ −1/pZ 1 0 dz 1 −p−1|z|p 1/p (2.1.7) whence λ1/p = π p := 2 R1 0 dz 1−|z|pp−11/p.
If u vanishes twice in (0, 1) then by symmetry the points where it happens are 1/3and 2/3, and consequently we get that u0(1/6) = u0(1/2) = 0. Proceeding as before we will find that λ1/p = 2π
p. Hence iterating we find that the
eigenvalues have the specific form
λn,p:= (nπp)p n ∈ N∗
1
such a hypothesis can be done without loss of generality, if u0(0) = 1 there is nothing to do, otherwise it is enough to consider the eigenfunction u/u0(0).
The eigenfunctions associated to each λn,pare the constant multiplies of
u(t) = sinp(λ1/pn,pt)
where the function y = sinp(s) is defined implicitely by the equation
s = Z y 0 dτ 1 −p−1τp 1/p and extended by symmetry to [πp
2 , πp].
Also in the semilinear PDE case n ≥ 2, p = 2, it is possible to prove by means of the spectral theorem for compact operators on reflexive Banach spaces that σ(−∆) consists of a sequence of eigenvalues λk % ∞, but in the quasilinear
case n ≥ 2, p 6= 2 a complete description of the spectrum is not available yet.
2.2
Index Theory
Index theory is the tool by means of which together with minimax methods we will be able to build up several sequences of eigenvalues for the quasi-linear p-Laplacian. We address the reader to the book of Struwe [Str08] for a deep insight on the matter.
An informal description of the methods is the following: looking for flux-invariant families of subsets, sometimes one finds a space with a poor topology; however, the properties of the functional and those of the manifold can help. Considering the action of a certain group on the space involved it is possible to classify different families of subsets with the same index, generating thus several critical values, at least one for every index value.
We start considering a particular index, the Krasnoselski genus, which will explain quite well the ideas behind index theory.
Let V be a Banach space together with G = Z2 = {id, −id} as group of
symmetries acting on V . Define
A = {A ⊂ V : A closed, A = −A}
as the class of closed and symmetric subsets of V , which are the G-invariant sets.
Definition 2.1. Let A ∈ A, A 6= ∅, the Krasnoselski genus of A is defined as γ(A) =
(
inf{m: ∃h ∈ C0(A, Rm\ {0}) h(−u) = −h(u)} ∞ if the above set is empty, or if 0 ∈ A
γ(∅) = 0
Proposition 2.1. Let A, A1, A2 ∈ A, h ∈ C0(V, V ) an odd map. Then the
1. γ(A) ≥ 0 and γ(A) = 0 ⇐⇒ A = ∅ 2. A1 ⊂ A2 =⇒ γ(A1) ≤ γ(A2)
3. γ(A1∪ A2) ≤ γ(A1) + γ(A2)
4. γ(A) ≤ γ(h(A))
5. if A ∈ A is compact and 0 /∈ A, then γ(A) < ∞ and there exists a neighborhood N of A in V such that N ∈ A and γ(A) = γ(N).
The krasnoselski genus is then a definite, monotone, sub-additive, supervariant and continuous map from A to N ∪ {∞}.
Proof. (1) Follows from definition.
(2) If γ(A) = +∞ there is nothing to prove. Otherwise, consider a map h : A2 → Rγ(A2)\ {0}which is continuous and odd, then h|A1 is still a continuous
and odd map, from which we get γ(A1) ≤ γ(A2).
(3) Let
h1 : A1 → Rm1\ {0} continuous and odd,
h2 : A2 → Rm2\ {0} continuous and odd
then by the Tietze extension theorem, we can extend h1 and h2 to continuous
and odd maps in C(V, Rmi\ {0}), but then
h(u) := (h1(u), h2(u))
is still a good map from V to Rm1+m2\ {0} which does not vanish on A
1∪ A2,
and we are done.
(4) Every odd map h ∈ C(h(A), Rm\ {0})define a continuous odd map h ◦ h
from A to Rm\ {0}.
(5) Since A is compact and 0 /∈ A, we can choose r > 0 such that A∩B(0, r) = ∅. The family { ˜B(u, r) = B(u, r) ∪ B(−u, r)} is a cover for A, hence by its compactness admits a finite sub-cover { ˜B(u1, r), . . . , ˜B(um, r)}. Consider a
partition of unity {ϕj}1≤j≤msubordinate to { ˜B(uj, r)}1≤j≤m. We can suppose
that each ϕj is even by considering ¯ϕj(u) = 12(ϕj(u) + ϕj(−u)).
Define h : V → Rm as h(u) = (h
1(u), . . . , hm(u)), where
hj(u) =
(
ϕj(u) if u ∈ B(uj, r)
−ϕj(u) if u ∈ B(−uj, r),
which is continuous, odd and not vanishing on A.
To build the neighborhood N of A, suppose again γ(A) = m < ∞ and let h ∈ C0(A, Rm \ {0}). We may assume, always by the Tietze theorem, that h ∈ C0(V, Rm). By the continuity of h we get that h(A) is compact, and there
exists an open neighborhood U of h(A) which is symmetric and compactly contained in Rm \ {0}. We can consider then N = h−1(U ), then 0 /∈ h(N)
and γ(N) ≤ m. By the monotonicity property and the fact that A ⊂ N, γ(N ) = γ(A), and we can conclude.
We introduce now the general index theory, which will follows exactly the same steps we made to construct Krasnoselski genus.
Let M be a complete differential manifold together with a compact action group G. Define
A = {A ⊂ M : A closed, g(A) = A ∀ g ∈ G} as the class of closed G-invariant subsets of M, and
Γ = {h ∈ C0(M, M ): h ◦ g = g ◦ h for every g ∈ G} the family of G-equivariant maps of M. If G 6= {id} let
Fix(G) = {u ∈ M : gu = u for every g ∈ G}
Definition 2.2. An index for (G, A, Γ) is a map i : A → N0∪ {∞} such that for every A, A1, A2∈ A, h ∈ Γ hold
1. (definiteness) γ(A) ≥ 0 and γ(A) = 0 ⇐⇒ A = ∅ 2. (monotonicity) A1⊂ A2 =⇒ γ(A1) ≤ γ(A2)
3. (sub-additivity) γ(A1∪ A2) ≤ γ(A1) + γ(A2)
4. (supervariance) γ(A) ≤ γ(h(A))
5. (continuity) if A ∈ A is compact and 0 /∈ A, then γ(A) < ∞ and there exists a neighborhood N of A in V such that N ∈ A and γ(A) = γ(N). Remark 2.1. The Krasnoselski genus γ is an index for G = {id, −id}, A = closed and symmetric subsets and Γ = continuous and odd maps.
As an example of index theory different from the genus, we present the Lusternick-Schnirelman category.
Let M be a topological space and consider a closed subset A ⊂ M. We say that Ahas category k ∈ N∪{∞} with respect to M (and we will write catM(A) = k)
if A has a cover of exactly k closed subsets which are homotopic equivalent to a point (i.e. contraible) in M, and if k is minimal in N with this property. If none of these covering exists, we will set catM(A) = ∞.
Proposition 2.2. Let G = {id}, A = {A ⊂ M, A closed } and Γ = {h ∈ C0(M, M ) : h homeomorphism} then cat
Proof. (1)-(3) follow from definition. Also supervariance (4) holds because homeomorphisms preservee topological properties. (5) is true because each covering of a compact subset made by contraible subsets admits a finite sub-covering.
Z2-bundles and cohomological index
In this section, we will follow the exposition given in [AP10].
We recall that a paracompact topological space is Hausdorff and such that every open cover has an open locally finite refinement, in particular, such spaces are normal, i.e. every couple of closed subsets has a pair of disjoint open neighborhoods. Here, every space we consider will be paracompact, so that we will omit to specify it.
Writing the group Z2 multiplicatively as {1, −1}, a Z2-space is a space X
together with a mapping µ : Z2× X → X, called a Z2-action on X, such that
µ(1, x) = x, µ(−1, x) = −x ∀ x ∈ X. The action is fixed-point free if
−x 6= x ∀ x ∈ X.
As expected, we say that a subset A of a Z2-space X is invariant if it is
symmetric, i.e. A = −A, and a map f : X → X0 between two such spaces is
equivariant if
f (−x) = −f (x) ∀ x ∈ X.
We say that two spaces X and X0 are equivalent if there exists an equivariant
homeomorphism between them, and denote by F the set of free Z2-spaces,
identifying equivalent spaces.
Example 2.1. Symmetric subsets of normed linear spaces that do not contain the origin are in F and odd maps between them are equivariant. In particular, Sn−1∈ F and the antipodal map is equivariant on it.
A principal Z2-bundle is a bundle for which every fiber consists of exactly
two antipodal points. We can define it as a triple ξ = (E, p, B) consisting of an E ∈ F, the total space, the base space B, and the bundle projection p : E → B, such that there are
1. an open covering {Uλ}λ∈Λ of B,
2. for each λ ∈ Λ, a homeomorphism φλ : Uλ× Z2→ p−1(Uλ) satisfying
φλ(b, −1) = −φλ(b, 1), pφλ(b, ±1) = b ∀ b ∈ B.
A bundle map f : ξ → ξ0 consists of an equivariant map f : E → E0 and a
E E0 B B0 f p ¯ f p0
commutes. Two bundles are equivalent if there are bundle maps from one to the other such that their composition is the identity bundle map.
We denote by PrinZ2 the set of principal Z2-bundles, identifying equivalent
bundles, and by PrinZ2B the set of principal Z2-bundles over B.
Each free Z2-space can be identified with a principal Z2-bundle as follows. Let
X = X/Z2be the quotient space of X ∈ F with each antipodal point identified
(the orbit space of X), and π : X → X the quotient map. Then P : F → PrinZ2, X 7→ (X, π, X)
is a one-to-one correspondence.
Example 2.2. P(Sn−1) = (Sn−1, π, RPn) where RPn is the projective space in dimension n and π is the projection which identifies the antipodal maps
Consider now a Z2-bundle ξ0 = (E0, p0, B0) and a map f : B → B0, such a
map automatically induces a bundle on B, called the pullback bundle, that we denote by f∗ξ0 = (f∗(E0), p, B)where
f∗(E0) = {(b, e0) ∈ B × E0 : f (b) = p0(e0)}, −(b, e0) = (b, −e0) and
p(b, e0) = b.
Hence setting ¯f : f∗(E0) → E0, (b, e0) 7→ e0, we see that f induces a bundle map from f∗(ξ0) to ξ0. Note that homotopic maps induce equivalent bundles.
We present now a construction by means of which it is possible to classify every principal Z2-bundle over a given space B.
Consider RP∞ and S∞, the infinite real projective space and infinite sphere,
direct limits of every n-dimensional projective space and every n-sphere re-spectively, for n → ∞, then note that for each ξ0 = (E0, p, B0) ∈ PrinZ
2 we
can consider the following mapping
T : [B, B0] → PrinZ2B, [f ] 7→ f
∗
ξ0
where [B, B0] are the homotopy class of maps from B to B0. Such a map
correlate every homotopy class of maps from the space B to B0 to the pullback
Z2-bundle given by such homotopy class. The interesting fact is that if we
Proposition 2.3. Setting ξ0 = (S∞, π, RP∞), called the universal principal Z2-bundle, T is a one-to-one correspondence.
A reference for the proof of this proposition is [Dol63].
Hence every principal Z2-bundle is obtainable as a pullback of the universal
bundle (S∞
, π, RP∞), and principal Z2-bundles over a given space B are
clas-sified by homotopy classes of maps from B to the infinite dimensional real projective space, the base space of the universal bundle, called the classifying space.
To sum up what we have done, for every free Z2-space X ∈ F (e.g.
symmet-ric subsets of a normed space), we are able to define a principal Z2-bundle
(X, π, X), considering the space of orbits given by the quotient X/Z2 and the
projection π, and there is a one-to-one correspondence P : F → PrinZ2.
More-over, there is a map f : X → RP∞, unique up to homotopy, and called the
classifying map, such that
T ([f ]) = P(X).
We are now in position to define the cohomological index for Z2-actions.
Let f : X → RP∞ be the classifying map of X ∈ F, and
f∗: H∗(RP∞) → H∗(X)
the induced homomorphism on singular cohomology rings with coefficients in Z2 2. We recall that H∗(RP∞) = Z2[ω], the polynomial ring generated by
ω ∈ H1(RP∞) 3, hence the Z2-cohomological index of X is defined as
ic(X) = sup{k ≥ 1 : f∗(ωk−1) 6= 0}.
Such index is well defined for the homotopy property of cohomology (homotopic maps induce the same homomorphism in cohomology) and taking ω0 = 1 ∈
H0(RP∞), f∗(ω0) = 1 ∈ H0(X)and i
c(X) ≥ 1if X 6= ∅. As always, ic(∅) = 0.
Example 2.3. Obviously, Sn−1/Z2 = RPn−1 and we can classify such a Z2 -bundle with the inclusion RPn−1 ,→ RP∞, from which we get isomorphisms
between the cohomology rings Hq for q ≤ n, hence i
c(Sn−1) = n.
Before proving that the cohomological index enjoys the properties of an index we have stated before, we need a bit of notation and facts about relative cohomology.
2this is the Alexander-Spanier cohomology. 3
in fact the Z2-cohomology groups of the n-dimensional projective space are isomorphic
to the group Z2for every i ≤ n, and this construction can be applied to the direct limit RP∞
because by definition contains each projective space of finite dimension. Then, to conclude one has to notice that the generator of the first cohomology ring is enough to build up every other cohomology ring of the graded ring H∗(RP∞) by the cup product of such generator by himself.
A neighborhood of a pair (A, B) in X consists of a couple (U, V ) in X such that U is a neighborhood of A and V is a neighborhood of B. The set Λ of the neighborhoods of a pair (A, B) is a directed set, in fact it is partially ordered by inclusion:
(Uλ, Vλ) (Uµ, Vµ) if iλµ : (Uµ, Vµ) ,→ (Uλ, Vλ)
and (Uλ∩ Uµ, Vλ∩ Vµ) ⊂ (Uλ, Vλ), (Uµ, Vµ) for each µ, λ.
Moreover, also the collection {Hq(U
λ, Vλ)}λ∈Λ of cohomology groups and the
induced maps i∗
λµ : Hq(Uλ, Vλ) → Hq(Uµ, Vµ) form a directed set. Indeed
i∗λλ= id∗(U
λ,Vλ) and i
∗
λν = (iλµiµν)∗ = i∗µνi∗λµ.
Finally, j∗
λ : Hq(Uλ, Vλ) → Hq(A, B) induced by inclusions jλ : (A, B) ,→
(Uλ, Vλ) satisfy jµ∗iλµ∗ = (iλµjµ)∗ = jλ∗, so their limit
j∗= lim−→ Λ jλ∗: lim−→ Λ Hq(Uλ, Vλ) → Hq(A, B) is defined
For a proof of the following Proposition we refer to [Spa08].
Proposition 2.4. If X is paracompact and A, B are closed, then j∗ is an isomorphism for all q
Proposition 2.5. The cohomological index ic : F → N enjoys the following
properties:
1. (Definiteness) ic(X) ≥ 0, and ic(X) = 0if and only if X = ∅.
2. (Supervariance) If f : X → Y is an equivariant map, then ic(X) ≤ ic(Y ).
3. (Monotonicity) X ⊂ Y ⇒ ic(X) ≤ ic(Y ).
4. (Subadditivity) If X ∈ F and A and B are closed invariant subsets of X such that X = A ∪ B, then
ic(A ∪ B) ≤ ic(A) + ic(B)
5. (Continuity) If X ∈ F and A is a closed invariant subset of X, then there is a closed invariant neighborhood N of A in X such that
ic(N ) = ic(A).
Moreover, when X is a metric space and A is compact, N may be chosen to be a δ-neighborhood Nδ(A).
