Bifurcation of critical points along gap-continuous families of subspaces

We consider the restriction of twice differentiable functionals on a Hilbert space to families of subspaces that vary continuously with respect to the gap metric. We study bifurcation of branches of critical points along these families, and apply our results to semilinear systems of ordinary differential equations.


Introduction
Let H be a real separable Hilbert space and J : H → R a C 2 -functional. We denote the derivative of J at u ∈ H by d u J ∈ L(H, R) and, henceforth, assume that d 0 J = 0, i.e. 0 ∈ H is a critical point of J . Usually, critical points of a functional J on a suitable H are studied since they correspond to solutions of a related differential equation. Accordingly, critical points of a restriction J | H ′ : H ′ → R may correspond to solutions of such a differential equation but under additional constraints. In [1] Abbondandolo and Majer study the Grassmannian of a Hilbert space H, i.e. the set of all closed subspaces of H. On this set there is a canonical metric which is induced by orthogonal projections, and consequently we can define paths {H t } t∈ [a,b] in it. Using that for each t ∈ [a, b] the element 0 ∈ H t is a critical point of the restriction J | Ht : H t → R, the goal of this paper is to study bifurcations from this branch of critical points (for the formal definition, see Definition 3.1). Here, we provide existence results for bifurcations in terms of the second derivative of J at the critical point 0, which are based on [8] and [15]. To this aim, we introduce a family of functionals f t : H → R, t ∈ [a, b], such that each f t involves the orthogonal projection onto the space H t , and is such that its critical points are the critical points of the restriction J | Ht . Consequently, 0 ∈ H is a critical point of any f t : H → R, t ∈ [a, b], and, by considering the second derivative d 2 0 f t of f t at 0 we can define a path {L t } t∈[a,b] of bounded selfadjoint operators by the Riesz Representation Theorem. The assumptions of our theorems ensure that each L t is actually a Fredholm operator, and we prove that bifurcation of critical points of f along {H t } t∈ [a,b] arise if the spectral flow of L : t → L t does not vanish. Let us recall that the spectral flow is an integer valued homotopy invariant for paths of selfadjoint Fredholm operators that was introduced by Atiyah, Patodi and Singer in [3], and whose relevance in bifurcation theory was discovered in [8]. For example, if all operators L t have a finite Morse index µ Morse (L t ), then the spectral flow of L is just the difference of the Morse indices at the endpoints, i.e. µ Morse (L a ) − µ Morse (L b ). Then a non-vanishing spectral flow of L corresponds to a jump in the Morse indices of L, which implies bifurcation of critical points of f by a well known theorem in bifurcation theory (cf. [14, §8.9], [11,§II.7.1]). However, if µ Morse (L t ) = +∞ for some t ∈ [a, b], then the spectral flow may depend on the whole path L and not only on its endpoints, which makes the theory more complicated. The paper is structured as follows. In Section 2, we introduce some preliminaries that we need in order to state our theorems. First, we recall some facts about the Grassmannian of a Hilbert space H, essentially following Abbondandolo and Majer's paper [1]. However, we also state and prove a folklore result which shows that the kernels of families of surjective bounded operators on H yield paths in the Grassmannian and which we use in the final section for creating examples. Second, we briefly recall the definition of the spectral flow from [8]. In Section 3, we introduce the path L and state our main theorems and a corollary, which we prove in Section 4. Finally, in Section 5 we apply our theory to a Dirichlet problem for indefinite semilinear ordinary differential operators.

Grassmannians and spectral flows
Here and in the following, let H be a real separable Hilbert space of infinite dimension, and we denote by L(H) the Banach space of all linear bounded operators on H and by I H ∈ L(H) the identity operator. Let us recall that a Fredholm operator T on a Hilbert space H is an operator T ∈ L(H) such that both its kernel and its cokernel are of finite dimension. We denote the open subset of all Fredholm operators in L(H) by Φ(H). Furthermore, a path {K t } t∈[a,b] in a metric space G is a continuous map K : t ∈ [a, b] → K t ∈ G.

The Grassmannian of a Hilbert space
In this section, we recall briefly the definition and some properties of the Grassmannian G(H) of H, i.e. the set of all closed linear subspaces of H (for all the details see the comprehensive exposition [1]). For every U ∈ G(H), there exists a unique orthogonal projection P U : H → H onto U and the distance d(U, V ) := P U − P V , U, V ∈ G(H), makes G(H) into a complete metric space (cf. also [10]). Moreover, one can show that G(H) is an analytical Banach manifold, and the map Lemma 2.1. The connected components of G(H) are the sets with n, k ∈ N ∪ {+∞} such that k + n = +∞.
Proof. Firstly, we note that, if P U − P V < 1 for U, V ∈ G(H), then dim U = dim V and dim U ⊥ = dim V ⊥ (cf. [10, I.4.6] Let us point out that a computation of all homotopy groups π i (G nk (H)) can be found in [1, Section 2]. The following lemma is essentially well known (e.g., cf. [6, Appendix A]), but we are not aware of a proof in the literature, and so we include it here for the sake of completeness. However, the reader may compare it with a related assertion on Banach bundles, which can be found e.g. in [23] (cf. also [21]), and on which our argument is based.
From the fact that the invertible elements in L(X) are open, we see that Consequently, by using a partition of unity, we may conclude that there exists a family M : Defining R t := M t A t ∈ L(H), we note that R t is a projection since Moreover, since M t is clearly injective, we infer that so that Q t := I H − R t is a continuous family of projections such that im Q t = ker A t . Thus, is an immediate consequence of the rank-nullity theorem in Linear Algebra.

The spectral flow
We denote by Φ S (H) ⊂ Φ(H) the subspace of all selfadjoint Fredholm operators, which is well known to consist of three connected components (cf. [4]). Two of them are given by Their elements are called essentially positive or essentially negative, respectively, and it is readily seen that both of these spaces are contractible. Elements of the remaining component ) are called strongly indefinite, and, in contrast to Φ + S (H) and Φ − S (H), this space has a non-trivial topology. Indeed, Φ i S (H) has the same homotopy groups as the stable orthogonal group (cf. [20]) and the spectral flow provides an explicit isomorphism between its fundamental group and the integers. There are several different, but equivalent, constructions of the spectral flow in the literature. Here, we follow the approach developed by Fitzpatrick, Pejsachowicz and Recht in [8], and we refer to the introduction of [15] for further references on the subject. We call two selfadjoint invertible operators in L(H) Calkin equivalent if S − T is compact. It is well known that in this case the relative Morse index is well defined and finite, where E − (·) and E + (·) denote the negative and positive subspaces of a selfadjoint operator for which 0 is an isolated point of the spectrum. From the second resolvent identity it follows that for Calkin equivalent operators S, T , also the difference of the associated resolvent operators is compact whenever it is defined, where σ(T ) and σ(S) denote the spectrum of T , respectively S. Finally, since the compact operators are closed in L(H), it may be concluded that also the difference of the spectral projections is compact, where a, b do not belong to σ(S)∪σ(T ) and Γ is the circle around a+b 2 in C intersecting the real axis at a and b. Here, S C and T C denote the complexification of operators and Re the real part of an operator on a complexified Hilbert space (cf. [22, Subsection 2.1] for more details). The group GL(H) of all invertible operators on H acts on Φ S (H) by mapping M ∈ GL(H) to M * LM , with L ∈ Φ S (H), which is called the cogredient action. One of the main theorems in [8] states that for any path L : From well known properties of the relative Morse index it follows that the spectral flow does not depend on the choices of J and {K t } t∈ [a,b] , and has the following main properties: (ii) if H 1 and H 2 are separable Hilbert spaces and the paths , and L a , L b are invertible, then the spectral flow of L is the difference of the Morse indices at its endpoints: Finally, let us remark that the spectral flow is even uniquely characterised by the properties i)-iv) above (cf. [7]). A further uniqueness theorem for the spectral flow, which is based on the different but equivalent construction [16], can be found in [13, Subsection 5.2].

Bifurcation along gap continuous paths of subspaces
Throughout this section, let H be a real Hilbert space and J : H → R a C 2 functional having 0 as a critical point. We denote by d u J ∈ L(H, R) the derivative of J at u ∈ H. Moreover, let T be the Riez representation of the Hessian d 2 0 J : H × H → R of J at 0, i.e., the unique selfadjoint operator T ∈ L(H) which satisfies and u n → 0 in H as n → +∞; ii) u n ∈ H tn and u n = 0 for all n ∈ N; iii) u n is a critical point of J | Ht n for all n ∈ N.
Since {H t } t∈[a,b] is a continuous path of subspaces, there exists a family P t , t ∈ [a, b], of orthogonal projections such that im P t = H t . We define which is a continuous path of selfadjoint operators in L(H), and call {H t } t∈[a,b] admissible if both the operators P a T P a : H a → H a and P b T P b : b] is admissible. Now, let us state our main theorems and a corollary, which we are proving in the next section.  Let us point out that L t ∈ Φ + S (H), t ∈ [a, b], and so in any of the following cases: • if in Theorem 3.2 it is n = +∞, since each L t is positive on the subspace H ⊥ t which is of finite codimension; • for all compact operator K in Theorem 3.3 by the same argument as in the previous item.
Finally, we will prove in the subsequent section a corollary of the proof of Theorem 3.2, which rephrases a well known fact from bifurcation theory in our setting. Let us point out that both Theorem 3.2 and Theorem 3.3 do not give any information about the location of the bifurcation point in the interval (a, b). Corollary 3.4. We assume that either the assumptions of Theorem 3.2 or the ones of Theorem 3.3 hold. If t * is a bifurcation point, then

Proofs of the main theorems
Our proofs are based on the main theorem of [15], which deals with the relation between the spectral flow and the bifurcation theory that was previously established in [8]. For completeness, we recall it. Let f : [a, b]×H → R be a continuous map such that each f t := f (t, ·) is C 2 and all its derivatives depend continuously on t ∈ [a, b]. Henceforth, we assume that 0 ∈ H is a critical point of all f t , and we call t * a bifurcation point of critical points of the functional f if there exist two sequences (t n ) n ⊂ [a, b] and (u n ) n ⊂ H \ {0} such that t n → t * in [a, b], u n → 0 in H and u n is a critical point of f tn for all n ∈ N.
The second derivatives of f t , t ∈ [a, b], define a path of selfadjoint operators and we denote the corresponding Riesz representations of d 2 0 f t by L t . The bifurcation theorem in [15] can be stated as follows: Now, in the setting of Section 3 we define a one-parameter family of functionals by Proof. If u is a critical point of f t , then In particular, taking v = P ⊥ t u, it is hence, P ⊥ t u = 0 and so u ∈ H t . Consequently, from (4.1) we obtain that whence, u is a critical point of the restriction of J to H t . Conversely, if u is a critical point of the restriction of J to H t , then u ∈ H t and Consequently, from Definition 3.1 and Lemma 4.2 it follows that t * ∈ [a, b] is a bifurcation point of J along {H t } t∈ [a,b] if and only if it is a bifurcation point for the family of functionals f t . By applying Theorem 4.1, for each t ∈ [a, b] we have to consider the Hessian of f t at the critical point 0 ∈ H, which is given by and which has the Riesz representation [a,b] are exactly the operators introduced in (3.2). Now, we deduce Theorems 3.2 and 3.3 from Theorem 4.1 but before we point out a direct consequence of the definition of Fredholm operators.
In what follows, we will apply Lemma 4.3 to L t | Ht : Firstly, assume that n = +∞. Then, by Lemma 4.3 the operator L t is Fredholm, since it is invertible on the subspace H ⊥ t and Fredholm on the finite dimensional space H t . Furthermore, by assumption L a and L b are invertible, and so it is enough to apply Theorem 4.1 in order to conclude that f admits a bifurcation point. Moreover, if {H t } t∈[a,b] is analytic, then P t and so L t depend analytically on t. Reasoning as in [15,Section 2] the set of all t such that ker L t = {0} is discrete. Moreover, it is readily seen that for any t ∈ [a, b], which implies that Hence, (3.3) follows from Theorem 4.1.
On the other hand, assume that k = +∞. Since again L a and L b are invertible by assumption, in order to apply Theorem 4.1 it is enough to show that L t is Fredholm for all t ∈ (a, b). To this aim, by Lemma 4.3 we just need to prove that P t T P t is Fredholm on H t . Clearly, the kernel and cokernel of the projection P t are H ⊥ t , which is of finite dimension k < +∞, and hence P t is a Fredholm operator. Whence, P t T P t is Fredholm, because the composition of Fredholm operators is still Fredholm (cf. [9, Theorem 3.2]).
Proof of Theorem 3.3. Also in the setting of Theorem 3.3 we can apply Theorem 4.1 once we prove that L t is Fredholm for all t ∈ [a, b]. Unluckily, since k = n = +∞, none of the arguments used in the proof of Theorem 3.2 can be applied. Instead, by the (new) hypothesis on T , taking any t ∈ [a, b] the assertion follows from the remark here below: which is a compact perturbation of I H since the set of compact operators is an ideal in L(H). Consequently, L t is Fredholm by a classical result of Riesz and Schauder (i.e., compact perturbations of the identity are Fredholm operators, cf. [9, Corollary XII.2.5]).
For the proof of Corollary 3.4, we need a special case of the Implicit Function Theorem in Banach spaces that we recall here for completeness (for more details, cf. [2, Subsection 2.2]). Theorem 4.4. Let X, Y be Banach spaces and F : [a, b] × X → Y a continuous map. Assume that the equation has a solution (t 0 , x 0 ) ∈ (a, b) × X, and that the derivative d x F t of F t := F (t, ·) with respect to x ∈ X exists and depends continuously on • every solution of (4.3) in U × V is of the form (t, f (t)).
Proof of Corollary 3.4. Both in the hypotheses of Theorem 3.2 and in those ones of Theorem 3.3, we know that a bifurcation point t * ∈ (a, b) exists. Arguing by contradiction, assume that On the other hand, we can define the map

An example
Throughout this section, for notational simplicity we put I := [0, 1] and we denote by H 1 0 (I, R n ) the Hilbert space of all absolutely continuous functions u : I → R n such that the derivative u ′ is square integrable, and by (H −1 (I, R n ), · H −1 ) its dual space. Our aim is investigating the existence of nontrivial solutions for the semilinear system of ordinary differential equations −(A(x)u ′ (x)) ′ + g(x, u(x)) = 0, x ∈ I, u(0) = u(1) = 0, (5.1) where A : I → GLS(n, R) is a smooth family of invertible symmetric matrices, and g : I × R n → R n , g = g(x, t), is a C 1 function such that g(x, 0) = 0 for all x ∈ I. Let us consider the functional J : where G : I × R n → R is any function such that ∇ ξ G(x, ξ) = g(x, ξ), (x, ξ) ∈ I × R n . It is well known (see, e.g., [19,Proposition B.34]) that J is of class C 2 in H 1 0 (I, R n ) and for any u, v ∈ H 1 0 (I, R n ). Whence, the critical points of J are precisely the weak solutions of problem (5.1). In particular, 0 ∈ H 1 0 (I, R n ) is a critical point, and one can show that the corresponding Hessian is given by where S(x) = ∂g ∂t (x, 0) is a family of symmetric matrices. Let us recall that, for every t ∈ I, we can consider the evaluation map It is a bounded linear operator which is surjective if t ∈ (0, 1). Moreover, ev t depends continuously on t in (0, 1). Indeed, for every t 0 ∈ (0, 1) and u ∈ H 1 0 (I, R n ), we have Consequently, for every 0 < a < b < 1 by Lemma 2.2 we get a continuous family of subspaces Definition 5.1. We say that t * ∈ (a, b) is a bifurcation point for (5.1) if there exist two sequences (t n ) n ⊂ [a, b] and (u n ) n ⊂ H 1 0 (I, R n ) such that (i) t n → t * in [a, b] and u n → 0 in H 1 0 (I, R n ) as n → +∞; (ii) u n ≡ 0 for each n ∈ N; (iii) taking any n ∈ N, the restriction u 0,n := u n | [0,tn] satisfies −(A(x)u ′ 0,n (x)) ′ + g(x, u 0,n (x)) = 0, x ∈ [0, t n ]; (iv) taking any n ∈ N, the restriction u 1,n := u n | [tn,1] satisfies −(A(x)u ′ 1,n (x)) ′ + g(x, u 1,n (x)) = 0, x ∈ [t n , 1], (v) u 0,n (t n ) = u 1,n (t n ) = 0 for each n ∈ N.
Let us remark that, according to Definition 5.1, the two restrictions u 0,n and u 1,n define a global solution of (5.1) if and only if u ′ 0,n (t n ) = u ′ 1,n (t n ). Proof. If t * ∈ (a, b) is a bifurcation point of (5.1), then (t n ) n ⊂ [a, b] and (u n ) n ⊂ H 1 0 (I, R n ) exist which satisfy the properties (i)-(v) in Definition 5.1. Hence, for all v ∈ H tn we have that Then, by (5.2) it follows that u n ∈ H 1 0 (I, R n ) is a non-trivial critical point of J | Ht n . Since u n → 0, t * is a bifurcation point of J along {H t } t∈[a,b] (see Definition 3.1). Conversely, let (t n ) n ⊂ [a, b] and (u n ) n ⊂ H 1 0 (I, R n ) \ {0} be such that u n ∈ H tn is a critical point of J | Ht n , with t n → t * and u n → 0 in H 1 0 (I, R n ). Setting u 0,n and u 1,n as in (iii) and (iv) of Definition 5.1, we see that Whence, u n satisfies both (iii) and (iv) of Definition 5.1, while (v) is an immediate consequence of the definition of H tn . Thus, t * is a bifurcation point of (5.1).
As from Lemma 5.2 the existence of bifurcation points of (5.1) can be reduced to the study of bifurcation points of the functional J on {H t } t∈[a,b] , we assume that the bilinear form d 2 0 J is non-degenerate both on H a and H b . This implies that the path {H t } t∈[a,b] is admissible. Moreover, it is easy to check that the corresponding operator T : Finally, we introduce the orthogonal projection P t onto H t . To this aim, we choose a smooth function χ : [0, 1] → [0, 1] such that χ(0) = χ(1) = 0 and χ | [a,b] ≡ 1. Then, taking Q t u(x) := u(x) − χ(x)u(t), we have that Q t defines a bounded projection in H 1 0 (I, R n ) onto H t , so we obtain a family of projections P t onto H t by P t = Q t Q * t (Q t Q * t + (I H 1 0 (I,R n ) − Q * t )(I H 1 0 (I,R n ) − Q t )) −1 which are orthogonal (cf. [5, Lemma 12.8 a)]). Consequently, writing down the path L t = P t T P t + P ⊥ t , we have everything at hand in order to claim the existence of a bifurcation for (5.1) by Theorem 3.2 if we can show that sf(L, [a, b]) = 0. However, in order to avoid too many technical computations, we restrict to the special case of positive definite matrices A(x). Then, L t ∈ Φ + S (H 1 0 (I, R n )) and according to (iv) in Section 2 we can deduce the existence of a bifurcation point if µ Morse (L a ) = µ Morse (L b ). Since these Morse indices can be computed directly from the quadratic form associated to the Hessian of J , we summarise our discussion for positive definite A in the following proposition. Proposition 5.3. Assume that the matrices A(x), x ∈ I, are positive definite, and that 0 < a < b < 1 are such that the restrictions of the Hessian d 2 0 J to both H a and H b are non-degenerate. If then there is a bifurcation point for (5.1).