Linking Theorems, Kato-Ponce Inequalities, and Applications in Mathematical Physics

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Università degli Studi di Pisa

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Matematica

Linking Theorems, Kato-Ponce

Inequalities, and Applications in

Mathematical Physics

Relatore

Prof. Vladimir Georgiev

Candidato

Francesco Paolo Maiale

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Introduction

The primary goal of the thesis is the study of the existence, uniqueness, regularity, and stability of solutions for several PDEs arising in physics such as the nonlinear Schrödinger equation

     ı∂tu + ∆u ± |u|p−1u = 0 if (x, t) ∈ Rd× R, u(0, x) = u0(x) ∈ ˙H1(Rd).

The first goal is to present the notions of solitary wave and soliton as solutions of minimisation problems

min

ϕ∈HJ (ϕ),

whose orbits satisfy precise properties. Next, we show that, when the system preserves two physical quantities, solitons exist in a very general framework, provided that the functionals meet certain assumptions.

In the second chapter, we construct the topological degree in a finite-dimensional setting and show that we can extend it to Banach spaces through compact operators. The purpose is to use this topological tool to prove deformation-type lemmas, which, in turn, leads us to min-max theorems (e.g., the mountain pass or the saddle theorems.)

Definition (Linking). Let H be a Hilbert space, let N ⊂ H be a Hilbert manifold with nonempty boundary, and let C ⊂ H. We say that ∂N and C link if

C ∩ h(N ) 6= ∅

for all "admissible homotopies" h ∈ Σ.

We mainly focus on linking-type theorems, which roughly assert that if we have ∂N and C that link, then the functional J admits a critical point at the level

c := inf

h∈Σu∈Nsup

J ◦ h(u),

provided that J is "well-separated" between N and C, and satisfies certain regularity properties. In the third chapter, we first present several tools in harmonic analysis, such as the Stein-Weiss

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kM (f )(x)kLp_(Rd₎._{d, p}kf k_Lp_(Rd₎,

the Fefferman-Stein maximal inequality kM (fn)k`qn(Z) Lp_(Rd₎.d, p, q kfnk`qn(Z) Lp_(Rd₎,

and the Littlewood-Paley atomic decomposition. Next, we employ all these tools to give a detailed proof of the well-known Coifman-Meyer multiplier theorem, which asserts that the bilinear operator of symbol σ, Aσ(F, G) := Z Rd Z Rd eıx(ξ+η)σ(ξ, η) ˆF (ξ) ˆG(η) dξ dη, is bounded from Lp (Rd_)×Lq (Rd_{) to L}r

(Rd_{). We next employ these results to derive a}

Kato-Ponce-type inequality that involves the Riesz potential |D|s. Namely, we show that k|D|s_{(f g) − f |D|}s_{(g) − g|D|}s_{(f )k}

Lr_(Rd₎.r, p, q, dk|D|s1(f )kLp_(Rd₎k|D|s2(g)k_Lq_(Rd₎ (0.1) for all f, g ∈ S(Rd), 1 < p, q < ∞, s = s1+ s2> 0 and 1_r =1_p+1_q. Furthermore, we present a brief

overview of the state-of-art of this kind of inequalities, pointing out the main gaps and the reason behind them.

In the second half of the thesis, we present some examples of PDEs arising from Physics, and we apply all the tools introduced so far to develop local/global well-posedness theories or investigate the existence of a solution under certain assumptions.

• Nonlinear Schrödinger Equation. We show that, in the energy subcritical case, we can always find hylomorphic solitons, if the nonlinearity is regular enough. We next deal with the critical case using a combination of two relatively recent tools: the profile decomposition and rigidity-type theorems.

• Heat Equation. We study the critical energy case using a slightly different profile decom-position and a rigidity theorem, which follows from uniqueness results fascinating enough by themselves. The somewhat simple structure of the heat equation allows us to provide more details about the method mentioned above.

• Dirac Equation. We investigate the existence of a radially symmetric solution of the Dirac equation with nonzero mass using linking-type theorems.

• Burgers Equation. We show that Kato-Ponce-type inequalities are significant when dealing with, e.g., hyperbolic PDEs for they can provide energy estimates using only integration by parts techniques.

In the appendix, we dedicate a chapter to illustrate Clifford algebras and their relation with

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We conclude the thesis with a few counterexamples to the central estimates presented such as the Young convolution inequality, the fractional Leibniz estimate, and so on.

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Variational Methods in Nonlinear

Analysis

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Chapter 1

Solitary Waves and Solitons

I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped-not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed.

John Scott Russell

In this chapter, the primary goal is to give a formal definition of the physical phenomena known as solitons and express some existence results via variational methods. The principal reference for the material presented here is the monograph [6] written by D. Fortunato and V. Benci.

1.1 Introduction

In this first section, we collect some specific notions of solutions to PDEs, such as solitary waves and solitons, whose importance in mathematical physics is well-known.

Definition 1.1 (Dynamical System). A dynamical system is a triple (G, X, Φ), where G is a group or, more generally, a semigroup1, X is a set, and Φ is the action of G on X.

There are several definitions of dy-namical system. See, e.g., [70].

Caution!

In our discussion, X will always denote the set of the states, that is, functions u : Rd −→ V , where (V, k · kV) is a reasonable normed space. Since we are interested only in the evolution of u

over time, we will mainly consider the action of G := R over X denoted by u(t, x) := Φtu0(x),

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where u0 is the initial state. We also require the states of X to have finite energy - so that they

decay sufficiently fast at ∞ - and that

X ⊂ L1_loc_(Rd; V ).

A dynamical system (G, X, Φ) satisfying the conditions above is usually referred to as field-theory type dynamical system, or FTP from now on.

Fix a field-theory type dynamical system (G, X, Φ). The next definition generalises the notion of compactness, and we will see that it is a fundamental ingredient in the well-known concentrated compactness principle due to Lions [57,58].

Definition 1.2 (T -compact). A set Γ ⊂ X is said to be T -compact if it is compact up to translation. Remark 1.1. A set Γ is T -compact if and only if for each sequence of states {un(x)}n∈N⊂ Γ we can

find a subsequence {unk(x)}k∈N of states and a sequence of real numbers {tk}k∈N ⊂ R, converging to some t∞∈ R, such that

unk(· − tk)

k→∞

−−−−→ u(· − t∞).

We are now ready to give the exact definition of a solitary wave in the abstract setting, however it is interesting to see the physical meaning through a simple example. A solitary wave is a field, whose energy travels as a localized packet preserving this localization in time. For example,

u(t, x) := A(x − vt − x0) eı (v·x−ωt), (1.1)

is a solitary wave depending on constants x0, v ∈ Rd and ω ∈ R, provided that A ∈ L2(Rd).

Definition 1.3 (Solitary Wave). Let u ∈ X be an admissible state. The orbit of u is given by the closure of the time-evolution, that is,

O(u) = {Φtu(x) : t ∈ R} X

.

A state u ∈ X is a solitary wave if and only if its orbit O(u) is T -compact and 0 /∈ O(u).

The T -compactness property usually requires a fair amount of work to be checked, even for (1.1), but there is a particular wave - important in physics - for which the compactness is automatic.

In physics, a stationary wave is a wave whose amplitude may depend on space, but its phase remain constant. For example, the function

u(t, x) = A(x)e−ıωt (1.2)

is a stationary wave for every A ∈ L2

(Rd_{), and it is easy to verify that it is also a solitary wave since}

the closure of the orbit is compact. Example 1.1. Let X := L1

(RN_{). For any given state u ∈ X, the exponential evolution}

u(t, x) = u(etx)

is not a solitary wave because 0 belongs to its orbit. The reason is that ku(t, x)kL1_(RN₎

t→∞

−−−→ 0

since u decays fast enough (by assumption X is FTP), and thus 0 belongs to the orbit. In a similar fashion, the action

u(t, x) = etu(etx) is not a solitary wave since the orbits are not compact.

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Example 1.2. The sum of two solitary waves is not, in general, a solitary wave and, as we will see later, this is the main difference with solitons. For example, the sum of two waves of the form (1.1) with same amplitude traveling in opposite directions, i.e.,

u(t, x) = [A(x − vt) + A(x + vt)] eı(v·x−ωt),

is not a solitary wave, as the translation compactness property fails. We will come back to this property in the final section, giving a physical explanation of the aforementioned phenomena.

We mentioned earlier that the T -compactness is usually hard to prove. The next result estab-lishes an equivalence between T -compactness and the fact that a space-translation plus a compact perturbation produce the evolution of u.

Lemma 1.4. A state u ∈ X is a solitary wave if and only if 0 /∈ O(u) and there exist a compact set K ⊂ X and a vector x(t) ∈ RN such that

u(t, x) = u(x − x(t)) + w(t, x), (1.3)

where w(t, ·) ∈ K for all t ∈ R.

Proof. A straightforward application of the definitions proves the result, but it is interesting to notice that x(t) is given by −θ(Φtu), while K is given by

T−θ(Φtu)Kθ− T−θ(Φtu)u, where T is the translation operator and θ a function such that

Kθ= {Tθ(z)z : z ∈ O(u)}

is compact.

Definition 1.5 (Invariant Set). A set Γ ⊂ X is invariant if and only if u(t, ·) ∈ Γ _{for all t ∈ R and all u ∈ Γ.}

Definition 1.6 (Stable Set). Let (X, d) be a FTP dynamical system with a metric structure. A set Γ ⊂ X is stable if and only if for all > 0 we can find δ := δ() > 0 such that

d(u, Γ) ≤ δ =⇒ d(u(t, ·), Γ) ≤ for all t > 0 and for all u ∈ Γ.

Definition 1.7 (Soliton). Let X be a FTP dynamical system. A state u ∈ X is a soliton if and only if it belongs to a set Γ ⊂ X satisfying the following properties:

(S1) The set Γ is both invariant and stable. (S2) The set Γ is T -compact and 0 /∈ Γ.

The set Γ is usually referred to as soliton manifold, despite the fact that it is not always an actual manifold.

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Remark 1.2. A soliton u ∈ Γ is necessarily a solitary wave, but the opposite "arrow" is generally false. Indeed, it is easy to notice that

O(u) ⊂ Γ,

and this inclusion is closed, which means that O(u) is also translation compact.

Remark 1.3. Let u be a solitary wave, and set Γ := O(u). The set Γ is obviously invariant since it contains the closure of the time-evolution, and the property (S2) is satisfied by definition. The orbit is also stable, so it follows that

u solitary wave =⇒ O(u) is a soliton manifold and u is a soliton.

Note, however, that the notion of soliton is a more general since every u ∈ Γ is a solitary wave and, in general, we have Γ ⊃ O(u).

1.1.1 Solitons and Group Representations

Let G be a continuous group, whose elements A(α) are expressed as a function of a set of continuous real-valued2 _{parameters {α}}

α∈∆ = {(α1, α2, . . . , αk)}α∈∆. The parameters are chosen in such a

way that the identity element of G is given by

A(0, . . . , 0) = e.

Definition 1.8 (Lie Group). A Lie group G is a continuous group, of parameter α, satisfying the following properties:

(a) Closure. For every α and β it turns out that

A(α)A(β) = A(γ),

where γ = f (α, β) and f is a differentiable function with respect to both variables such that f (γ, 0) = γ and f (0, γ) = γ.

(b) Inverse. For every α it turns out that

A(α)−1= A(α0), where the function α 7−→ α0 is differentiable.

(c) Associativity. For every α, β and γ it turns out that

A(α) (A(β)A(γ)) = (A(α)A(β)) A(γ),

Remark 1.4. Let G be a Lie group. The associative property immediately implies that f (α, f (β, γ)) = f (f (α, β), γ) for every α, β, γ.

Remark 1.5. There is an equivalent definition of real Lie group, which relies on the theory of smooth manifolds. We say that a group G is a Lie group if and only if G has a smooth manifold structure such that

G × G 3 (x, y) 7−→ xy ∈ G,

G 3 x 7−→ x−1∈ G are both smooth functions.

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Definition 1.9 (Action Invariant). Let (G, X, Φ) be an FTP dynamical system, and let H be a Lie group contained in Hom(X). We say that (G, X, Φ) is invariant under the action of H if

h · u(t, x) = h(u)(t, x) for all u ∈ X and all h ∈ H.

Proposition 1.10. Let (G, X, Φ) be an FTP dynamical system invariant under the action of a Lie group H. If u ∈ X is a soliton, then h · u is also a soliton for all h ∈ H.

Proof. Let Γ be the soliton manifold associated to u. Then h · Γ := {h · v : v ∈ Γ} is also a soliton manifold, and therefore h · u is a soliton.

This property is fundamental in mathematical physics because we usually deal with dynamical systems induced by the Euler-Lagrange equations associated to a Lagrangian L, which is usually invariant under a representation {Tg}g∈G of a group3.

Corollary 1.11. Let S be the family of all orbits, that is, S_{:= {u(t, x) : t ∈ R, u ∈ X} ,} and let ρ be the mapping defined by

ρ(g)(u) := [Tg(u(t, x))]_t=0.

Then ρ(g)u is a soliton for all g ∈ G and all soliton u ∈ X.

1.2 Existence of Hylomorphic Solitons

In this section, we introduce the notion of hylomorphic solitons, and we develop an abstract frame-work that allows us to prove both the existence and uniqueness under suitable assumptions.

1.2.1 Hylomorphic Solitons

Let (G, X, Φ) be an FTP dynamical system, and suppose that the energy E and the hylenic charge C are constant of motions4_.

Definition 1.12 (Hylomorphic Soliton). A soliton u ∈ X is hylomorphic if and only if the soliton manifold Γ is given by

Γ(e0, c0) := {u ∈ X : E(u) = e0, |C(u)| = c0} ,

where

e0= min{E(u) : |C(u)| = c0}.

3_{Here G is usually the Galileo group, the Poincaré group, the Lorentz group, SU(k, C), or SO(k, R).}

4_{In the abstract framework "energy" and "hylenic charge" simply denote two functionals, but in practice these are} quantities with a specific physical meaning!

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Notice that, as a consequence of the definition of e0, a hylomorphic soliton u is (by Lagrange

multiplier theorem) a solution of the nonlinear eigenvalue problem

E0(u) = λC0(u). (1.4)

Obviously, not every solution to (1.4) is a hylomorphic soliton, but these are important enough to deserve a specific name in the literature.

Definition 1.13 (Ground State). A solution u of (1.4) is called a ground state solution if it minimizes the energy E(u) on the setMc0 := {v ∈ X : |C(v)| = c0}.

Remark 1.6. A ground state u is not, in general, a soliton. The set Γ given above is obviously stable, but we need to prove that:

(G1) The set Γ(e0, c0) is stable.

(G2) The set Γ(e0, c0) is T -compact and 0 /∈ Γ(e0, c0).

The first property is, in general, delicate to prove. However, if it is the only one that does not hold true, then u is a solitary wave but not a soliton.

Remark 1.7. Suppose that u is a hylomorphic soliton, and suppose that the dynamical system is induced by a Lagrangian L invariant under the action of a group G. We known that

ρ(g)u is a soliton for all g ∈ G,

but it is not hard to see that, in general, these are not hylomorphic solitons since the energy changes, that is,

E(ρ(g)u) > E(u), and thus ρ(g)u /∈ Γ(e0, c0).

Definition 1.14 (Traveling Hylomorphic Soliton). A solitary wave u ∈ X is a traveling hylomorphic soliton if and only if u = ρ(g)u0and u0 is a hylomorphic soliton.

1.2.2 Existence Results

The first problem that arises in the abstract framework is to study the minimisation of the energy E(·) onMc, and also under which conditions the resulting set Γ(e, c) is stable.

We first recall some basic notions concerning functionals. A sequence (un)n∈N⊂ X is said to be

G-compact if and only if there exist (gk)k∈N ⊂ G and a subsequence (nk)k∈N such that

gkunk

k→∞

−−−−→ ¯u ∈ X.

Definition 1.15 (G-Compact). A G-invariant functional J is said to be G-compact if all its mini-mizing sequences are G-compact.

Definition 1.16 (Splitting). A functional J on X has the splitting property if and only if J (un) = J (u) + J (wn) + o(1)

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Definition 1.17 (Vanishing). A sequence (un)n∈N⊂ X is G-vanishing if and only if

gnun * 0

for all sequences (gn)n∈N⊂ G, where G is a unitary group acting on X.

Note that sequences converging strongly to 0 are vanishing, but this is false if we assume that un converges to 0 weakly. For example, given a solitary wave u0∈ X, the sequence

Φtnu0 is non-vanishing for all (tn)n∈N⊂ R converging to +∞.

Definition 1.18 (Hylomorphic Condition). The hylenic ratio of the abstract problem is defined as

Γ(u) = E(u) |C(u)|. We say that the hylomorphic condition is satisfied if and only if

inf u∈X E(u) |C(u)| < Λ0, where Λ0:= inf n lim inf

n→∞ Γ(un) : un is a vanishing sequence

o .

We are now ready to give the statements of the existence results, provided that X is an FTP dynamical system and G is the group of translations.

Theorem 1.19. Assume that E and C satisfy the following assumptions:

(a) The functionals E and C are of class C1and map bounded sets into bounded sets. In addition, we have E(0) = C(0) = 0 and E0(0) = C0(0) = 0.

(b) The functionals E and C are G-invariant.

(c) The functionals E and C satisfy the splitting property. (d) The energy is positive (E(u) > 0) and it turns out that

kunk n→∞ −−−−→ ∞ =⇒ E(un) n→∞ −−−−→ ∞, E(un) n→∞ −−−−→ 0 =⇒ kunk n→∞ −−−−→ 0.

Moreover, assume that the hylomorphic condition is satisfied at all u ∈ X. Then there exists a collection of hylomorphic solitons.

Theorem 1.20. Assume that E and C satisfy the following assumptions:

(a) The functionals E and C are of class C1and map bounded sets into bounded sets. In addition, we have E(0) = C(0) = 0 and E0(0) = C0(0) = 0.

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(c) The functionals E and C satisfy the splitting property. (d) The energy is positive (E(u) > 0) and it turns out that

kunk n→∞ −−−−→ ∞ =⇒ E(un) n→∞ −−−−→ ∞, E(un) n→∞ −−−−→ 0 =⇒ kunk n→∞ −−−−→ 0.

In addition, assume that the hylomorphic condition is satisfied at all u ∈ X and kE0(u)k + kC0(u)k = 0 ⇐⇒ u = 0.

Then for all δ ∈ (0, δ∞) there exists a hylomorphic soliton uδ. Furthermore, if δ1 < δ2, the

corre-sponding solitons are distinct and we have that Λ(uδ1) < Λ(uδ2) and |C(uδ1)| > |C(uδ2)|.

The proofs of this results are very complicated and require plenty of technical results. The interested reader is encouraged to consult [6] for a complete discussion (containing the case of positive charge that requires different assumptions.)

1.2.3 Stability Result

We conclude the section by stating a stability result that is used to prove the theorems above. We mention it also because it shows the connection between variational methods and the stability of Γ. Theorem 1.21. Let (X, Φ) be a dynamical system and let Γ be an invariant set. Suppose that there exists a Lyapunov function defined on an open neighborhood of Γ satisfying

V (u) ≥ 0 and V (u) = 0 ⇐⇒ u ∈ Γ,

∂tV (Φtu) ≤ 0,

V (un) n→∞

−−−−→ 0 ⇐⇒ d(un, Γ) → 0.

Then the set Γ is stable.

This result is particularly crucial for mathematical physics since it allows us to determine the orbital stability of a solution using a Lyapunov function.

In particular, given a solution u(t, x) := Φtu0(x) of an initial value problem, we have that O(u0)

is stable if we can find a function V , defined on a neighborhood (w.r.t. a norm k · k) of the orbit, satisfying the following properties:

V (u) ≥ 0 and V (u0(t, x)) = 0, ∂tV (Φtu) ≤ 0, V (un) n→∞ −−−−→ 0 ⇐⇒ inf v∈O(u0) kun− vk n→∞ −−−−→ 0.

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1.3 Physical Interpretation of Solitary Waves

The goal of this final section is to give a brief overview of the history of solitary waves in physics, trying to give more concrete meaning to the notions introduced so far.

The most familiar example of waves is water waves. For example, the ones at the beach produced by cliffs or the wind, the ones created by a stone thrown at the water or by a ship departing. Surprisingly, despite the similarities, these waves belong to entirely different kinds.

Stokes (19th century) was one of the pioneers of the fluid dynamics. Indeed, he derived the equations describing the motion of a fluid (inviscid and incompressible) subject to the gravity force and bounded from below by an impermeable means, and from above by a free surface.

As mentioned in the introduction, the first reported observation of solitary waves in shallow water is due to Russel, who also tried to recreate them in his laboratory using a water tank (see, for example, the biography [27]). In the 1985, D. Korteweg and his doctoral student G. de Vries derived a partial differential equations (now known as KdV) that acted as a model of the solitary waves, whose modern form is given by

ut(t, x) + uxxx(t, x) − 6u(t, x)ux(t, x) = 0.

However, solitary waves were commonly considered a mere, yet exciting, curiosity in the field of nonlinear waves. This perspective changed altogether in 1965 when Zabusky and Kruskal proved that the KdV equation is the limit (in the continuum) of the one-dimensional anharmonic lattice, which was introduced by Fermi, Pasta, and Ulam to investigate the distribution of the energy on the possible oscillation of the lattice itself.

Furthermore, Zabusky and Kruskal developed a simulation of the collision of two solitary waves in a nonlinear crystal lattice. They found out that, after the collision, the waves preserved both the shape and the velocity, but the phases shifted (advancing the faster one and retarding the slower one). The term "soliton" was later introduced to refer to this class of solitary waves that is stable under "elastic" collisions.

In the next few decades, equations arising in mathematical physics admitting solitary wave solutions have been investigated thoroughly, with a particular emphasis on the completely integrable models (i.e., the ones that can be solved using the inverse scattering transform.

1.3.1 Example of Water Waves

Let us consider a solitary wave that propagates with speech c along the x-axis. The amplitude function (in the frame of reference moving at velocity cg) is a function A(ξ), where

ξ = x − ct.

We assume that the function A has a global maximum at ξ = 0 and decays fast enough on the tail ξ → ±∞. We note that it makes sense to look for solutions proportional to

Re eıkξ ,

in a neighbourhood of the tail, where k ∈ C is the wavenumber associated to the wave. It is interesting to notice that k cannot be real because the amplitude decays at infinity, and this gives a physical explanation of the fact that solitary waves with real phase moving at speed c exist only for a limited range of values of k of the spectrum, which is usually called gap.

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Remark 1.8. Recall that to find a relation between c and k it is enough to linearise the equation that describes the physical phenomenon, which yields to the so-called dispersive relation.

Example 1.3. The dispersive relation associated to water waves of a certain kind is given by c2 hg = f (q) := 1 + Bq2 q tanh q, (1.5)

where h is the constant depth of the unperturbed water, q = kh the dimensionless variable, and B := σ

ρgh2

Here σ denotes the superficial tension constant and ρ the density of the water.

N.B.

the Bond number . A necessary condition for solitary waves to exist is that no real k is a solution, with q = kh, of the equation (1.5). In particular, we have that:

(a) If B = 0, then a necessary condition for solitary waves to exist is that c2

hg > 1.

(b) If B > 1₃, then a necessary condition for solitary waves to exist is that c2

hg < 1.

(c) If B ∈ (0, 1₃), then there exists cm> 0 such that (1.5) has a real solution for all c such that

|c| > cm. In particular, a necessary condition for solitary waves to exist is that

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Chapter 2

Topological Methods in Calculus of

Variations

In this chapter, we aim to present an application of topological tools in calculus of variations and, more specifically, in PDEs analysis. The idea is to introduce a function

deg : Rm× C0(Ω ⊂ Rn; Rm) × Pob(Rn) −→ Z

that maps a triplet (p, f, Ω), where p /_{∈ f (∂Ω) and Ω is an open bounded subset of R}n_{, to an integer}

number that represents, in some sense, the number of solutions to the equation f (x) = p for x ∈ Ω.

Note that the topo-logical degree deg will often give us in-formations useful to distinguish whether there are infinite so-lutions, at least one solution or no solu-tion at all.

Caution

The most important property, though, is the invariance under homotopy, which asserts that deg(p, H(0, ·), Ω) = deg(p, H(1, ·), Ω),

provided that certain assumptions are satisfied (i.e., that H is an admissible homotopy.) Next, we will show that the topological degree provides us easy proofs for the saddle-point theorem and the mountain pass theorem, both of which have important applications in PDEs analysis.

In conclusion, we introduce the notion of linking and present the abstract framework for Hilbert spaces. Linking theorems are nothing but a generalisation of the mountain pass theorem, and allows us to deal with unbounded (from above and below) functionals.

2.1 Topological Degree Theory in Finite-Dimensional Spaces

In this section, we construct the so-called topological degree for continuous maps defined on a bounded open set Ω ⊂ Rn. The idea is to define it for functions of class C2, and then extend it to continuous functions via uniform density, that is,

deg(p, f, Ω) = lim

k→∞deg(p, fk, Ω),

where fk is a sequence of C2 functions such that kfk− f k∞ → 0. The approach presented here

is slightly technical, but, luckily, most of the properties of the topological degree (included the homotopy-invariance) are straightforward to prove.

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In [2, Chapter 2] there is an analytic approach to the topological degree, which is easier to prove to be well-defined, but yields to slightly more technical proofs for the properties mentioned above.

2.1.1 Topological Degree for C

2

_{-Regular Maps}

Let Ω ⊂ Rn be an open bounded set, and let I ⊃ [0, 1] be an interval. Consider a C2 function F : I × Ω −→ Rn,

and set Φ0(x) := F (0, x) and Φ1(x) := F (1, x). We first recall some basic notions.

Definition 2.1. Let Φ ∈ C1_{(Ω ⊂ R}n_{; R}m) be a differentiable map. We say that x ∈ Ω is a critical point for Φ if the differential at x, denoted by dΦ(x), is not surjective. Moreover, if we define

ZΦ:= {x ∈ Ω | dΦ(x) is not surjective } , (2.1)

then we say that y ∈ Rmis a critical value if

Φ−1(y) ∩ Zf 6= ∅.

Remark 2.1. Let Φ ∈ C1

(Ω ⊂ Rn

; Rm_{). A point x ∈ Ω is critical if and only if the determinant of}

the Jacobian of Φ at x equals zero, that is,

x ∈ ZΦ ⇐⇒ |JΦ(x)| = 0.

Definition 2.2 (Proper Map). Let X and Y be metric spaces. A continuous map Φ : A ⊂ X −→ Y is proper if, for any sequence (xn)n∈N⊂ A such that

dY (Φ(xn), y) n→∞

−−−−→ 0

for some y ∈ Y , there exists a converging subsequence (xnk)k∈N in X.

Lemma 2.3. Let Ω ⊆ Rn be an open subset, and let Φ : Ω −→ Rn be a proper map of class C1(Ω) ∩ C0 Ω .

If y is a regular value that does not belong to the image of the border of Ω, then it turns out that:

(1) The cardinality of the fiber is finite, i.e., #Φ−1_{(y) < ∞.}

(2) If y ∈ Φ(Ω) and Φ−1(y) = {x1, . . . , xk}, then there is a neighbourhood V of y and there are

neighbourhoods U1, . . . , Uk of x1, . . . , xk such that

U1, . . . , Uk⊂ Ω and V ∩ Φ (∂Ω ∪ Zf) = ∅.

Moreover, the restriction Φ_U

i : Ui−→ V is a diffeomorphism for all i = 1, . . . , k and Φ−1(V ) ⊂ k [ i=1 Ui. Proof.

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(a) The set Φ−1(y) is compact (because Φ is proper and {y} is compact) and discrete (as a consequence of the local invertibility theorem), and therefore finite.

(b) Let Φ−1(y) := {x1, . . . , xk}. The local invertibility theorem asserts that we can always find

U1, . . . , Uk open neighbourhoods of x1, . . . , xk and V1, . . . , Vk of y such that

Φ_U

i : Ui −→ Vi

is a diffeomorphism for all i = 1, . . . , k. We claim that there exists a neighbourhood V of y such that Φ−1(V ) ⊆ k [ i=1 Ui.

We argue by contradiction. Suppose that there exists sequence (yn)n∈N⊂ Y converging to y

and a sequence (zn)n∈N⊂ Ω satisfying Φ(zn) = yn and

zn ∈/ k

[

i=1

Ui

for all n ≥ N . Since Φ is proper we can always find a subsequence (znk)k∈Nsuch that znk→ x. The function Φ is continuous, and thus Φ(x) = y, which is the desired absurd since

Φ−1(y) = {x1, . . . , xk} ⊂ k

[

i=1

Ui.

In conclusion, the reader may check that

Ui 7→ Ui0:= Ui∩ Φ−1(V )

are the requested neighbourhoods satisfying all the properties.

Lemma 2.4 (Sard’s Lemma). Let Φ : Ω ⊆ Rn

−→ Rm _{be a function of class C}k_{(Ω), where}

k ≥ max{n − m + 1, 1}. Then the Lebesgue measure of the set Φ(ZΦ) is zero.

Note that Sard’s Lemma is the reason why we need to de-fine the topological degree for C2 func-tions. Indeed, if F is defined on I × Ω with values in Rn, then F (ZF) is a null-set if k ≥ 2. Caution!

Remark 2.2. The required regularity cannot be lowered unless we add some extra assumptions on the function f . We refer the reader to [5] for the sharp statement.

Lemma 2.5. Let F be as above, and let y ∈ Rn _{be a point such that}

y ∈ F I × Ω \ F (ZF ∪ (I × ∂Ω)) ,

and assume that y /∈ Φ0(ZΦ0) ∪ Φ1(ZΦ1). Then the following assertions hold: (a) The subset F−1_{(y) ⊆ Ω is a 1-manifold of class C}2 _{with no border.}

(b) If (0, x) ∈ F−1(y), then the tangent line to F−1_{(y) at (0, x) does not lie on the border {0}×R}n_.

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(c) Let Γ be a connected component of F−1(y) ∩ ([0, 1] × Ω). If Γ ∩ ({0, 1} × Ω) 6= ∅,

then there exists a curve γ : [a, b] −→ [0, 1] × Ω of class C2_{, such that α([a, b]) = Γ, α}0_{(τ ) 6= 0}

for all τ ∈ [a, b] and

{α(a), α(b)} = Γ ∩ ({0, 1} × Ω) .

(d) If α is the curve given in the previous point, then α0(a) and α0_{(b) do not belong to {0, 1} × R}n_.

Lemma 2.6. Let F be as above, and let y ∈ Rn be a point such that y ∈ F I × Ω \ F (ZF ∪ (I × ∂Ω)) ,

and assume that y /∈ Φ0(ZΦ0) ∪ Φ1(ZΦ1). Then the cardinality of the fiber is equal modulo 2, that is,

# Φ−1₀ (y) = # Φ−1₁ (y) (mod. 2). (2.2)

Proof. It is a straightforward corollary ofLemma 2.5. In fact, each connected component Γ whose intersection with the boundary is nontrivial is diffeomorphic to a compact interval:

Γ ∼= [a, b] =⇒ |∂ Γ| = 2. Since each Γ contributes either with

• two points to {0} × Ω; or • two points to {1} × Ω; or

• one point to {0} × Ω and one point to {1} × Ω.

Definition 2.7 (Topological Degree). Let Ω ⊂ Rn

be an open and bounded set, and let Φ : Ω −→ Rn

be a function of class C2_{. The topological degree modulo 2 at y /}_{∈ Φ(∂Ω) is defined as}

deg₂(y, Φ, Ω) :=     

1 if odd cardinality and y /∈ Φ(ZΦ),

0 if even cardinality and y /∈ Φ(ZΦ),

limk→∞deg2(yk, Φ, Ω) if y ∈ Φ(ZΦ),

where yk is a sequence of regular points converging to y, which exists by Sard’s Lemma.

Remark 2.3. The topological degree modulo 2 is a function deg₂_{(y, Φ, Ω) : R}n\ Φ(∂Ω) −→ {0, 1} continuous and well-defined. Moreover, if y /∈ Φ(∂Ω), then

deg₂(y, Φ, Ω) = 1 =⇒ y ∈ Int Φ(Ω).

Proof. We argue by contradiction. If y /∈ Φ(Ω), then y /∈ Φ Ω and this clearly implies that y is a regular value whose topological degree is equal to 0. On the other hand, if y ∈ Φ(Ω), then there exists a neighbourhood U 3 y such that, for any y0∈ U , we have y0_{∈ Φ (∂ Ω) and deg}_/

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Figure 2.1: The idea behind the proof ofLemma 2.6.

2.1.2 Topological Degree for C

0

_{-Regular Maps}

In this section, we denote by Ω an open subset of Rn

and by Φ : Ω −→ Rn _{a continuous map, unless}

we need it to be more regular (in which case, we will specify it).

Definition 2.8. Let y ∈ Rn _{be any point. The set of all the continuous maps with the property}

that y does not belong to the image of the border of Ω is denoted by Fy=Φ ∈ C0 Ω; Rn | y /∈ Φ (∂Ω) .

In a similar fashion, for all k ∈ N, the subset of all the functions of class Ck is denoted by Fk

y =Φ ∈ C 0

Ω; Rn | y /∈ Φ (∂Ω) ∩ Ck

(Ω; Rn) .

Remark 2.4. The set Fy is a normed space endowed with k · k∞, and the inclusion Fyk ⊂ Fy is

dense with respect to the uniform norm.

Proof. We claim that, given d(y, Φ(∂ Ω)) := δ > 0, we have

B Φ, δ 2 ⊂ Fy.

Indeed, if Ψ ∈ Fy is a function arbitrarily near Φ, that is, a function such that kΦ − Ψk∞ < δ/2,

then it follows from the reverse triangular inequality that

|Ψ(x) − y| ≥ δ −δ 2 > 0.

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Lemma 2.9. Fix y ∈ Rn. The map F2

y 3 Φ 7−→ deg2(y, Φ, Ω) ∈ {0, 1}

is continuous, and thus locally constant.

Corollary 2.10. If C is a connected component of Fy, then deg2(·, Φ, Ω) is a constant function on

the whole intersection

C ∩ C2

Ω, Rn = C ∩ Fy2.

Definition 2.11 (Topological Degree). Let Φ ∈ C0

Ω; Rn_{, and let y ∈ R}n_{\Φ(∂Ω). The topological}

degree modulo 2 of Φ is given by

deg2(y, Φ, Ω) := lim

k→+∞deg2(y, Ψk, Ω),

where Ψk is a sequence of C2 maps converging to Ψ with respect to the uniform norm satisfying the

property y /∈ Ψk(∂Ω) definitively (i.e., for all k ≥ N .)

2.1.3 Properties of the Topological Degree Modulo 2

In this section, we state and prove some of the main properties of the topological degree modulo 2, which, somehow, explain the reason why we did so much work to introduce this (powerful) tool. Lemma 2.12 (Homotopy Invariance). Let H : [0, 1] × Ω −→ Rn _{be an admissible homotopy, that}

is, assume that y /∈ H ([0, 1] × ∂Ω). Then

deg₂(y, H(0, ·), Ω) = deg₂(y, H(1, ·), Ω) .

Proof. The curve

[0, 1] 3 t 7−→ H(t, ·) ∈ Fy

is well-defined since, by assumption, it does not intersect the image of the border of Ω. We conclude by applyingCorollary 2.10to H(0, ·) and H(1, ·).

Lemma 2.13 (Solution Property). Let Φ ∈ C0 _{Ω; R}n. Suppose that deg₂(y, Φ, Ω) 6= 0.

Then the equation Φ(x) = y admits (at least) one solution, that is, y ∈ Φ(Ω).

Proof. We argue by contradiction. If y /∈ Φ(Ω), then y /∈ Φ Ω, which means that we can find a positive constant ρ > 0 such that

B(y, 2ρ) ∩ Φ Ω_{= ∅.}

By density we can find a function Ψ ∈ C2 _{Ω; R}n, in the same connected component of Φ, satisfying kΨ − Φk∞< ρ.

Recall now that in Remark 2.3 we have already proved that the solution property holds for a C2

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Proposition 2.14. Let Φ, Ψ ∈ C0 _{Ω; R}n, and let y ∈ Rn\ Φ(∂Ω) = Ψ(∂Ω). Assume that Φ_∂Ω≡ Ψ

_∂Ω.

Then the topological degree modulo 2 depends only on the value on the border, that is, deg₂(y, Φ, Ω) = deg₂(y, Ψ, Ω) .

Proof. The reader can easily check that the map

H(t, x) = tΦ(x) + (1 − t)Ψ(x) is an admissible homotopy since H(t, x) = Φ(x) for all x ∈ ∂Ω.

2.1.4 Brouwer Topological Degree

The topological degree modulo 2 is a useful tool in Topology, but the problem is that it is not good enough to count, in some sense, the number of solutions of an equation for it can only distinguish between even and odd.

Example 2.1. Let id : Sn−1 −→ Sn−1 _{be the identity map, and let ı : S}n−1 _{−→ S}n−1 _{be the}

antipodal map defined by

ı(x) := −x,

where Sn−1⊂ Rn_{is the (n−1)-dimensional sphere. We ask whether or not there exists an admissible}

homotopy H : [0, 1] × Sn−1−→ Sn−1 _{connecting these two maps, that is,}

H(0, ·) = id(·) and H(1, ·) = ı(·).

The answer is yes if and only if n is an even natural number, but the topological degree modulo 2 is not enough to prove our claim for it cannot distinguish 1 from −1. This is roughly the reason why we need to introduce a more powerful tool, which is known as (Brouwer) topological degree.

Let Ω ⊂ Rn be a bounded open subset, let Φ ∈ C1 _{Ω; R}n, and let y ∈ Rn\ Φ (∂Ω ∪ ZΦ). The

leading goal of this section is to study the value of the sum X

x∈f−1_(y)

sgn |JΦ(x)| . (2.3)

Lemma 2.15 (Dog on Leash). Let v : [a, b] −→ R × Rn be a path of class C2([a, b]) satisfying v(τ ) 6= 0 for all τ ∈ [a, b].

Assume that Ba := {v(a), z1, . . . , zn} is a basis of Rn+1= R × Rn. Then there are n differentiable

paths β1, . . . , βn : [a, b] −→ R × Rn such that βi(a) = zi, and

Bτ := {v(τ ), β1(τ ), . . . , βn(τ )}

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Lemma 2.16. Let I ⊃ [0, 1] be an open interval of R, let F : I × Ω −→ Rn be a map of class C1, and let α : [a, b] −→ I × Ω be a curve of class C1 such that

α0(τ ) 6= 0, F ◦ α(τ ) = c and α(τ ) /∈ ZF, ∀ τ ∈ I.

Let z1, . . . , zn∈ R × Rn be vectors such that Ba:= {α0(a), z1, . . . , zn} is a basis of Rn+1. Then the

following properties hold true:

(a) There are vectors z0₁, . . . , z0_n∈ R × Rn _{such that B}

b := {α0(b), z10, . . . , z0n} is a basis, with the

same orientation as Ba.

(b) If F1, . . . , Fn are the components of F with respect to the canonical base, then the determinant

of the (n × n)-minor

(F_i0(α(a))(zj))_{i, j=1, ..., n}

is nonzero, and it has the same sign as of the determinant of the (n × n)-minor F_i0(α(b))(z_j0)

i, j=1, ..., n.

(c) If α0(b) /∈ {0} × Rn_{, then we may choose the z}0

is in {0} × Rn.

We are now ready to state the main result of the section that allows us to show that the topological degree is a well-defined notion. Let I ⊃ [0, 1] and F ∈ C2 _{I × Ω; R}n, and set

Φ0(x) := F (0, x) and Φ1(x) := F (1, x).

Furthermore, let us pick a point y ∈ Rnsuch that y /∈ F (I × ∂Ω ∪ ZF) and y /∈ Φ0(ZΦ0) ∪ Φ1(ZΦ1). Lemma 2.17. In this setting, it turns out that

X x∈Φ−1₀ (y) sgn|JΦ0(x)| = X x∈Φ−1₁ (y) sgn|JΦ1(x)|,

where |JΦi(x)| denotes the determinant of the Jacobian of Φi at x ∈ Ω.

Proof. We have proved already (seeLemma 2.5) that

Y := F−1(y) ∩ ([0, 1] × ∂ Ω)

is a smooth 1-manifold, with a finite number of connected components. Let Γ be one of them, and assume that it intersects {0, 1} × Ω. Then Γ is diffeomorphic to an interval, that is, there exists a diffeomorphism

γ : [a, b]−∼→ [0, 1] such that γ ([a, b]) = Γ, γ0(τ ) 6= 0 for all τ ∈ [a, b], and

Γ ∩ ({0, 1} × Ω) = {α(a), α(b)} . There are now only two possible outlines to discuss:

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(1) Suppose that α(a) = (0, xa) and α(b) = (0, xb). We claim that

sgn|JΦ0(xa)| + sgn|JΦ0(xb)| = 0.

Let us set z1 := (0, e1), . . . , zn := (0, en). Since γ0(τ ) is non-vanishing in [a, b], and it does

not intersect {0} × Ω (see Lemma 2.5), then

B = {γ0(a), z1, . . . , zn}

is a basis of R × Rn_{. By}_{Lemma 2.16}_{there are vectors z}0

1, . . . , zn0 ∈ {0} × Rn such that

B0_{= {γ}0_{(b), z}0

1, . . . , zn0}

is also a basis of R × Rn_{, with the same orientation as B, such that the determinant of}

(F_i0(γ(a))(zj))_{i, j=1, ..., n}

is nonzero and it has the same sign of the determinant of F_i0(γ(b))(z0_j)_{i, j=1, ..., n}. Since F_i0(γ(a)(zj) = Φ00, i(xa)(ej), it turns out that

det [F_i0(γ(a))(zj)]_{i, j=1, ..., n}= |JΦ0(xa)|.

If we set z_i0:= (0, e0_i) for i = 1, . . . , n, then it is easy to prove that F_i0(γ(b)(z_j0) = Φ0_{0, i}(xb)(e0j),

and thus

detF0

i(γ(b))(zj0)

i, j=1, ..., n= ±|JΦ1(xb)|.

To find the right sign, recall that B and B0 have the same orientation. On the other hand, since α0(a) and α0(b) have opposite orientations, then {z1, . . . , zn} and {z10, . . . , zn0} necessarily have

the opposite orientations, and thus sgn|JΦ0(xa)| = sgndetF

0

i(γ(b))(zj0)

i, j=1, ..., n= −sgn|JΦ1(xb)|. (2) Suppose that α(a) = (0, xa) and α(b) = (1, xb). We claim that

sgn|JΦ0(xa)| = sgn|JΦ1(xb)|.

The proof of this assertion is left to the reader. It suffices to mimic the proof of the previous case, employing the obvious fact that here α0(a) and α0(b) have the same orientation.

Definition 2.18 (Topological Degree). Let Ω ⊂ Rn be an open bounded set, and let Φ : Ω −→ Rn be a function of class C2. The topological degree at y /∈ Φ (∂Ω ∪ ZΦ) is defined by setting

deg (y, Φ, Ω) := X

x∈F−1_(y)

sgn |JΦ(x)| .

We will now show that the topological degree is constant on all connected components of Rn_\

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Proposition 2.19. Let y0, y1 ∈ C ⊆ Rn\ Φ(∂Ω), where C is a connected component, and suppose

that y0, y1∈ Φ (Z/ Φ). Then

deg (y0, Φ, Ω) = deg (y1, Φ, Ω) .

Lemma 2.20. The map

F2

y 3 Φ 7−→ deg2(y, Φ, Ω) ∈ Z

is continuous, and thus locally constant on Fy.

Definition 2.21 (Topological Degree). Let Φ ∈ C0 _{Ω; R}n, let y ∈ Rn\ Φ(∂Ω). The topological degree of Φ at y is defined as follows:

deg(y, Φ, Ω) := lim

k→+∞m→+∞lim deg(ym, Φk, Ω),

where ymis a sequence of regular points converging to y and Φka sequence of C2functions converging

to Φ in the uniform norm.

Remark 2.5. The main properties of the topological degree modulo 2 (see Subsection 2.1.3) can be proved in a similar way for deg(y, Φ, Ω).

We are finally ready to prove that the identity and the antipodal map on the (n − 1)-dimensional sphere are homotopic if and only if n is even as a consequence of the next result.

Theorem 2.22. Let Ω ⊂ Rn _{be an open bounded set, and let H : [0, 1] × ∂Ω −→ ∂Ω be a homotopy}

between id∂Ω: ∂Ω 3 x 7−→ x ∈ ∂Ω and ı _∂Ω: ∂Ω 3 x 7−→ −x ∈ ∂Ω. If n is odd, then Ω ∪ (−Ω) ⊂ H([0, 1] × ∂Ω), and therefore such a homotopy cannot exist.

2.2 Leray-Schauder Topological Degree

In this section, we define, following [2], the Leray-Schauder topological degree via a limit process involving the Brouwer topological degree introduced previously.

Definition 2.23 (Compact Operator). Let X and Y be Banach spaces. An operator - not necessarily linear - T : X −→ Y is compact if and only if it satisfies the following properties:

(a) The operator T is continuous1.

(b) The image T (E) of every bounded subset E ⊆ X is relatively compact.

Definition 2.24 (Perturbation). Let Ω be an open bounded subset of a Banach space X. We say that an operator S : Ω −→ X is a compact perturbation of the identity if and only if

S = idΩ− T,

where T is a compact operator.

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The reason why we introduce compact perturbations of the identity is that we can always write a compact operator as the limit - with respect to the operator norm - of a sequence of nonlinear finite-rank operators (see [13]), that is,

T ∈ Lc(Ω, X) =⇒ lim

k→+∞kT − TkkL= 0,

where Tk(Ω) ⊂ Ek and dim Ek < ∞. In Hilbert spaces, we can say something more precise:

Lemma 2.25. Let X be a Hilbert space. Then the closure of the finite-rank operators is the ideal of the compact operators, that is,

Lc(X) = Lf(X).

Proof. Any finite-rank operator is compact, and the ideal Lc(X) is closed; hence there is a

straight-forward inclusion

Lc(X) ⊇ Lf(X).

Step 1. Let T : X −→ X be a compact operator. We claim that the closure of the image T (X) is a separable Hilbert space. To prove this, we first notice that

X= [

n∈N

B(0, n).

Now every ball of the form B(0, n) is a bounded subset of X, which means that its image via T is relatively compact in X. On the other hand, a relatively compact set is always separable and

T (X) = [

n∈N

T (B(0, n))

is a countable union of separable set, meaning that it is also separable.

Step 2. The closure of the image is separable, and therefore we can always consider a sequence of projections (Pn)n∈N over n-dimensional spaces such that

[

n∈N

PnX= T (X) ∼= `2.

A standard convergence criterion shows that that PnT → T strongly (in norm). The projections are

onto a finite-dimensional subspaces, and thus PnT ∈ Lf(X) for all n ∈ N. Then T belongs to the

closure of Lf(X), that is,

Lc(X) ⊆ Lf(X).

The idea is to define the Leray-Schauder topological degree of a compact perturbation S = idΩ−T

via an approximation sequence, that is,

deg(p, S, Ω) = lim

k→+∞deg(p, idΩ− Tk, Ω),

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Proposition 2.26. Let Φ ∈ C0_{(Ω; R}m_{) be a continuous map with Ω ⊂ R}n and n ≥ m. Identify Rmwith Rm× {0} ⊂ Rn. Let g(x) := x − Φ(x), and let gmbe the restriction of g to Ω ⊂ Rm. Then

p ∈ Rm\ g(∂Ω) =⇒ deg(g, Ω, p) = deg(gm, Ω ∩ Rm, p).

Remark 2.6. Consider a function g(x) = x − Φ(x), where Φ sends Ω into a finite-dimensional subspace E of X. Let p ∈ X \ g(Ω), and let E1 be a subspace containing both E and p. We define

deg(g, Ω, p) = deg(g1, Ω ∩ E1, p).

The reader may check that this definitions is independent of the choice of E1, and thus the definition

is well-posed.

We are now ready to define the Leray-Schauder topological degree. Let p /∈ S(∂Ω). The image of ∂Ω via S is closed, and thus the distance r between p and ∂Ω is strictly positive. Let

Tk ∈ Lf(Ω, X) =⇒ kTk− T k k→∞

−−−−→ 0, and let M be a natural number big enough to have the inequality

kTk− T k ≤

r

2 for all k ≥ M .

The topological degree of idΩ− Tk is well-defined for all k ≥ M , and thus it makes sense to define

the topological degree of S as

deg(p, S, Ω) := deg(p, idΩ− TM, Ω). (2.4)

Remark 2.7. The definition (2.4) does not depend on the particular approximation sequence Tk.

Proof. Let Ti, i = 1, 2, be two elements satisfying the condition

kTi− T k ≤

r 2, and also the initial condition

Ti(Ω) ⊂ Ei, where dim Ei< ∞.

If E is the space generated by hE1, E2i, then we use the result stated above to infer that

deg(p, Si, Ω) = deg(p, Si

_Ω∩E, Ω ∩ E) for i = 1, 2. We now consider the homotopy

h(t, ·) := tS1

_Ω∩E+ (1 − t)S2

_Ω∩E,

and we notice that it is admissible on Ω ∩ E, which is a finite-dimensional space. It follows from the homotopy-invariance property that

deg(p, S1

_Ω∩E, Ω ∩ E) = deg(p, S2

_Ω∩E, Ω ∩ E), which is exactly what we wanted to prove.

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To conclude this section, we state the homotopy-invariance property for the Laray-Schauder topological degree, which requires the homotopy to satisfy an additional property.

Lemma 2.27. Let H be a separable Hilbert space, and let Ω ⊂ H be an open bounded subset. Let Σ be the set of the homotopies h(t, x) ∈ C [0, 1] × Ω, H such that

h(t, ·) = idH(·) + Wt(·),

where Wt is a compact operator for every t ∈ [0, 1]. Then the Leray-Schauder topological degree is

invariant under Σ, that is, if h(t, x) 6= y for all (t, x) ∈ [0, 1] × ∂Ω, then deg (h(0, ·), Ω, y) = deg(h(1, ·), Ω, y).

2.3 Topological Methods in Calculus of Variations

The primary goal of this section is to prove several deformation-type results and implement them to obtain the well-known mountain-pass theorem and saddle-point theorem.

2.3.1 Deformation Lemmas

Let X be a (eventually infinite-dimensional) Hilbert space, and let f : X −→ R be a functional. Definition 2.28 (Sublevel Set). Let β ∈ R. The sublevel set at the level β of f is defined by

f(β):= {x ∈ X : f (x) ≤ β} .

The central purpose of this section is to show that f(c−δ)_{is a strong deformation retract of the}

sublevel f(c+δ)_{, provided there are no critical points at the level c.}

Definition 2.29 (Gradient). The gradient of a differentiable functional f : A ⊂ X −→ R is the unique vector, denoted by grad f (u), satisfying

df (u)[v] = (grad f (u), v)_X (2.5)

for all v ∈ TuA, where (·, ·)Xis the scalar product on X and df (u) is the differential of f at u.

Remark 2.8. If A ⊂ X is a submanifold of codimension one, then the gradient restricted to A is given by the formula

gradAf (u) = grad f (u) −

(grad f (u), νA(u))X

kνA(u)k2

νA(u),

where νA(u) is the vector normal to A at the point u.

We are now ready to present a notion of compactness, which, as the reader may check by them-selves, is a much weaker notion than the usual one.

Definition 2.30 (Palais-Smale). Let c ∈ R, and let f : X −→ R be a functional. A Palais-Smale sequence at the level c is a sequence (un)n∈N⊂ X satisfying

     f (un) n→+∞ −−−−−→ c, grad f (un) n→+∞ −−−−−→ 0.

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Definition 2.31 (Palais-Smale Condition). Let c ∈ R, and let f : X −→ R be a functional. We say that f satisfies the Palais-Smale condition at the level c, and we denote it by f ∈ (PS)_c, if every Palais-Smale sequence at the level c is precompact, that is,

     f (un) n→+∞ −−−−−→ c grad f (un) n→+∞ −−−−−→ 0 =⇒ ∃(nk)k∈N⊂ N : unk k→∞ −−−−→ u ∈ X.

Definition 2.32 (Retract). Let A ⊆ B be subspaces of a metric space X.

(a) We say that A is a retract of B if there exists a continuous map r : B −→ A, called retraction, such that r_A= idA.

(b) We say that A is a deformation retract of B in X if there exists a homotopy H between the retraction r : B −→ A above and the inclusion ı : A ,→ B.

(c) We say that A is a strong deformation retract of B in X if it is a deformation retract of B in X, and the homotopy H has the additional property of not moving A, that is,

H(t, ·)_A= idA for all t ∈ [0, 1].

Remark 2.9. If the functional f : X −→ R satisfies the (PS)c condition for every c ∈ [a, b], and if

there are no critical values of f in that interval, then there exists > 0 such that σ := inf

u∈f−1_(I₎kgrad f (u)k > 0, where I= [a − , b + ].

Proof. We argue by contradiction. Namely, assume that for all > 0 we have inf

u∈f−1_(I )

kgrad f (u)k = 0.

Then there is a decreasing sequence (n)n∈N⊂ R, which converges to 0, and a sequence (un)n∈N⊂ X

such that

kgrad f (un)k_X≤ n and f (un) ∈ [a − n, b + n] .

It follows that, up to subsequences, we have f (un)

n→∞

−−−−→ c and grad f (un) n→∞

−−−−→ 0.

In particular, the sequence (un)n∈N is Palais-Smale at the level c, and therefore by assumption is

precompact. Then there exists u ∈ X such that unk

k→∞

−−−−→ u

for some subsequence (nk)k∈N ⊂ N, and it is easy to check that u is a critical point. This is the

desired contradiction since f (u) ∈ [a, b] is a critical value.

Lemma 2.33 (First Deformation Lemma). Let f : X −→ R be a functional satisfying the (PS)c

condition for every c ∈ [a, b], and assume that

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Then there exists > 0 such that, for all η, δ ∈ [0, ], the sublevel f(a−η) is a strong deformation retract of f(b+δ), that is, there exists a homotopy

H : [0, 1] × f(b+δ)−→ f(b+δ)

satisfying the following properties:

H(0, ·) = id_f(b+δ)(·),

H(1, ·) ∈ f(a−η), H(t, ·)

f(a−η)= idf(a−η)(·).

Proof. The argument is slightly involved. Hence we shall split the proof into six steps to make it more clear to the reader.

Step 1. The assumptions ofRemark 2.9 are satisfied, and thus we can find > 0 such that σ := inf

u∈f−1_(I )

kgrad f (u)k > 0.

We are now ready to show that the sublevel f(α)_{is a strong deformation retract of the sublevel f}(β)

for all α ∈ [a − , a] and all β ∈ [b, b + ].

Step 2. Let u ∈ f(β)_{. Consider the initial value problem}

   v0(t) = −_{kgrad f (v(t))k}grad f (v(t))2 X , v(0) = u. (2.6)

The strategy is the following one. We will prove that there exists a unique solution of the initial problem (2.6), which can also be extended up to f(α)_.

Step 3. The right-hand side of (2.6) satisfies the assumptions of the Cauchy-Lipschitz theorem, according to which there exists a unique solution

v : [0, ζ] −→ X

of class C1_{of the initial value problem (}_2.6_{), where ζ > 0 is a parameter small enough.}

Step 4. Suppose now that f (u) ≥ α - otherwise the thesis is trivially true -. We observe that d dt(f ◦ v) (t) = grad f (v(t)) , − grad f (v(t)) kgrad f (v(t))k2 ! = −1,

which means that f is decreasing along the solutions of (2.6) and f (u) − α ≤ f (v(t)) ≤ f (u).

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Step 5. Let I be the maximal interval where the solution v is defined. We claim that sup I > f (u) − α.

We argue by contradiction. Suppose that sup I ≤ f (u) − α. Then kgrad f (v(t))k ≥ σ =⇒ kv0(t)k ≤ 1

kgrad f (v(t))k ≤ 1 σ,

which means that v0 is a bounded function. It follows that v is Lipschitz-continuous, and the limit lim

t→sup Iv(t)

exists. In conclusion, a straightforward application of the Cauchy-Lipschitz theorem proves that the maximum of I exists and is equal to f (u) − α.

Step 6. Let us set S(u, t) := v(t), where v is the solution of (2.6) with initial data u. We consider H(t, u) := S (u, t · (f (u) − α)) : [0, 1] ×f(β)\ f(α)_{−→ f}(β)_,

an we extend it up to f(α)_{by continuity (e.g., by setting it equal to the identity for any u such that}

f (u) ≤ α). The reader should check that this is the sought homotopy.

Theorem 2.34 (Unusual Minimum Theorem). Let f : X −→ R be a function bounded from below, that is, such that α := infx∈Xf (x) > −∞. If f satisfies the (PS)α condition, then f admits a

minimum.

Proof. We argue by contradiction. If α is not a critical value, then there exists > 0 such that f(α−) is a deformation retract of f(α+),

as a consequence of theDeformation Lemma 2.33. The contradiction is now clear: the first set is empty, while the second one is nonempty.

Lemma 2.35 (Second Deformation Lemma). Let f : X −→ R be a functional, let c be a critical value of f and set

Zc:= {u ∈ X | grad f (u) = 0, f (u) = c }

to be the set of critical points. Assume that the (PS)c condition holds, and assume that there is an

open neighbourhood V of Zc. Then there exist a real number > 0 and a homotopy

H : [0, 1] ×f(c+)\ V−→ f(c+)

satisfying the following properties:

H(0, ·) = id_f(c+)_\V(·),

H(1, ·) ∈ f(c−),

H(t, ·)_f(c−)_\V = idf(c−)_\V(·).

Proof. The argument is rather involved. Hence we shall split the proof into six steps to make it more clear for the reader.

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Step 1. The Palais-Smale condition at the level c implies that Zcis a compact set; the reader may

prove this fact as a simple exercise. In particular, it turns out that d (Vc, Zc) > 0,

and thus there exists an open neighbourhood U of Zc such that

U ⊆ V and δ := d(U , V ) > 0.

Furthermore, since there are no critical points in X \ Zc, we can find 0> 0 such that

σ := inf {kgrad f (u)k : u ∈ X \ U , f (u) ∈ [c − 0, c + 0]} > 0.

Step 2. Let u ∈ f(c+0)_{\ V . Consider the initial value problem}    v0(t) = − grad f (v(t)) kgrad f (v(t))k2, v(0) = u. (2.7)

The strategy is the following one. We will prove that there exists a unique solution of the initial problem (2.7), which can also be extended up to f(c−)_{without intersecting Z}

c at any point.

Step 3. The right-hand side of (2.7) satisfies the assumptions of the Cauchy-Lipschitz theorem, according to which there exists a unique solution

v : [0, δ] −→ X

of class C1_{of the initial value problem (}_2.7_{), where δ > 0 is a parameter small enough.}

Step 4. Let I be an interval such that min I = 0, and suppose that the solution v is defined on the whole I. We claim that, if there is t ∈ I such that

v(t) ∈ U and f (v(t)) ≥ c − 0,

then the time can be estimated from below by

t ≥ δ · σ. (2.8)

This claim follows immediately from the fact that

t ≥ t0:= inf {τ ∈ I | v(τ ) ∈ U } .

Indeed, employing the fact that v(0) = u ∈ f(c+0)\ V , the length of the curve from 0 to t

0 can be

easily estimated as follows: Z t0 0 kv0(τ )k dτ = Z t0 0 1 kgrad f (v(τ )) kdτ ≤ t0 σ,

On the other hand, the length of the curve needs to be at least equal to δ to intersect U , and hence t0

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Step 5. Let be a real number such that 0 < < min 0, 1 2σ · δ .

If u ∈ f(c+)_{\ V and u /}_{∈ f}(c−)_{, then the solution curve v reaches f}(c−)_{\ U , which means that there}

is a time t such that

U (t) ∈ f(c−) and U (s) /∈ U for all s ≤ t. Moreover, from the estimate

kv0k ≤ 1

kgrad f (v(t))k ≤ 1 σ,

it turns out that the curve can be extended in such a way that v is well-defined on the whole interval J := [0, f (u) − (c − )] .

Step 6. Let us set S(u, t) := v(t), where v is the solution of (2.7) with initial data u. We consider

H(t, u) := S (u, t · (f (u) − (c − ))) : [0, 1] ×f(c+)\f(c−)∪ V−→ f(c+)_,

an we extend it up to f(c−) by continuity (e.g., by setting it equal to the identity for any u such that f (u) ≤ c − ). The reader should check that this is the sought homotopy.

2.3.2 Saddle Point and Mountain Pass Theorems

In this section, we want to state and prove two fundamental results in nonlinear PDEs, namely the Saddle Point Theorem and the Mountain Pass Theorem, and show the minimax version of both. Theorem 2.36 (Saddle Point). Let X be a Hilbert space, and let f ∈ C1

(X; R) be a differentiable functional. Suppose that

X= X0⊕ X1,

where X0, X1⊂ X are vector spaces, with dim X0< +∞. Assume also that:

(i) There exists ρ0> 0 such that, if we set S0:= SX(0, ρ0) ∩ X0, then

sup

u∈S0

f (u) < inf

u∈X1

f (u). (2.9)

(ii) The functional f satisfies the Palais-Smale condition (PS)_c at every level c ∈ [a, b], where a := inf

u∈X1

f (u) and b := sup

u∈B0 f (u).

Then there exists a critical point u ∈ X for the functional f , such that f (u) ∈ [a, b].

Despite the lengthy statement, the proof is elementary and results directly from the following technical lemma, which relies on the topological degree theory developed in the previous sections.

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Lemma 2.37. If Φ : B0−→ X is a continuous function such that

Φ_S

0 ≡ idS0, then the image of the ball intersects the space X1, that is,

Φ(B0) ∩ X16= ∅.

Proof. Let P0: X −→ X0 be the orthogonal projection onto X0with kernel equal to X1, and let us

consider the continuous composition

P0◦ Φ : B0−→ X0.

Let u ∈ S0. Then P0◦ Φ(u) = P0(u) = u, and thus the restriction of P0◦ Φ to the sphere S0 is the

identity map. Therefore (see, e.g.,Proposition 2.14), we have the inclusion B0⊂ P0◦ Φ (B0) ,

which, in turn, proves the existence of a point u ∈ B0 such that P0◦ Φ(u) = 0. Since the kernel of

P0is X1, this is equivalent to saying that

Φ(u) ∈ Ker P0= X1.

Proof. We argue by contradiction. Let > 0 be a small constant such that (i) still holds: sup

u∈S0

f (u) < inf

u∈X1

f (u) − .

ByDeformation Lemma 2.33 we can find a retraction r : f(b)_{−→ f}(a−) _satisfying

B0⊆ f(b), S0⊆ f(a−) and X1∩ f(a−)= ∅.

The map Φ := r_B

0 satisfies the assumptions ofLemma 2.37, and hence Φ (B0) ∩ X16= ∅.

In particular, there exists a point u ∈B0 such that Φ(u) > a − strictly, and this is absurd since,

as a consequence of the assumption (2.9), we have Φ(B0) ⊆ f(a−), and

Φ(B0) ⊆ f(a−) ⇐⇒ Φ(u) ≤ a − .

Theorem 2.38 (Minimax). Let X be a Hilbert space, and let f ∈ C1

(X; R) be a differentiable functional. Suppose that

X= X0⊕ X1,

where X0, X1⊂ X are vector spaces, with dim X0< +∞. Assume that there exists ρ0> 0 such that,

if we set S0:= SX(0, ρ0) ∩ X0, then sup u∈S0 f (u) < inf u∈X1 f (u). Set c := inf ( sup x∈B0 f ◦ Φ (x) : Φ ∈ C0 B0; X , Φ _S 0 = idS0 ) . Then the following assertions hold:

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(1) The value of c can be estimated by inf u∈X1 f (u) ≤ c ≤ sup u∈B0 f (u).

(2) Assume also that f satisfies the Palais-Smale condition (PS)_c. Then there exists a critical point u ∈ X for the functional f at the level c (i.e., f (u) = c).

We are now ready to state and prove the mountain pass theorem, which roughly asserts that, to travel along a path γ between two low points on a mountain, we must go up and then down. The "smoothness" of the path indicates that there must be a critical point somewhere in between. Theorem 2.39 (Mountain Pass). Let X be a Hilbert space, and let S := SX(0, ρ) for some ρ > 0.

Suppose that there are u0, u1∈ X such that

u0∈ int Bρ and u1∈ B/ ρ.

Let γ : [0, 1] −→ X be a curve from u0 to u1, and let f ∈ C1(X; R) be a functional satisfying

f (u0), f (u1) < inf u∈Sf (u).

If f satisfies the Palais-Smale condition (PS)_c at every level c ∈

inf

u∈Sf (u), sup_t∈If ◦ γ(t)

,

then there exists a critical level c - in the same interval - for the functional f .

Proof. We argue by contradiction. By assumption there exists > 0 such that f (u0), f (u1) < inf

u∈Sf (u) − ,

and, since we assumed that no critical values belong to the interval above, as a consequence of the

Deformation Lemma 2.33, we can find a retraction r : f(b)−→ f(a−)_{. Then}

γ ([0, 1]) ⊆ f(b) and r(ui) = ui since u0, u1∈ f(a−),

and this is absurd since

(r ◦ γ(I)) ∩ S = ∅.

Indeed, the composition r ◦ γ(t) belongs to f(a−) _{for every t ∈ [0, 1], and - at the same time - we}

also have that f(a−)_{∩ S = ∅.}

Theorem 2.40 (Minimax). Let X be a Hilbert space, and let S := SX(0, ρ) for some ρ > 0. Suppose

that there are u0, u1∈ X such that

u0∈ int Bρ and u1∈ B/ ρ.

Let f ∈ C1

(X; R) be a functional such that

f (u0), f (u1) < inf u∈Sf (u), and set s := inf sup t∈I f ◦ γ(t) : γ ∈ C0(I; X) , γ(0) = u0, γ(1) = u1 . Then the following assertions hold true:

Linking Theorems, Kato-Ponce Inequalities, and Applications in Mathematical Physics

Università degli Studi di Pisa

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Matematica

Linking Theorems, Kato-Ponce

Inequalities, and Applications in

Mathematical Physics

Relatore

Candidato

Introduction

Contents

I

Variational Methods in Nonlinear Analysis

1

III

Applications to PDEs Analysis

99

IV

Appendix

149

Part I

Variational Methods in Nonlinear

Analysis

Chapter 1

Solitary Waves and Solitons

1.1

Introduction

1.1.1

Solitons and Group Representations

1.2

Existence of Hylomorphic Solitons

1.2.1

Hylomorphic Solitons

1.2.2

Existence Results

1.2.3

Stability Result

1.3

Physical Interpretation of Solitary Waves

1.3.1

Example of Water Waves

Chapter 2

Topological Methods in Calculus of

Variations

2.1

Topological Degree Theory in Finite-Dimensional Spaces

2.1.1

Topological Degree for C

-Regular Maps

2.1.2

Topological Degree for C

-Regular Maps

2.1.3

Properties of the Topological Degree Modulo 2

2.1.4

Brouwer Topological Degree

2.2

Leray-Schauder Topological Degree

2.3

Topological Methods in Calculus of Variations

2.3.1

Deformation Lemmas

2.3.2

Saddle Point and Mountain Pass Theorems

_{-Regular Maps}

_{-Regular Maps}