Solitons on Manifolds

Vieri Benci^{1}

Dipartimento di Matematica Applicata U. Dini Facoltá di Ingegneria Universitá di Pisa

Italy

Fabio Giannoni^{1} Dipartimento di Energetica

Facoltá di Ingegneria Universitá dell’Aquila

Italy

Paolo Piccione^{1} Dipartimento di Energetica

Facoltá di Ingegneria Universitá dell’Aquila

Italy

1The authors were partially supported by MURST

Abstract

We prove existence and multiplicity result for solutions of a class of Lorentz invariant equations on a compact Riemannian manifold M. Such solutions, that arise as extremals for a suitable positive functional on the space of smooth functions from IR^{3} to M, are classified by the elements of the third homotopy group of M. A physical interpretation of this invariant is given in terms of the quantic numbers of a traveling particle.

1 Preliminaries and Statement of the Results

The aim of this paper is to study the existence and the multiplicity of solutions of a Lorentz invariant equation on a compact Riemannian manifold. Such solutions have the energy traveling as localized packets, preserving their form under perturbation, and so they describe a particle-like behavior.

This kind of solutions occurs naturally in several fields of mathematical physics, like classical and quantum field theory, non linear optics, fluid mechanics and plasma physics. Details may be found in the references [12, 15, 23, 26].

The solitonic behavior may be found when there exist infinitely many first integral of the motion, like for instance in the KdV equation, or when there exist topological constraint in the state space. In this paper we will deal exclusively with the second case, by considering topological solitons in a curved manifold of dimension greater than one.

A classical 1-dimensional model for a soliton solution is given by the socalled sine-Gordon equation, given by:

u_{tt} − c^{2} u_{xx} + sin u = 0,  (1)
lim_{x→−∞} u(x, t) = 0,  (2)
lim_{x→+∞} u(x, t) = 2 k π, k ∈ Z,  (3)

where c is the velocity of light. The asymptotic conditions (1) and (2) allow to have solutions with finite energy E(u) given by:

E(u) = \frac{1}{2} ∫_{−∞}^{+∞} (u_{t}^{2} + c^{2} u_{x}^{2}) dx + ∫_{−∞}^{+∞} (1 − cos u) dx.

The static solutions u(x, t) = ϕ(x) of the sine-Gordon equation above solve the elliptic equation:

−c^{2} ϕ̈ + sin ϕ = 0,  (4)
lim_{x→−∞} ϕ(x) = 0,  (5)
lim_{x→+∞} ϕ(x) = 2 k π, k ∈ Z.  (6)

Any solution of the static problem defines a solution of the sine-Gordon equation traveling at the constant velocity v, by setting:

u(x, t) = ϕ (\frac{x − v t}{\sqrt{1 − \frac{v^{2}}{c^{2}}}} ) .

The solutions to the equation (4) have a variational characterization, namely they are critical point for the energy functional:

E(ϕ) = ∫_{−∞}^{+∞} (\frac{1}{2} c^{2}ϕ̇^{2} + 1 − cos ϕ) dx,

defined on a suitable function space that will be denoted by Λ

The asymptotic conditions (6) split the function space Λ into infinitely many connected components, denoted by Λ_{k}, where:

Λ_{k} = {ϕ ∈ Λ | lim_{x→+∞} ϕ(x) = 2 k π}, k ∈ Z.

One can prove that E possess local minima on the connected components Λ_{1} and Λ_{1} and these solutions have a solitonic behavior.

The situation is drastically different for dimensions of the base space higher than 1, where the existence of non trivial minimal point for the energy functional E fails. For instance, let’s consider the 3-dimensional analogue of (4), given by the scalar equation:

u_{tt}(x, t) − c^{2}Δu(x, t) + ∇u(x, t) = 0,

where Δ and ∇ denote respectively the Laplacian and the gradient operator with respect to the space variable x ∈ IR^{3}.

The corresponding energy functional for the static solutions has the form:

E(ϕ) = ∫_{IR^{3}} (\frac{c^{2}}{2} |∇ϕ| + V (ϕ)) dx.  (7)

If V is bounded from below, Derrick’s Theorem (see [12]) says that every minimal point for (7) is necessarily trivial, i.e. constant.

On the other hand, if one considers more general potentials V which are unbounded from below (see for instance [8, 25]), then the typical critical points for E are saddle points, and this kind of extremizers fails to be stable by small perturbations, hence they do not represent an observable physical situation (see [2, 7]).

In order to overcome the lack of stability for the critical points, in [6] the authors consider a correction of the functional (7) given by the positive functional:

E(ϕ) = ∫_{IR^{3}} [\frac{c^{2}}{2} |∇ϕ|^{2} + ε \frac{c^{6}}{6} |∇ϕ|^{6} + V (ϕ)] dx.  (8)

Under some regularity and growth conditions on the function V, the authors prove the existence of sufficiently many critical points for (8) in a suitable space Λ of functions ϕ : IR^{3} ⟼ Ω, where Ω is an open, connected subset of IR^{n}.

The purpose of this paper is to extend the results of Benci, Fortunato and Pisani to the case where one replaces the open set Ω with any compact Riemannian manifold M. We will consider manifolds M with a distinguished base point p, that plays the role of the 0 in the open set Ω. In our approach to the problem, the disadvantage of losing the linear structure on the target space is compensated by the fact that, using the non-flat metric and the topological structure of M, we are able to prove the same multiplicity results of [6] without using any potential function V. We will consider a functional E of the form:

E(ϕ) = ∫_{IR^{3}} β(∥dϕ∥^{2}) ds,  (9)

where β(λ) = a_{1}λ + a_{2}λ^{2} + a_{3}λ^{3} is a polynomial of the third degree (a_{3} > 0), ϕ : IR^{3} ⟼ M is an absolutely continuous function vanishing at infinity varying in a suitable space, denoted by H_{2,6}(IR^{3}, M), where E is naturally defined. The space H_{2,6}(IR^{3}, M) will be defined formally in Section 3, where we will discuss the entire variational structure of our problem. For simplicity, in our treatment we will consider the polynomial β(λ) = λ + λ^{3}.

For k ∈ IN, we denote by S^{k} the unit sphere in IR^{k+1}, with ω distinguished point in S^{k}, and by π_{k}(M) the k-th homotopy group of the manifold M, which is the group of homotopy classes of all maps in the set:

C_{ω,p}(S^{k}, M) = {ϕ : S^{k} ⟼ M continuous : ϕ(ω) = p}.

The space H_{2,6}(IR^{3}, M) ⊂ C_{0}(IR^{3}, M) has the same homotopy type of C_{0}(IR^{3}, M) ≃ C_{ω,p}(S^{k}, M) (see Proposition A.5). Hence, every ϕ ∈ H_{2,6}(IR^{3}, M) identifies an element ϕ^{#} of π_{3}(M); for α ∈ π_{3}(M), we denote by Λ_{α} the set of all maps ϕ ∈ H_{2,6}(IR^{3}, M) that belong to the homotopy class α.

Let A(M) be the subset of π_{3}(M) consisting of all elements α such that the energy functional E attains its minimum on Λ_{α}.

The main result of the paper is the following:

Theorem 1.1. The set A(M) is a generator set for the group π_{3}(M).

2 Topological Invariants.

The Quantic Numbers

In this section we define a local topological invariant for the functions in H_{2,6}(IR^{3}, M), that comes from the restriction onto suitable closed subsets of IR^{3}. The model for this invariant is the topological degree of maps.

We will not use any embedding of M into IR^{N}; M will be thought as a metric space, with metric dist(·, ·).

We fix a normal, convex subset V of p ∈ M; in particular, V is contractible in M. Let δ_{0} > 0 be such that the closure \overline{B(p, δ_{0})} of the δ_{0}-ball around p satisfies:

\overline{B(p, δ_{0})} ⊂ V,

and let ρ > 0 be defined by:

ρ = dist(∂V, \overline{B(p, δ_{0})}) > 0.

Finally, we denote by U the open ball B(p, ρ + \frac{δ_{0}}{2} ); since U ⊂ V, then for every point q ∈ U there exists a unique minimal geodesic between p and q, and it is entirely contained in U.

For each q ∈ U, we denote by γ_{q} : [0, 1] ⟼ U the unique minimal geodesic (parameterized on the interval [0, 1] by a multiple of the arc-length) such that γ_{q}(0) = p and γ_{q}(1) = q. Using the exponential map, it is easy to see that the map q ⟼ γ_{q} ∈ C([0, 1], U) is continuous in the compact–open topology, i.e., if q_{n} ⟼ q, then γ_{q_{n}} tends to γ_{q} uniformly.

Definition 2.1. For ϕ ∈ H_{2,6}(IR^{3}, M), we define the support of ϕ, denoted by K(ϕ) the closed set:

K(ϕ) = \overline{\{x ∈ IR^{3} : ϕ(x) ∉ V \}}.  (10)

Since V is contractible, and every map with values in a contractible space is homotopically trivial, if ϕ^{#} ≠ 0 then K(ϕ) ≠ ∅.

We denote by ≃ the homotopy equivalence of functions.

The homotopical invariant ϕ^{#} is localized in K(ϕ) in the following sense:

Lemma 2.2. Let ϕ_{1}, ϕ_{2} ∈ H_{2,6}(IR^{3}, M). If K(ϕ_{1}) = K(ϕ_{2}) and ϕ_{1}|_{K(ϕ_{1})} ≡ ϕ_{2}|_{K(ϕ_{2})}, then ϕ_{1} ≃ ϕ_{2}.

Proof. Define A to be the open set ϕ_{1}^{−1}(V) ⊂ IR^{3}; by definition A∪K(ϕ_{1})^{∘} = IR^{3}, where ∘ denotes the interior.

The restriction ϕ_{1}|_{A} and ϕ_{2}|_{A} are homotopic, because they are both homotopic to the constant map ϕ ≡ p.

Clearly, ϕ_{1}|_{K(ϕ_{1})} ≃ ϕ_{2}|_{K(ϕ_{2})}, and ϕ_{1} ≃ ϕ_{2} on IR^{3}.

For all ϕ ∈ H_{2,6}(IR^{3}, M) (or more in general for all ϕ ∈ C_{0}(IR^{3}, M)), we denote by K(ϕ) a collection of closed subsets of IR^{3}, given by:

K(ϕ) = \{K ⊂ IR^{3} closed : ϕ(∂K) ⊂ \overline{V} \}.

Since lim_{|x|→+∞} ϕ(x) = p ∈ V, then every ball in IR^{3} centered in 0 and with radius sufficiently large belongs to K(ϕ). Moreover, every connected component of the support K(ϕ) belongs to K(ϕ).

Let ϕ ∈ H_{2,6}(IR^{3}, M) be fixed and K ∈ K(ϕ). By continuity, there exists an open neighborhood N of ∂K such that ϕ(N) ⊂ U. Define K̃ to be the open set:

Ñ = K ⋃ N.

By the Uryson Lemma, there exists a continuous function c : IR^{3} ⟼ [0, 1] such that:

c(x) = \{ \begin{matrix}1, & if x ∈ K; \\ 0, & if x ∉ Ñ. \end{matrix}

Now, we define a function ϕ̃: IR^{3} ⟼ M by:

ϕ̃(x) = \{ \begin{matrix}γ_{ϕ(x)}(c(x)), & if ϕ(x) ∈ Ñ; \\ p & if ϕ(x) ∉ Ñ. \end{matrix}

The continuity of ϕ̃ is easily estabilished, considering that q ⟼ γ_{q} is continuous in the compact–open topology, that c is continuous and that γ_{ϕ(x)}(0) = p for all x. Moreover, ϕ̃ ≡ ϕ in K, because c ≡ 1 in K and γ_{ϕ(x)}(1) = ϕ(x) for all x ∈ U.

Observe that ϕ̃ ∈ C_{0}(IR^{3}, M), hence its homotopy class defines an element ϕ̃^{#} of π_{3}(M).

Definition 2.3. For all ϕ ∈ H_{2,6}(IR^{3}, M) and every K ∈ K(ϕ), we set:

(ϕ|_{K})^{#} = ϕ̃^{#}

The main properties of the localized invariant (ϕ|_{K})^{#} are described in the following:

Proposition 2.4. (1) (ϕ|_{K})^{#}) is well defined, i.e., it does not depend on the choice of the open neighborhood N of ∂K and the continuous function c;

(2) for all closed set K ⊂ IR^{3} such that K(ϕ) ⊂ K, it is K ∈ K(ϕ) and (ϕ|_{K})^{#} = ϕ^{#}. In particular, (ϕ|_{K(ϕ)})^{#} = ϕ^{#};

(3) if K_{1}, K_{2} ∈ K(ϕ) and K_{1}^{∘} ∩ K_{2}^{∘} = ∅, then

(ϕ|_{K_{1}∪K_{2}})^{#} = (ϕ|_{K_{1}})^{#} + (ϕ|_{K_{2}})^{#}.

Proof. To prove part (1), let ϕ̃_{i}, i = 1, 2, be two different extensions of ϕ outside K corresponding to two different choices of pairs (N_{1}, c_{1}) and (N_{2}, c_{2}) as in the definition of the local invariant (ϕ|_{K})^{#}.

We show that ϕ̃_{1} and ϕ̃_{2} are homotopic, hence they define the same element in π_{3}(M).

Observe indeed that ϕ̃_{1} = ϕ̃_{2} = ϕ on K, and ϕ̃_{1} = ϕ̃_{2} = p outside U. For a, b ∈ U, let γ_{a,b} : [0, 1] ⟼ U denote the unique minimal geodesic from a to b, parameterized by a multiple of arc length.

Again, using the exponential map it is easy to see that (a, b) ⟼ γ_{a,b} ∈ C([0, 1], U) is continuous in the compact open topology.

We define a homotopy H : [0, 1] × M by:

H(λ, x) = γ_{ϕ̃_{1}(x),ϕ̃_{2}(x)}(λ) ∀ x ∈ U, λ ∈ [0, 1],

and

H(λ, x) = ϕ̃_{1}(x) = ϕ̃_{2}(x), if x ∉ U.

The map H is clearly continuous. Moreover, it is:

H(0, x) = ϕ̃_{1}(x), H(1, x) = ϕ̃_{2}(x), ∀ x ∈ U,

hence H is a homotopy between ϕ̃_{1} and ϕ̃_{2}.

Part (2) follows from the fact that, if K(ϕ) ⊂ K, then, by definition of ϕ̃, we have that K(ϕ̃) = K(ϕ) and ϕ̃ ≡ ϕ on K(ϕ).

Finally, for part (3), observe that if K_{1}^{∘} ∩ K_{2}^{∘} = ∅, then we can find the corresponding open neighborhoods N_{1} and N_{2} disjoint, in such a way that also Ñ_{1} = K_{1} ∪ N_{1} and Ñ_{2} = K_{2} ∪ N_{2} are disjoint open sets. The function ϕ̃ corresponding to the union K_{1} ∪ K_{2} is then given by ϕ̃|Ñ_{i} = ϕ̃_{i}, i = 1, 2, and ϕ̃(x) ≡ p if x ∉ Ñ_{1}∪Ñ_{2}. Since p is the base point for the maps representatives of elements in π_{3}(M), we have ϕ̃^{#} = ϕ̃_{1}^{#} + ϕ̃_{2}^{#} in π_{3}(M).

3 The Variational Framework

Let (M, ⟨·, ·⟩_{x}) be an m-dimensional (m ≥ 1) complete Riemannian manifold of class C^{4}, and let p denote a distinguished point in M. Using Nash’s embedding Theorem (see [20]), we can assume that M is isometrically embedded in some Euclidean space, ι : M ⟼ IR^{N}, and we will also assume that ι(p) = 0.

Even though some of the formulas written in this Section depend on the particular choice of the embedding ι, due to the compactness of M one can check (see Appendix A) that passing to a different embedding does not change the differentiable structure of our framework. Moreover, the regularity results proven Theorem 3.2 ensure that the extremals of our variational problem satisfy an intrinsic differential equation, defined only in terms of the Riemannian structure of M (see (18)).

For x ∈ IR^{N} and ρ ∈ IR^{+}, B(x, ρ) will denote the open ball centered in x and of radius ρ in IR^{N}. If ϕ : IR^{n_{1}} ⟼ IR^{n_{2}} is an absolutely continuous map, we denote by dϕ(x) : IR^{n_{1}} ⟼ IR^{n_{2}} , the linear map given by the derivative of ϕ at the point x. We denote by ∥dϕ(x)∥ the Hilbert-Schmidt norm of dϕ(x) as a linear operator from IR^{n_{1}} to IR^{n_{2}} with respect to the Euclidean norms of IR^{n_{1}} and IR^{n_{2}}:

∥dϕ(x)∥^{2} = ∑_{i=1}^{n_{1}} ∥dϕ(x) e_{i}∥^{2},

where \{e_{i}\}_{i} is any orthonormal basis of IR^{n_{1}}. In particular, for e_{i} = \frac{∂}{∂x_{i}}, we have:

∥dϕ∥^{2} = ∑_{i=1}^{n_{1}} ∥\frac{∂ϕ}{∂x_{i}} ∥^{2}.

We consider the Banach space H given by the completion of the vector space C_{c}^{∞}(IR^{3}, IR^{N}) of all smooth maps maps from IR^{3} to IR^{N} with compact support, with respect to the norm:

∥ϕ∥_{2,6} = ∥dϕ∥_{L^{2}} + ∥dϕ∥_{L^{6}},  (11)

where

∥dϕ∥_{L^{p}} = (∫_{IR^{3}} ∥dϕ(x)∥^{p} dx)^{\frac{1}{p}} .

By definition of ∥ · ∥_{2,6}, the derivative dϕ belongs to the intersection

L^{2}(IR^{3}, IR^{3N}) ∩ L^{6}(IR^{3}, IR^{3N}),

where we identify IR^{3N} with the set of all linear maps from IR^{3} to IR^{N}. By the inequality of Sobolev, Gagliardo and Nirenberg (see [10]), there exists a positive constant C_{1} such that, for ϕ ∈ H it is:

∥ϕ∥_{L^{6}} ≤ C_{1} · ∥dϕ∥_{L^{2}}.

This implies that H ⊂ W^{1,6}(IR^{3}, IR^{N}) with continuous immersion. Then, by Morrey’s inequality (see [10]), it is also:

H ⊂ L^{∞}(IR^{3}, IR^{N})  (12)

with continuous injection; moreover, there exists a positive constant C_{2} such that, for every ϕ ∈ H it is:

|ϕ(x) − ϕ(y)| ≤ C_{2} · ∥dϕ∥_{L^{6}} · |x − y|^{\frac{1}{2}} .  (13)

If we denote by C^{0,α}(IR^{3}, IR^{N}) the space of all Hölder continuous maps with Hölder constant α, then by (13) we also have a continuous injection:

H ⊂ C^{0, \frac{1}{2}} (IR^{3}, IR^{N}).  (14)

Observe also that, by (12), for all ϕ ∈ H it is:

lim_{|x|→∞} ϕ(x) = 0,

because every ϕ ∈ H is uniform limit of smooth functions with compact support. We denote by C_{0}(IR^{3}, IR^{N}) the set of continuous functions from IR^{3} to IR^{N} vanishing at infinity; we also have:

H ⊂ C_{0}(IR^{3}, IR^{N}),  (15)

so that ϕ(IR^{3}) ∪ {0} is a compact subset of IR^{N} for all ϕ in H. Remark 3.1. Observe that the combination of the Sobolev–Gagliardo–Nirenberg inequality with Morrey’s theorem is a very special coincidence due to the dimension 3 of the base space. This fact is crucial for the proof of our results without the need of a potential in our variational problem. The three– dimensional case, which is the one with more physical interest, appears to be mathematically the most natural environment for the study of solitonic waves. Different results were found also in the one dimensional case, where the problem of the existence of homoclinic orbits is studied by global variational methods (see [17] and the references therein).

The infinite dimensional space where our problem is studied is given by:

H_{2,6}(IR^{3}, M) = {u ∈ H | u(IR^{3}) ⊂ M}.

As shown in the Appendix A, the space H_{2,6}(IR^{3}, M) has the structure of a Banach manifold. For ϕ ∈ H_{2,6}(IR^{3}, M), the tangent space T_{ϕ}H_{2,6}(IR^{3}, M) can be identified with the Banach space:

T_{ϕ}H_{2,6}(IR^{3}, M) = {ζ ∈ H_{2,6}(IR^{3}, IR^{N}) | ζ(x) ∈ T_{ϕ(x)}M ∀ x}.

We consider the following functional on H_{2,6}(IR^{3}, M):

E(ϕ) = ∫_{IR^{3}} (∥dϕ(x)∥^{2} + ∥dϕ(x)∥^{6}) dx.  (16)

If Ω is an open subset of IR^{3}, we define on H_{2,6}(IR^{3}, M) a localized functional E_{Ω} by restricting the integral in (16) to Ω:

E_{Ω}(ϕ) = ∫_{Ω} (∥dϕ(x)∥^{2} + ∥dϕ(x)∥^{6}) dx.

We also define the pseudo-distance d_{Ω} in H_{2,6}(IR^{3}, M) by:

d_{Ω}(ϕ_{1}, ϕ_{2}) = (∫_{Ω} ∥dϕ_{1}(x) − dϕ_{2}(x)∥^{2} dx)^{\frac{1}{2}} + (∫_{Ω} ∥dϕ_{1} − dϕ_{2}∥^{6} dx)^{\frac{1}{6}} .

It is proven in Appendix A that different embeddings of M into IR^{N} give rise to equivalent metrics d_{Ω}.

Observe that, if ϕ ∈ H_{2,6}(IR^{3}, M), for almost every x ∈ IR^{3} the differential dϕ(x) belongs to space L(IR^{3}, T_{ϕ(x)}M) ≃ (IR^{3})^{∗} ⊗ T_{ϕ(x)}M ≃ IR^{3} ⊗ T_{ϕ(x)}M of all linear maps from IR^{3} to T_{ϕ(x)}M. Here the ∗ denotes the dual space. This space is endowed with a natural Hilbert structure induced by the Euclidean structure on IR^{3} and the Riemannian structure on T_{ϕ(x)}M; the inner product in (IR^{3})^{∗}⊗T_{ϕ(x)}M will be denoted by ⟨·, ·⟩_{0}. The norm relative to this product is the Hilbert–Schmidt norm of a linear operator from IR^{3} to T_{ϕ(x)}M.

Using standard computations (see Appendix A) in Calculus of Variations on Riemannian manifolds, one proves that E is a functional of class C^{2}. Moreover, for ϕ ∈ H_{2,6}(IR^{3}, M) and for ζ ∈ T_{ϕ}(IR^{3}, M), the Gateaux derivative E′(ϕ)[ζ] is given by:

E′(ϕ)[ζ] = ∫_{IR^{3}} (2 + 6 ∥dϕ(x)∥^{2}) · ⟨∇^{ϕ}ζ(x), dϕ(x)⟩_{0} dx.  (17)

In (17), ∇^{ϕ}ζ denotes the covariant derivative of the vector field ζ along the function ϕ induced by the Levi–Civita connection of M. For the reader’s convenience, a fairly detailed discussion about the Banach manifold structure, the differentiability of the functional E, together with some basic notions from covariant calculus along functions, is postponed to the Appendix A.

Using local coordinates on M, if Γ_{ij}^{k}(p) denote the Christoffel symbols for the metric of M, the Euler–Lagrange equations of the functional E for the function ϕ = (ϕ^{1}, …, ϕ^{m}) can be written as:

Δϕ^{k}(x) + 3 div (∥dϕ(x)∥^{4} · ∇ϕ^{k}(x))  (18)

+ ∑_{a,b=1}^{m} ∑_{i,j=1}^{3} Γ_{ab}^{k}(ϕ(x)) \frac{∂ϕ^{a}}{∂x_{i}} (x) \frac{∂ϕ^{b}}{∂x_{j}} (x) = 0, k = 1, 2, …, m

where Δ is the Laplacian operator in IR^{3} and ∇ϕ^{k} denotes the gradient of ϕ^{k} (see Proposition A.7).

An integration by parts in (17) shows that if ϕ is a strong solution (in the C^{2}-sense) of (18), then ϕ is a critical point of E in H_{2,6}(IR^{3}, M), i.e.

E′(ϕ)[ζ] = 0, ∀ ζ ∈ T_{ϕ}H_{2,6}(IR^{3}, M).  (19)

The converse is given by the following regularity result, whose proof is basically contained in [1] and [18]:

Theorem 3.2. Assume that ϕ satisfies (19). Then ϕ is a map of class C^{2}, and therefore it satisfies (18).

Proof. Computing the Hessian of E in local coordinates and using well known embedding theorems shows that every critical point ϕ of E minimizes locally E. This means that every point x_{0} ∈ IR^{3} has an open neighborhood Ω such that E_{Ω}(ϕ) ≤ E_{Ω}(ϕ′) for all ϕ′ ∈ H_{2,6}(IR^{3}, M).

By (13), ϕ is Hölder continuous and we can apply Theorem 4.3 of [18] in local coordinates to get that dϕ is also Hölder continuous. Finally, the methods used in [1] allow to prove that dϕ is of class C^{1}.

Remark 3.3. Some estimates on the Hessian of the functional E at ϕ ≡ 0 show that, if ρ ∈ IR^{+} is chosen sufficiently small, then the only critical point of E with image in B(0, ρ_{0}) is the null map. Clearly, this remains true for the localized functional E_{B(0,ρ_{0})}. This observation is essential for the study of the behavior of the Palais–Smale sequences which is done in Section 4.

4 Palais–Smale Sequences for the functional E

In this section we will study the Palais–Smale sequences for the functional E defined in (16). We recall that, if c ∈ IR, a sequence {ϕ_{n}}_{n} ⊂ H_{2,6}(IR^{3}, M) is said to be a Palais–Smale sequence at the level c for the functional E if:

lim_{n→∞} E(ϕ_{n}) = c, and lim_{n→∞} ∥E′(ϕ_{n})∥ = 0,

where ∥E′(ϕ_{n})∥ is the norm in the dual Banach space of T_{ϕ_{n}}H_{2,6}(IR^{3}, M).

Before stating the main result of the section, the following observation is needed:

Remark 4.1. As an immediate consequence of the continuous embedding of H_{2,6}(IR^{3}, M) is L^{∞}, we have the existence of two positive constants δ_{0} and Δ_{∗} such that, for all ϕ ∈ Λ, if for some x ∈ IR^{3} it happens that ϕ(x) ∉ B(0, δ_{0}), then E(ϕ) ≥ Δ_{∗}. It follows that:

inf_{Λ^{∗}} E ≥ Δ_{∗} > 0.  (20)

The main result of the section is contained in the following Proposition, and it is about the behavior of the Palais–Smale sequences of the functional E:

Proposition 4.2. Let b > Δ_{∗} and {ϕ_{n}}_{n} ⊂ H_{2,6}(IR^{3}, M) be a Palais–Smale sequence at the level b for E.

Then, there exists ρ ∈ IN, \overline{ϕ}_{1}, …, \overline{ϕ}_{ρ} ∈ Λ_{∗} critical points of E, x_{n}^{1}, …, x_{n}^{l} in IR^{3} and R_{n}^{1}, …, R_{n}^{l} > 0 such that, up to passing to subsequences, the following facts hold:

lim_{n→∞} R_{n}^{i} = +∞, ∀ i = 1, …, l;  (21)
lim_{n→∞} d_{B(x_{n}^{i},R_{n}^{i})}(ϕ_{n}, \overline{ϕ}(· + x_{n}^{i})) = 0, ∀ i = 1, …, l;  (22)
lim_{n→∞} |x_{n}^{i} − x_{n}^{j}| = 0, if i ≠ j;  (23)
lim_{n→∞} E_{B(x_{n}^{i},R_{n}^{i})}(ϕ_{i}) = E(\overline{ϕ}_{i}), ∀ i = 1, …, l;  (24)

∑_{i=1}^{l} E(\overline{ϕ}_{i}) = b;  (25)

lim_{n→∞} d_{IR^{3}\(⋃_{i=1}^{l} B(x_{n}^{i},R_{n}^{i}))}(ϕ_{n}, 0) = 0.  (26)

Remark 4.3. From (26) of Proposition 4.2 it follows that, if ϕ_{n} is a Palais– Smale sequence for E, then:

lim_{n→∞} sup {d(ϕ_{n}(x), 0) : x ∈ IR^{3} \ ⋃_{i=1}^{l} B(x_{n}^{i}, R_{n}^{i})} = 0.

Then, it is not too difficult to see that, for n sufficiently large, ϕ_{n}^{#} is generated by \overline{ϕ}_{i}^{#}, i = 1, 2, …, l. (see [5])

To carry out the proof of Proposition 4.2, the following preliminaries Lemmas are needed.

Lemma 4.4. Let u_{n} be a sequence in H. Let R_{n} be a sequence in IR^{+}∪{+∞} such that lim_{n→∞} R_{n} = R_{0} ∈ IR^{+}. Assume that:

lim_{n→∞} E′_{B(0,R_{n})}(u_{n}) = 0,  (27)
E_{B(0,R_{n})}(u_{n}) ≤ c eventually,  (28)

where c is a constant independent of n.

Then there exists u ∈ H(\overline{B(0, R_{0})}) such that:

lim_{n→∞} d_{B(0,R)}(u_{n}, u) = 0, ∀ R ∈]0, R_{0}[;  (29)
E′_{B(0,R_{0})}(u)[V] = 0, ∀ V ∈ T_{u}H with supp(V) ⊂ B(0, R_{0}), (30)
E_{B(0,R_{0})}(u) ≤ lim inf_{n→∞} E_{B(0,R_{n})}(u_{n}).  (31)

Lemma 4.5. Let u_{n} a sequence in H and R_{n} a sequence in IR^{+} ∪ {+∞} be as in Lemma 3.2. Assume that (27) and (28) hold, and that the following two conditions hold:

sup_{x∈B(0,R_{n})} d(u_{n}(x), 0) ≤ ρ_{0}, ∀ n ∈ IN,  (32)
lim_{n→∞} (sup {d(u_{n}(x), 0) : x ∈ ∂B(0, R_{n})}) = 0.  (33)

Then lim_{n→∞} d(u_{n}, 0) = 0.

Proof of Lemma 4.4. By (28), u_{n} can be extended to a bounded sequence ũ_{n} in H(\overline{B(0, R_{0}}). Then, there exists u ∈ H(\overline{B(0, R_{0}}) such that u_{n} converges weakly to u.

The extension ũ_{n} can be done in such a way that

lim_{n→∞} E_{B(0,R_{0})\B(0,R_{n})}(ũ_{n}) = 0,

from which (31) follows immediately. Since E is a functional of class C^{1}, (30) follows from (27) and (29). Thus, we are left with the proof of (27).

Fix R ∈]0, R_{0}[. By (27) we have:

|E′(u_{n})[ξ]| = |∫_{B(0,R)}(1 + 3 ∥du_{n}∥^{2} · ⟨du_{n}, ∇^{u_{n}}ξ⟩ dx| ≤ ε_{n} · ∥∇^{u_{n}}ξ∥, (34)

for all ξ ∈ H_{2,6}(IR^{3}, IR^{N}) with ξ|_{∂B(0,R)} = 0.

For all y ∈ M, let P (y) denote the orthogonal projection from IR^{N} to the tangent space T_{y}M.

Now, let u be as above and define

ρ_{n} = R − \sqrt{∥u_{n} − u∥_{L^{∞}}},

c_{n}(x) = θ_{n}(|x|) · (u_{n}(x) − u(x)),

where

θ_{n}(s) = \{ \begin{matrix}0 & if 0 ≤ s ≤ ρ_{n}; \\ \frac{s−ρ_{n}}{R−ρ_{n}} & if ρ_{n} < s ≤ R; \\ 1 & if s > R. \end{matrix} .

Since u_{n} tends to u uniformly, then c_{n} tends to 0 uniformly and so does dc_{n}. Finally, we consider:

ξ_{n} = P (u_{n})(u_{n} − u − c_{n}).

By the above construction, we have that ξ_{n} ≡ 0 on ∂B(0, R), and ∥ξ_{n}∥ is bounded independently of n. Then, by (34), we get:

lim_{n→∞} ∫_{B(0,R)}(1 + 3 ∥du_{n}∥^{2}) · ⟨du_{n}, dP (u_{n})[du_{n}]u_{n} − u − c_{n}⟩ dx + (35)
∫_{B(0,R)}(1 + 3 ∥du_{n}∥^{2})⟨du_{n}, P (u_{n})(du_{n} − du − dc_{n}⟩ dx = 0,

because u_{n} is uniformly bounded. Since u_{n} − u − c_{n} tends to 0 uniformly, from (35) we deduce that:

lim_{n→∞} ∫_{B(0,R)}(1 + 3 ∥du_{n}∥^{2}) · ⟨du_{n}, P (u_{n})(du_{n} − du − dc_{n})⟩ dx = 0.

Now,

⟨du_{n}, P (u_{n})(du_{n} − du − dc_{n})⟩ = ⟨du_{n}, du_{n} − du − dc_{n}⟩,

and dc_{n} tends to 0 uniformly. Then,

lim_{n→∞} ∫_{B(0,R)}(1 + 3 ∥du_{n}∥^{2})⟨du_{n}, du_{n} − du⟩ = 0.  (36)

Moreover, since u_{n} tends to u weakly, we have:

lim_{n→∞} ∫_{B(0,R)}(1 + 3 ∥du∥^{2})⟨du, du_{n} − du⟩ = 0.  (37)

Now, there exists ν = ν(k)^{1} such that, for every pair v_{1}, v_{2} it is:

⟨(1 + 3 ∥dv_{2}∥^{2}) v_{2} − (1 + 3 ∥dv_{1}∥^{2}) v_{1}, v_{2} − v_{1}⟩ ≥ ν(k)(1 + 3 ∥dv_{1} − dv_{2}∥^{2}).

Hence, combining (36) and (37) gives (29).

Proof of Lemma 4.5. It will be sufficient to prove that:

lim_{n→∞} sup {d(u_{n}(x), 0) : x ∈ B(0, R_{n}) = 0.}  (38)

Indeed, if (38) holds, by the same arguments used in the proof of Lemma 4.4, it is possible to conclude that

lim_{n→∞} ∫_{B(0,R_{n})} ∥du_{n}∥^{6} dx = 0.

Up to passing to a subsequence, we can assume that lim_{n→∞} R_{n} = R_{0} > 0. By Lemma 4.4 there exists u ∈ H(\overline{B(0, R_{0})}) such that (29) is satisfied.

Now, if R_{0} < +∞, by (33) we have:

lim_{|x|→R_{0}} u(x) = 0,  (39)

while if R_{0} = +∞ (39) is a consequence of (28). Moreover, by Lemma 4.4, u is a critical point of E_{B(0,R_{0})}, and by Remark 3.3 we get a contradiction.

2

1chi é k?

2che cos’é che dá una contradizione?

Proof of Proposition 4.2. If u is a strong solution of (18) on Ω and its behavior on the straight lines .....^{3}, it is not difficult to see that, if u vanishes on a ball, then it vanishes on the whole set Ω. Then, the uniform Hölder continuity of the curves in H, together with Lemmas 3.2 and 3.3, allows to repeat the argument used in the proof of Proposition 3.2 of [17] (concerning the one dimensional case).

5 Proof of the Results

A The Manifold Structure of H_{2,6}(IR^{3}, M). Smoothness of the Functional E. The Euler–Lagrange Equations.

The purpose of this Appendix is to give a formal presentation of regularity property of our main function space and the energy functional E introduced in Section 1. These results and the techniques used to prove them are of technical nature, and they may be skipped without jeopardizing the general understanding of the main issue discussed in the paper. Nevertheless, the problem of determining a Hilbert manifold structure for a family of maps defined on a non-compact base space is by no means secondary. Even though some authors may find it a bit awkward, it must be said that the regularity problem for maps on a non-compact base in its full generality has not been satisfactorily investigated in the classical literature (see [13, 22]).

We will assume throughout this section that M is a compact, m-dimen-sional Riemannian manifold, with m ≥ 4, and p ∈ M is a distinguished point. Whenever M is embedded (isometrically) into some Euclidean space IR^{N} we will implicitly assume that 0 = p ∈ M. Also, all the maps from M to itself will be assumed to fix p.

We fix some notation first. If V_{1} and V_{2} are (real) vector spaces, we denote by L(V_{1}, V_{2}) the real vector space of all linear maps from V_{1} to V_{2}. By V_{1}^{∗} we mean the dual space of V_{1}; then L(V_{1}, V_{2}) is canonically isomorphic to the tensor product V_{1}^{∗} ⊗ V_{2}. If V_{1} and V_{2} are endowed respectively with the (positive definite) inner product ⟨·, ·⟩_{1} and ⟨·, ·⟩_{2}, then V_{1}^{∗} ⊗V_{2} inherits inner product structure ⟨·, ·⟩_{HS}, which, in terms of linear operators T_{1} and T_{2} from V_{1} to V_{2} is written as:

3cosa ci va qui?

⟨T_{1}, T_{2}⟩_{HS} = ∑_{i=1}^{dim(V_{1})} ⟨T_{1}e_{i}, T_{2}e_{i}⟩_{2} = ∑_{i=1}^{dim(V_{1})} ⟨T_{2}^{∗}T_{1}e_{i}, e_{i}⟩_{1} = tr(T_{2}^{∗}T_{1}),

where {e_{i}}_{i} is any orthonormal basis of V_{1}, T_{2}^{∗} is the adjoint map and tr is the usual trace. The product ⟨·, ·⟩_{HS} and the associated norm are called the Hilbert–Schmidt product and norm of linear operators between inner product spaces.

If V_{1} is an inner product space, V_{2} is vector space and B : V_{1} ⊗ V_{1} ⟼ V_{2} is a bilinear map, one defines the trace tr(B) as the vector in V_{2}:

tr(B) = ∑_{i=1}^{dim(V_{1})} B(e_{i}, e_{i}),  (40)

where {e_{i}}_{i} is any orthonormal basis of V_{1}. As for the usual trace, it is easily seen that the right hand side of (40) does not depend on the choice of the orthonormal basis of V_{1}.

We recall that the topology of H_{2,6}(IR^{3}, M) was defined in Section 3 using an isometric embedding of M into some Euclidean space IR^{N}. The first step in the proof of the manifold structure for H_{2,6}(IR^{3}, M) is to show that the topological structure does not depend on the particular choice of an embedding.

Lemma A.1. The topological structure of H_{2,6}(IR^{3}, M) does not depend on the isometric embedding of M into a Euclidean space.

Proof. Given an (isometric) embedding i of M into a Euclidean space IR^{N}, we define a distance in the space C_{c}^{∞}(IR^{3}, M) of all smooth function with compact support from IR^{3} to i(M) by:

dist(u, v) = (∫_{IR^{3}} ∥du(x) − dv(x)∥^{2})^{\frac{1}{2}} + (∫_{IR^{3}} ∥du(x) − dv(x)∥^{6})^{\frac{1}{6}} , (41)

where ∥·∥ denotes the Hilbert–Schmidt norm of a linear operator from IR^{3} to IR^{N}. Observe that the metric (41) does indeed depend on the choice of the embedding; however, we will show that two different embeddings give rise to equivalent metrics. Before doing that, we show that the topological structure of H_{2,6}(IR^{3}, M) as a subset of H_{2,6}(IR^{3}, IR^{N}) is completely determined by the distance (41).

Indeed, we claim that the closure C_{2,6}(IR^{3}, M) of C_{c}^{∞}(IR^{3}, M) with respect to (41) gives H_{2,6}(IR^{3}, M). To see this, first observe that clearly C_{2,6}(IR^{3}, M) ⊆ H_{2,6}(IR^{3}, M).

To prove the opposite inclusion, observe that if u_{n} : IR^{3} ⟼ IR^{N} is a sequence of smooth functions with compact support that converges in the norm (11) to a function u ∈ H_{2,6}(IR^{3}, M), then, by the Morrey’s inequality, in particular u_{n} converges uniformly to u. Hence, by the compactness of M, for n large enough the image u_{n}(IR^{3}) is contained in an open tubular neighborhood U of M in IR^{N}, which may assumed to be contractible onto M.

If r : U ⟼ M is any smooth retraction, then the sequence v_{n} = r ∘ u_{n} is in C_{c}^{∞}(IR^{3}, M) and converges to u in the distance (41). This says that C_{2,6}(IR^{3}, M) ⊇ H_{2,6}(IR^{3}, M).

To conclude the proof, it suffices to show that if i_{i} : M ⟼ IR^{N_{i}}, i = 1, 2 are two different isometric embeddings, then the corresponding distances d_{i} given by (41) are equivalent.

To this aim, observe first that we can assume N_{1} = N_{2} = N. Moreover, if N is taken sufficiently large, by the stable equivalence (see [3]) we can assume that the normal bundles of i_{1}(M) and i_{2}(M) in IR^{N} are isomorphic. Then, since M is compact, the map i_{1} ∘ i_{2}^{−1} can be extended to a diffeomorphism I from a tubular neighborhood U_{2} of i_{2}(M) to a tubular neighborhood U_{1} of i_{1}(M). To see this, one simply extends i_{1} ∘ i_{2}^{−1} isometrically in the directions which are normal to M.

Since I is an isometry on M (so that ∥dI∥ ≡ 1 on M), then we can assume that ∥dI∥ ∈ [\frac{1}{2}, \frac{3}{2}] in U_{2}. This implies immediately that if u_{i} : IR^{3} ⟼ i_{2}(M), i = 1, 2, are smooth functions, setting ũ_{i} = i_{1} ∘ i_{2}^{−1} ∘ u_{i} : IR^{3} ⟼ i_{1}(M), one has:

\frac{1}{2} d_{2}(u_{1}, u_{2}) ≤ d_{1}(ũ_{1}, ũ_{2}) ≤ \frac{3}{2} d_{2}(u_{1}, u_{2}).

Thus, d_{1} and d_{2} are equivalent and we are done.

By similar arguments, i.e. considering a diffeomorphism of open sets containing the image of two different isometric embeddings of M into IR^{N}, we will prove later that also the differential structure of M is independent of the embedding (see Proposition A.3).

In order to prove the differential structure of our function space and the regularity of our functional, we give here a very brief account of the theory of covariant differentiation along maps. A detailed exposition of the theory may be found in classical textbooks of differential geometry, like for instance [11].

We recall that E is a vector bundle over the manifold X, a connection ∇ on E is an IR-linear map ∇ : Γ(E) ⟼ Γ(T X^{∗} ⊗ E) satisfying the product rule:

∇(f · W) = df ⊗ W + f · ∇W,  (42)

where f is any smooth function on X, W ∈ Γ(E), Γ(·) denotes the space of (smooth) sections of E, and T X^{∗} is the cotangent bundle of X. Here, T X^{∗} ⊗ E denotes the vector bundle over X with fiber (T X^{∗} ⊗ E)_{x} = T_{x}X^{∗} ⊗ E_{x} ≃ L(T_{x}X, E_{x}).

Roughly speaking, a connection ∇ over a vector bundle E allows to compute the directional derivative ∇_{v}W, where v is a tangent vector to X and W is a section of E, i.e. a vector field on X with values in E. The condition (42) is then translated into the familiar product rule for derivatives:

∇_{v}(f · W) = df[v] · W + f ∇_{v}W.

If M is a Riemannian manifold and T M is the tangent bundle, the Levi– Civita connection of M is the unique symmetric connection on T M which is compatible with the metric of M (see [24]).

If ϕ : IR^{3} ⟼ M is a smooth map, we denote by ϕ^{∗}(T M) the vector bundle on IR^{3} given by:

ϕ^{∗}(T M) = {(x, v) ∈ IR^{3} × T M : v ∈ T_{ϕ(x)}M},  (43)

with projection π(x, v) = x. A section W of the bundle ϕ^{∗}(T M) is a vector field along the map ϕ, i.e. a smooth map W : IR^{3} ⟼ T M such that W (x) ∈ T_{ϕ(x)}M for all x ∈ IR^{3}. These vector fields are also called infinitesimal variations of the map ϕ: every such field W defines a variation ϕ_{t} of ϕ, with t ∈] − δ, δ[, by ϕ_{t}(x) = exp(t · W (x)), where exp is the exponential map of M.

Observe that if W is a vector field on M, i.e. a smooth section of the bundle T M, then the composition W ∘ ϕ defines a smooth section of the bundle ϕ^{∗}(T M); however, if ϕ is not an immersion, not all the sections of ϕ^{∗}(T M) are of this form.

It can be proven by elementary methods that, if ∇ is the Levi–Civita connection of the Riemannian manifold M, then there exists a unique connection ∇^{ϕ} in the vector bundle ϕ^{∗}(T M) such that:

∇_{V}^{ϕ} (W ∘ ϕ) = ∇_{dϕ[V]}W,  (44)

for all vector field W on M and all vector fields V in IR^{3}. Namely, the two conditions (42) and (44) imply that locally ∇^{ϕ} has the form:

∇_{v}^{ϕ}W = ∑_{i=1}^{dim(M)} (df_{i}[v] · W_{i} ∘ ϕ + f_{i} · ∇_{dϕ[v]}W_{i})  (45)

where v ∈ T M, W = ∑ f_{i} · W_{i}, {W_{i}}_{i} is a (local) basis of smooth sections of T M, and f_{i} are smooth functions. Conversely, it is easily checked that (45) defines a connection ∇^{ϕ} with the required properties.

If ϕ : IR^{3} ⟼ M is a smooth map, then the differential dϕ maps vector fields in IR^{3} into vector fields along ϕ, hence it is a section of the bundle (IR^{3})^{∗} ⊗ ϕ^{∗}(T M). The connection ∇^{ϕ} on ϕ^{∗}(T M) induces a connection \overline{∇} on (IR^{3})^{∗} ⊗ ϕ^{∗}(T M), given by:

(\overline{∇}_{V} ω)[ζ] = ∇_{V}^{ϕ} (ω[ζ]) − ω(∇_{V}^{IR^{3}}ζ),  (46)

where V is a vector field in IR^{3}, ζ : IR^{3} ⟼ T M is a vector field along ϕ, ω(x) : IR^{3} ⟼ T_{ϕ(x)}M is a linear map for all x ∈ IR^{3} and ∇^{IR^{3}} is the Levi–Civita connection of the Euclidean space IR^{3}. For x ∈ IR^{3} and ζ a section of ϕ^{∗}(T M), when we omit the lower index ∇^{ϕ}ζ(x) we mean the map in L(IR^{3}, T_{ϕ(x)}M) given by v ⟼ (∇_{v}^{ϕ}ζ)(x). The symbol ⟨·, ·⟩_{0} will denote the Hilbert–Schmidt inner product in L(IR^{3}, T_{ϕ(x)}M). The covariant derivative \overline{∇} dϕ is called the second fundamental form of ϕ. We will think of \overline{∇} dϕ as a bilinear map:

\overline{∇} dϕ [V_{1}, V_{2}],

that produces a vector field along ϕ when V_{1} and V_{2} are vector fields in IR^{3}. Observe that, by (46), \overline{∇} dϕ is C^{∞}(IR)-linear in both variables.

We define a vector field τ(ϕ) along ϕ by taking the trace of \overline{∇} dϕ:

τ(ϕ) = tr(\overline{∇} dϕ) = ∑_{i=1}^{3} \overline{∇} dϕ [\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{i}}] = ∑_{i=1}^{3} ∇_{∂_{i}}^{ϕ}dϕ [\frac{∂}{∂x_{i}}] = ∑_{i=1}^{3} ∇_{∂_{i}}^{ϕ} \frac{∂ϕ}{∂x_{i}} . (47)

The vector field τ(ϕ) is called the tension of ϕ. If we choose local charts (y_{1}, y_{2}, …, y_{m}) on M around the point ϕ(x), then, denoting by ϕ^{k} and τ^{k}(ϕ) respectively the k-th components of ϕ and τ(ϕ), it is easily computed:

τ^{k}(ϕ)(x) = Δϕ^{k}(x) + ∑_{a,b=1}^{m} ∑_{i,j=1}^{3} Γ_{ab}^{k}(ϕ(x)) \frac{∂ϕ^{a}}{∂x_{i}} (x) \frac{∂ϕ^{b}}{∂x_{j}} (x),  (48)

where Δ is the Laplace operator in IR^{3} and Γ_{ab}^{k} are the Christoffel symbols of the Riemannian metric of M.

If ϕ is an absolutely continuous map, one extends the concepts of the bundle ϕ^{∗}(T M) and the connection ∇^{ϕ} in a canonical way. Similarly, one defines the second fundamental form \overline{∇}dϕ and the tension field τ(ϕ) for C^{1}- functions having absolutely continuous derivative.

For ϕ ∈ H_{2,6}(IR^{3}, M), we denote by V (ϕ) the set of all absolutely continuous vector fields along ϕ, i.e. the set of all absolutely continuous sections of the vector bundle ϕ^{∗}(M).

Before proving the main result of the Section, we need the following simple criterion for the differentiability of maps between Banach spaces, which is basically contained in [22], Theorem 11.3.

If E and F are Banach spaces, we denote by L(E, F) the set of all continuous linear maps from E to F; if Λ is a family of linear maps from F to some other vector space, we say that Λ separates the points of F if for all x, y ∈ F, with x ≠ y, there exists λ ∈ Λ such that λ(x) ≠ λ(y). Equivalently, Λ separates the points of F if for all x ≠ 0 in F there exists λ ∈ Λ such that λ(x) ≠ 0.

Lemma A.2. Let E and F be real Banach spaces, f : E ⟼ F a map, and g : F ⟼ L(E, F) a continuous map. Suppose that there exists a family Λ of continuous linear maps from F to IR^{s} that separates the points of F and such that, for all x, v ∈ E and for all λ ∈ Λ, it is:

\frac{d}{dt}|_{t=0}λ(f(x + tv)) = λ(g(x)v).

Then, f is a map of class C^{1} and df = g.

Proof. Let x ∈ E be fixed. For all h ∈ E, we set r(h) = f(x + h) − f(x) − g(x)h; since g(x) is linear and continuous, to prove that f is differentiable at x and df(x) = g(x) it suffices to show that lim_{h→0} r(h) · ∥h∥^{−1} = 0.

To this aim, consider the curve γ(t) = g(x + th)h in F. Since γ is continuous, then there exists the F -valued integral:

I = ∫_{0}^{1} γ(t) dt ∈ F.

We claim that I = f(x + h) − f(x). Indeed, if it were I ≠ f(x + h) − f(x), by the separability condition on Λ there would exist λ ∈ Λ such that

λ(I) ≠ λ(f(x + h)) − λ(f(x)).  (49)

By the continuity of λ, it is:

λ(I) = λ (∫_{0}^{1} γ(t) dt) = ∫_{0}^{1} λ(γ(t)) dt = ∫_{0}^{1} λ(g(x + th)h) dt. (50)

On the other hand, our differentiability condition on the function q(t) = λ(f(x + th)) implies that q′(t) = λ(g(x + th)h), hence, from (50) we get:

λ(f(x + h)) − λ(f(x)) = q(1) − q(0) = ∫_{0}^{1} q′(t) dt =
= ∫_{0}^{1} λ(g(x + th)h) dt = λ(I).  (51)

The identity (51) contradicts (49), hence r(h) = ∫_{0}^{1} (g(x + th)h − g(x)h) dt. Since the evaluation at h is a continuous function in L(E, F), then we have:

r(h) = (∫_{0}^{1} (g(x + th) − g(x)) dt) h,

from which we get:

∥r(h)∥ ≤ ∥h∥ ∫_{0}^{1} ∥g(x + th) − g(x)∥ dt.

By the continuity of g, the latter integral tends to 0 as h → 0, so that ∥r(h)∥ · ∥h∥^{−1} goes to 0 as h → 0. This implies that df(x) = g(x). Since g is continuous, then f is C^{1} and we are done.

Observe that in the proof of Lemma A.2 we have never used the fact that the functions in Λ have image in IR^{s}, that could be substituted in the statement of the Lemma by any other Banach space G. As a matter of fact, it is also unnecessary to fix the image space of the functions in Λ, and it would be enough to have a different space G_{λ} for each λ in Λ.

We are now ready to describe the differentiable structure of H_{2,6}(IR^{3}, M):

Proposition A.3. Let M be an m-dimensional compact Riemannian manifold of class C^{n}, with n ≥ 3. The set H_{2,6}(IR^{3}, M) has the structure of a complete Banach manifold of class C^{n−2} modeled on the Banach space H_{2,6}(IR^{3}, IR^{m}).

For ϕ ∈ H_{2,6}(IR^{3}, M), the tangent space T_{ϕ}H_{2,6}(IR^{3}, M) is identified with the space:

T_{ϕ}H_{2,6}(IR^{3}, M) =  (52)
= {ζ ∈ V (ϕ) | ∫_{IR^{3}} (⟨∇^{ϕ}ζ, ∇^{ϕ}ζ⟩_{0} + ⟨∇^{ϕ}ζ, ∇^{ϕ}ζ⟩_{0}^{3}) dx < +∞}.

If M is isometrically embedded in the Euclidean space IR^{N}, then the tangent space at ϕ can be written as:

T_{ϕ}H_{2,6}(IR^{3}, M) = {ζ ∈ T_{ϕ}H_{2,6}(IR^{3}, IR^{N}) | ζ(x) ∈ T_{ϕ(x)}M ∀ x}. (53)

Proof. Using the Nash embedding theorem, we assume temporarily that M is isometrically embedded in IR^{N} as a submanifold of class C^{n}, and we assume that 0 ∈ M. Let k = N − m be the codimension of M in IR^{N}. We consider the Banach space H_{2,6}(IR^{3}, IR^{N}).

By the compactness of M, there exists an open tubular neighborhood U of M in IR^{N} and a map of class C^{n} r : U ⟼ M which is a retraction, i.e. r(x) = x for all x ∈ M. The map r is an orthogonal projection onto M.

For A ⊂ IR^{N} we denote by \overline{A} the closure of A in IR^{N}. The set:

H_{2,6}(IR^{3}, IR^{N}; U) = \{ϕ ∈ H_{2,6}(IR^{3}, IR^{N}) : \overline{ϕ(IR^{3})} ⊂ U\},

is an open subset of IR^{3}. Namely, since \overline{ϕ(IR^{3})} is a compact subset of IR^{N} for all ϕ ∈ H_{2,6}(IR^{3}, IR^{N}), then inf_{x∈IR^{3}} dist(ϕ(x), ∂U) > 0. So, the same is true for all ϕ̃ in a small neighborhood of ϕ in H_{2,6}(IR^{3}, IR^{N}), because of the continuous injection of H_{2,6} in L^{∞} (see (12)). It follows that H_{2,6}(IR^{3}, IR^{N}; U) is an open submanifold of H_{2,6}(IR^{3}, IR^{N}).

We need the following:

Lemma A.4. Let U be an open subset of IR^{N} that contains 0. Let Θ : U → IR^{k} be a map of class C^{n} (n ≥ 3) such that Θ(0) = 0 and such that ∥d^{k}Θ∥ admits a continuous extension to the closure \overline{U} for all k = 0, 1, …, n. Then, the map T : H_{2,6}(IR^{3}, IR^{N}; U) ⟼ H_{2,6}(IR^{3}, IR^{k}) defined by:

T (ϕ) = Θ ∘ ϕ

is of class C^{n−2}, and

dT (ϕ)[ζ] = dΘ(ϕ) ∘ ζ,  (54)

for all ϕ ∈ H_{2,6}(IR^{3}, IR^{N}; U) and all ζ ∈ T_{ϕ}H_{2,6}(IR^{3}, IR^{N}).

Proof of Lemma A.4. First of all we observe that, by the regularity of Θ and the fact that Θ(0) = 0 and ∥dΘ∥ is bounded on U, the map T is well defined, i.e. T (ϕ) ∈ H_{2,6}(IR^{3}, IR^{k}) for all ϕ ∈ H_{2,6}(IR^{3}, IR^{N}; U).

By our hypothesis, we can extend Θ to a map of class C^{n} defined on IR^{N} and such that ∥d^{k}Θ∥ is bounded on IR^{N} for all k = 0, 1, …, n. In this way, we can extend the map T to the entire space H_{2,6}(IR^{3}, IR^{N}); we will prove the Lemma for this extension, and the conclusion will follow easily by restricting T on the open subset H_{2,6}(IR^{3}, IR^{N}; U).

To prove the continuity of T, we consider the Banach space

B = H_{2,6}(IR^{3}, IR^{k}) ⊕ C,

where C is the set of constant functions from IR^{3} to IR^{k}, and the norm of B is given by:

∥(ϕ, y)∥_{B} = ∥ϕ∥_{2,6} + ∥y∥,

where y ∈ IR^{k} is the value of the constant function.

We define a map T̃ : C_{c}^{∞}(IR^{3}, IR^{N}) ⟼ B by:

T̃(ϕ) = (Θ ∘ ϕ − Θ(0), Θ(0)).  (55)

We temporarily ignore the fact that Θ(0) = 0 and we prove the C^{1}-regularity of an extension of the map T̃ without using this fact. Observe indeed that (55) defines T̃ independently of the value Θ(0). This technicality is needed to be able to use the induction on the derivatives of Θ, that in general do not satisfy the equality d^{k}Θ(0) = 0 for k ≥ 1.

Since C_{c}^{∞}(IR^{3}, IR^{N}) is dense in H_{2,6}(IR^{3}, IR^{N}), to prove that T̃ admits a continuous extension to H_{2,6}(IR^{3}, IR^{N}) we show that T̃ is locally uniformly continuous (observe that T̃ is not linear).

To this aim, let ϕ_{1}, ϕ_{2} ∈ C_{c}^{∞}(IR^{3}, IR^{N}) be fixed, and let K denote a compact convex subset of IR^{N} containing the images of ϕ_{1} and ϕ_{2}. Using the triangular inequality and the Mean Value Theorem, one sees easily that, for all x ∈ IR^{3}, the following inequality holds:

∥dT (ϕ_{1}(x)) dϕ_{1}(x) − dT (ϕ_{2}(x)) dϕ_{2}(x)∥ ≤  (56)
≤ c_{1} · ∥dϕ_{1}(x) − dϕ_{2}(x)∥ + c_{2} · ∥dϕ_{2}(x)∥ · ∥ϕ_{1}(x) − ϕ_{2}(x)∥,

where c_{1} = max_{K} ∥dT ∥ and c_{2} = max_{S} ∥d^{2}T ∥, and S is the segment between ϕ_{1}(x) and ϕ_{2}(x). From (56) it follows immediately that T̃ is locally uniformly continuous, and so it admits a continuous extension to H_{2,6}(IR^{3}, IR^{N}).

Finally, we define \overline{T} : B ⟼ B by:

\overline{T} (ϕ, y) = (Θ ∘ ϕ − Θ(0), Θ(0) + Θ(y)).

As T̃ is continuous, then also \overline{T} is continuous, and this was established using the C^{2}-regularity of Θ and the boundedness of the first two derivatives of Θ, without using the assumption that Θ(0) = 0.

To prove that \overline{T} is of class C^{1}, we use the criterion of Lemma A.2, by exhibiting explicitly the differential of \overline{T}. Namely, for each (ϕ, y) ∈ B, we consider the map g(ϕ, y) : B ⟼ B given by:

g(ϕ, y)(ζ, w) = (dΘ(ϕ(·)) ∘ ζ, dΘ(y)[w]).  (57)

Since dΘ is a map of class C^{n−1} and its first two derivatives are bounded in IR^{3}, then, by the previous argument, the map (ϕ, y) ⟼ g(ϕ, y) is continuous from B to L(B, B).

Let Λ = {λ_{x} : x ∈ IR^{3}} be the family of linear maps λ_{x} : B ⟼ IR^{4} defined by:

λ_{x}(ϕ, y) = (ϕ(x), y).

Clearly, each λ_{x} is linear and continuous in B. Infact, the evaluation at the point x is continuous in H_{2,6}(IR^{3}, IR^{N}), because it is continuous in the C^{0}- topology. Moreover, Λ separates the points of B, because the evaluations at the points of IR^{3} clearly separates the points of H_{2,6}(IR^{3}, IR^{N}).

For every x ∈ IR^{3}, we have:

λ_{x}(g(ϕ, y)(ζ, w)) = (dΘ(ϕ(x))[ζ(x)], dΘ(y)[w]),  (58)

for all (ϕ, y) and (ζ, w) in B.

On the other hand, using the definition of \overline{T} and the differentiability of Θ, we have:

\frac{d}{dt}|_{t=0}λ_{x}((ϕ, y)+ t(ζ, w)) = \frac{d}{dt}|_{t=0}λ_{x} ((ϕ + tζ, y + tw)) =
= \frac{d}{dt}|_{t=0}λ_{x}(Θ ∘ (ϕ + tζ), Θ(0) + Θ(y + tw)) =  (59)
= \frac{d}{dt}|_{t=0}(Θ(ϕ(x) + tζ(x)) − Θ(0), Θ(0) + Θ(y + tw)) =
= (dΘ(ϕ(x))[ζ(x)], dΘ(y)[w]).

From (58) e (59), using Lemma A.2, we have that \overline{T} is a C^{1}-map, whose differential is given by d\overline{T} (ϕ, y) = g(ϕ, y) of (57).

By induction, we have that \overline{T} is of class C^{n−2}, and so is the map T of the hypothesis, which is obtained by composing \overline{T} and the projection onto the first coordinate from B to H_{2,6}(IR^{3}, IR^{N}).

Observe that the differential of T is obtained by taking the first coordinate of the differential of \overline{T}, which gives (54) and concludes the proof.

Going back to the proof of Proposition A.3, we assume for simplicity that the normal bundle of M in IR^{3} is trivial; the general case will follow from this considering that the normal bundle of M can be written as the clutching of a finite number of trivial bundles over open submanifolds of M. The set U can be considered an open subset of the normal bundle of M, with r : U ⟼ M the projection. We choose a trivialization of U given by a map of class C^{n} ψ = (ψ^{1}, ψ^{2}) : U ⟼ M × IR^{k}, satisfying

ψ(x + λv) = (x, λψ^{2}(v)),

for all x ∈ M, v orthogonal to T_{x}M and λ real number sufficiently small. By construction, ψ^{1} = r. Up to shrinking the open set U, we can assume that the map ψ^{2} : U ⟼ IR^{k} satisfies the continuity hypotheses of Lemma A.4, because ψ^{2} is of class C^{n} in U, 0 ∈ M and ψ(0) = (0, 0). It follows that the map F : H_{2,6}(IR^{3}, IR^{N}; U) ⟼ H_{2,6}(IR^{3}, IR^{k}) defined by:

F(ϕ) = ψ^{2} ∘ ϕ,

is of class C^{n−2}, and

dF(ϕ)[ζ](x) = dψ^{2}(ϕ(x))[ζ(x)],  (60)

for all ϕ ∈ H_{2,6}(IR^{3}, IR^{N}), all ζ ∈ T_{ϕ}H_{2,6}(IR^{3}, IR^{N}) and all x ∈ IR^{3}. Observe that, by construction, H_{2,6}(IR^{3}, M) = F^{−1}(0). By the Implicit Function Theorem, in order to prove that H_{2,6}(IR^{3}, M) is

a C^{n−2}-submanifold of H_{2,6}(IR^{3}, IR^{N}) it suffices to show that dF is surjective on the points of H_{2,6}(IR^{3}, M) and that Ker(dF(ϕ)) has a closed complement in T_{ϕ}H_{2,6}(IR^{3}, IR^{N}).

To settle the first question, we simply observe that, by construction, the restriction of dϕ^{2}(z) to the orthogonal space T_{z}M^{⊥} gives an isomorphism with image IR^{k}, for all z ∈ M.

Then, if w ∈ H_{2,6}(IR^{3}, IR^{k}) ≃ T_{F(ϕ)}H_{2,6}(IR^{3}, IR^{k}), setting:

ζ(x) = (dψ^{2}(ϕ(x)))^{−1} [w(x)],

we have that ζ ∈ T_{ϕ}H_{2,6}(IR^{3}, IR^{N}) and, from (60), dF(ϕ)[ζ] = w.

As to the splitting condition on the kernel of dF, observe that this space consists of vector fields that are tangent to M, and it is complemented by the vector fields that are normal to M. This proves that H_{2,6}(IR^{3}, M) is a C^{n−2}-submanifold of H_{2,6}(IR^{3}, IR^{N}).

At each point ϕ ∈ H_{2,6}(IR^{3}, M), by the Implicit Function Theorem the tangent space T_{ϕ}H_{2,6}(IR^{3}, M) is given by the kernel of the differential dF(ϕ); as we have observed:

T_{ϕ}H_{2,6}(IR^{3}, M) = Ker(dF(ϕ)) =
= {ζ ∈ T_{ϕ}H_{2,6}(IR^{3}, IR^{N}) : ζ(x) ∈ T_{ϕ(x)}M, ∀ x ∈ IR^{3}},

obtaining (53). Observe that, since dim(T_{ϕ(x)}M) = dim(M) = m, then the Banach space model for H_{2,6}(IR^{3}, M) is H_{2,6}(IR^{3}, IR^{m}). Observe also that, arguing as in the proof of Lemma A.1 and using the fact that, by Lemma A.4, the composition on the left with a diffeomorphism is a C^{1}-map, we have that the differentiable structure of H_{2,6}(IR^{3}, M) does not depend on the isometric embedding of M in IR^{N}. Then, (52) is an intrinsic rewriting of (53).

Finally, the completeness of H_{2,6}(IR^{3}, M) follows easily from the completeness of H_{2,6}(IR^{3}, IR^{N}) and the fact that M is closed. Namely, every Cauchy sequence u_{n} ∈ H_{2,6}(IR^{3}, M) is convergent to u in H_{2,6}(IR^{3}, IR^{N}). Since the convergence is also uniform and M is closed, then u(IR^{3}) ⊂ M and u_{n} converges to u in H_{2,6}(IR^{3}, M).

The continuous embedding of H_{2,6}(IR^{3}, M) into C_{0}(IR^{3}, M) implies that the two spaces have the same homotopy type:

Proposition A.5. The inclusion map J : H_{2,6}(IR^{3}, M) ⟼ C_{0}(IR^{3}, M) is a homotopy equivalence.

Proof. The proof goes along the same lines of the proof of Theorem 13.14 of [22]. Namely, using the fact that and so IR^{3} × M is closed in IR^{3} × IR^{N}, one reduces the proof to the case where M is isometric to a Euclidean space. Observe that the compactness of M plays no role in this proof. Then, the map J is a continuous map between Banach spaces with dense image, because J(H_{2,6}(IR^{3}, M)) contains the set of compactly supported smooth functions. Hence, J is a homotopy equivalence (see Lemma 13.12 of [22]).

We recall the definition of the functional E : H_{2,6}(IR^{3}, M) ⟼ IR^{+} given in Section 1:

E(ϕ) = ∫_{IR^{3}} β (⟨dϕ(x), dϕ(x)⟩_{0}) dx,  (61)

where β(λ) = λ + λ^{3}.

Proposition A.6. The functional E : H_{2,6}(IR^{3}, M) ⟼ IR^{3} is of class C^{n−2}. For ϕ ∈ H_{2,6}(IR^{3}, M) and ζ ∈ T_{ϕ}H_{2,6}(IR^{3}, M), the Gateaux derivative of E at the point ϕ and in the direction ζ is given by:

E′(ϕ)[ζ] = 2 ∫_{IR^{3}} (1 + 3⟨dϕ(x), dϕ(x)⟩_{0}^{2}) · ⟨∇^{ϕ}ζ(x), dϕ(x)⟩_{0} dx. (62)

Proof. We identify the space IR^{3N} with the space L(IR^{3}, IR^{N}) endowed with the Hilbert–Schmidt norm of linear operators. Let’s consider the Banach space L_{2,6}(IR^{3}, IR^{3N}) given by the completion of C_{c}^{∞}(IR^{3}, IR^{N}) with respect to the norm:

∥u∥_{L_{2,6}} = ∥u∥_{L^{2}} + ∥u∥_{L^{6}}.

The map d : ϕ ⟼ dϕ is clearly a smooth map from H_{2,6}(IR^{3}, IR^{N}) to L_{2,6}(IR^{3}, IR^{3N}), because it is linear and continuous.

For p > 1, the map E_{p} : L^{p}(IR^{3}, IR^{3N}) ⟼ IR^{+} given by E_{p}(u) = ∥u∥_{L^{p}}^{p} is smooth, hence E_{2,6} = E_{2} + E_{6} is smooth on L^{2} ∩ L^{6}. Since L_{2,6}(IR^{3}, IR^{3N}) is a linear subspace of L^{2}(IR^{3}, IR^{3N}) ∩ L^{6}(IR^{3}, IR^{3N}) (not closed), and the topology of L_{2,6} is generated by the topologies induced by L^{2} and L^{6}, then the restriction of E_{2,6} to L_{2,6}(IR^{3}, IR^{3N}) is also smooth.

So, the composite map E_{2,6} ∘ d = E is smooth on the Banach space H_{2,6}(IR^{3}, IR^{N}), and, by Proposition A.3, E is a functional of class C^{n−2} on the submanifold H_{2,6}(IR^{3}, M).

Since (62) defines a formula which is H_{2,6}-continuous in ϕ and in ζ, in order to prove that (62) holds it suffices to consider the case of ϕ ∈ C_{0}^{∞}(IR^{3}, M) and ζ a smooth section of ϕ^{∗}(T M) with compact support.

To this aim, we consider the functional Ẽ given by:

Ẽ(ϕ) = ∫_{IR^{3}} ⟨dϕ(x), dϕ(x)⟩_{0} dx.

We have just proven the regularity of Ẽ; if ϕ_{t}(x) = exp(t·ζ(x)) is the variation associated to the vector field ζ, then we have:

Ẽ′(ϕ)[ζ] = \frac{d}{dt}|_{t=0}Ẽ(ϕ_{t}) = \frac{d}{dt}|_{t=0} ∫_{IR^{3}} ⟨dϕ_{t}(x), dϕ_{t}(x)⟩_{0} dx..

Since ϕ and ζ have compact support, we have:

Ẽ′(ϕ)[ζ] = ∫_{IR^{3}} \frac{d}{dt}|_{t=0}⟨dϕ_{t}, dϕ_{t}⟩_{0} dx =  (63)
= 2 ∫_{IR^{3}} ⟨∇^{ϕ} \frac{d}{dt}ϕ_{t}, dϕ_{t}⟩_{0}|_{t=0} dx = 2 ∫_{IR^{3}} ⟨∇^{ϕ}ζ(x), dϕ(x)⟩_{0} dx.

Finally, (62) follows immediately from (61) and (63).

Proposition A.7. The critical points of E satisfy the Euler–Lagrange equation of the functional E, given by:

τ(ϕ) + 3 div (⟨dϕ, dϕ⟩_{0}^{2} · dϕ) = 0,  (64)

where τ(ϕ) is the tension of the map ϕ (see (47) and (48) ) and ⟨dϕ, dϕ⟩_{0} is the Hilbert–Schmidt inner product given by:

⟨dϕ, dϕ⟩_{0} = ∑_{i=1}^{3} ⟨\frac{∂ϕ}{∂x_{i}} , \frac{∂ϕ}{∂x_{i}} ⟩.

Using local coordinates on the manifold M, the differential equation (64) is written as:

Δϕ^{k}(x) + ∑_{a,b=1}^{m} ∑_{i,j=1}^{3} Γ_{ab}^{k}(ϕ(x)) \frac{∂ϕ^{a}}{∂x_{i}} (x) \frac{∂ϕ^{b}}{∂x_{j}} +
+3 div (∥dϕ(x)∥^{4} · ∇ϕ^{k}(x)) = 0, k = 1, …, m, (65)

where Δ and ∇ are respectively the Laplacian and the gradient operators in IR^{3} and Γ_{ab}^{k} are the Christoffel symbols of the metric of M.

Proof. The regularity for the critical points of E is proven in Theorem 3.2, hence they are solutions of the Euler–Lagrange equations of E in a strong sense.

To obtain an explicit form of the Euler–Lagrange equations for the functional E, we need an integration by parts in (62). By definition of ⟨·, ·⟩_{0}, we have:

∫_{IR^{3}} ⟨∇^{ϕ}ζ(x), dϕ(x)⟩_{0} dx = ∑_{i=1}^{3} ∫_{IR^{3}} ⟨∇_{∂_{i}}^{ϕ}ζ(x), dϕ(x)\frac{∂}{∂x_{i}} ⟩ dx =

= ∑_{i=1}^{3} ∫_{IR^{3}} ⟨∇_{∂_{i}}^{ϕ}ζ(x), \frac{∂ϕ}{∂x_{i}} (x)⟩ dx.  (66)

Now, since ζ has compact support, for every i = 1, 2, 3 it is:

0 = ∫_{IR^{3}} \frac{∂}{∂x_{i}} ⟨ζ, \frac{∂ϕ}{∂x_{i}} ⟩ dx = ∫_{IR^{3}} (⟨∇_{∂_{i}}^{ϕ}ζ, \frac{∂ϕ}{∂x_{i}} ⟩ + ⟨ζ, ∇_{∂_{i}}^{ϕ} \frac{∂ϕ}{∂x_{i}} ⟩) dx.

Recalling (47), we therefore get:

∫_{IR^{3}} ⟨∇^{ϕ}ζ(x), dϕ(x)⟩_{0} dx = − ∫_{IR^{3}} ⟨ζ(x), τ(ϕ)(x)⟩ dx.  (67)

By a similar argument, we get:

∫_{IR^{3}} ⟨dϕ, dϕ⟩_{0}^{2}⟨∇^{ϕ}ζ, dϕ⟩_{0} dx = − ∑_{i=1}^{3} ∫_{IR^{3}} ⟨ζ, ∇_{∂_{i}}^{ϕ} (⟨dϕ, dϕ⟩_{0}^{2} · \frac{∂ϕ}{∂x_{i}} ) ⟩ dx =
= − ∫_{IR^{3}} ⟨dϕ, dϕ⟩_{0}^{2} · ⟨ζ, τ(ϕ)⟩ dx +  (68)

− ∑_{i=1}^{3} ∫_{IR^{3}} ⟨ζ, \frac{∂}{∂x_{i}} (⟨dϕ, dϕ⟩_{0}^{2}) · \frac{∂ϕ}{∂x_{i}} ⟩ dx.

From (62), (67) and (68), passing to a strong equality we obtain the Euler– Lagrange equations of E:

(1 + 3⟨dϕ, dϕ⟩_{0}^{2}) τ(ϕ) + 3 ∑_{i=1}^{3} \frac{∂}{∂x_{i}} (⟨dϕ, dϕ⟩_{0}^{2}) · \frac{∂ϕ}{∂x_{i}} = 0.  (69)

The equations (64) and (65) follow immediately from (69) and (48)).

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