Bifurcation of critical points along gapcontinuous families of subspaces
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Abstract
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.
Keywords
Bifurcation gap metric Grassmannian spectral flow semilinear ordinary differential equationMathematics Subject Classification
47J15 58E07 14M15 34B15 34C231 Introduction
Let H be a real separable Hilbert space and \({\mathcal {J}}:H\rightarrow {\mathbb {R}}\) a \(C^2\)functional. We denote the derivative of \({\mathcal {J}}\) at \(u\in H\) by \(\mathrm{d}_u{\mathcal {J}}\in {\mathcal {L}}(H,{\mathbb {R}})\) and in what follows we assume that \(\mathrm{d}_0{\mathcal {J}}=0\), i.e. \(0\in H\) is a critical point of \({\mathcal {J}}\). Usually, critical points of functionals \({\mathcal {J}}\) on Hilbert spaces H are studied as they can be solutions of differential equations. Correspondingly, critical points of a restriction \({\mathcal {J}}\mid _{H'}:H'\rightarrow {\mathbb {R}}\) to a subspace \(H'\subset H\) may yield solutions of differential equations under additional constraints.
In [2] Abbondandolo and Majer studied the Grassmannian of a Hilbert space H, i.e. the set of all closed subspaces of H. As there is a canonical metric on this set, which is induced by orthogonal projections, we can define paths \(\{H_t\}_{t\in [a,b]}\) in it. Clearly, for each \(t \in [a,b]\) the element \(0\in H_t\) is a critical point of the restriction \({\mathcal {J}}\mid _{H_t}:H_t\rightarrow {\mathbb {R}}\) as \(\mathrm{d}_0{\mathcal {J}}=0\), and the aim of this paper is to investigate bifurcation from this branch of critical points in a sense that we will introduce below in Definition 3.1. Our main results show the existence of bifurcation in terms of the second derivative of \({\mathcal {J}}\) at the critical point 0, which are based on [9, 16]. To this aim, we introduce a family of functionals \(f_t:H\rightarrow {\mathbb {R}}\), \(t\in [a,b]\), such that each \(f_t\) involves the orthogonal projection onto the space \(H_t\), and such that its critical points are the critical points of the restriction \({\mathcal {J}}\mid _{H_t}\). Consequently, \(0\in H\) is a critical point of any \(f_t:H\rightarrow {\mathbb {R}}\), \(t\in [a,b]\), and by considering the second derivative \(\mathrm{d}^2_0f_t\) of \(f_t\) at 0 we can define a path \(\{L_t\}_{t\in [a,b]}\) 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 \(\{H_t\}_{t\in [a,b]}\) arises if the spectral flow of \(L:t \mapsto L_t\) 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 et al. [4]. Its relevance to bifurcation theory was discovered in [9]. For example, if all operators \(L_t\) have a finite Morse index \(\mu _{\mathrm{Morse}}(L_t)\), then the spectral flow of L is just the difference of the Morse indices at the endpoints, i.e. \(\mu _{\mathrm{Morse}}(L_a)  \mu _{\mathrm{Morse}}(L_b)\). Hence, a nonvanishing 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. [15, §8.9] or also [12, §II.7.1]. However, if \(\mu _{\mathrm{Morse}}(L_t)=+\infty \) for some \(t \in [a,b]\), 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 Sect. 2, we introduce some preliminaries that we need to state our theorems. We recall some facts about the Grassmannian of a Hilbert space H, essentially following Abbondandolo and Majer’s paper [2]. 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 in our examples. In Sect. 2 we briefly recall the definition of the spectral flow from [9]. In the third section, we introduce the path L and state our main theorems and a corollary, which we prove in Sect. 4. Finally, we apply our theory to a Dirichlet problem for semilinear ordinary differential operators in Sect. 5.
2 Grassmannians and spectral flows
As before, we let H be a real separable Hilbert space of infinite dimension, we denote by \({\mathcal {L}}(H)\) the Banach space of all linear bounded operators on H equipped with its standard norm \(\Vert \cdot \Vert \) and by \(I_H\in {\mathcal {L}}(H)\) the identity operator. Let us recall that a Fredholm operator T on a Hilbert space H is an operator \(T\in {\mathcal {L}}(H)\) such that both its kernel and its cokernel are of finite dimension. We denote the open subset of all Fredholm operators in \({\mathcal {L}}(H)\) by \(\Phi (H)\).
2.1 The Grassmannian of a Hilbert space
In this section, we recall briefly the definition and some properties of the Grassmannian \({\mathcal {G}}(H)\) of H, i.e. the set of all closed linear subspaces of H, where we refer for more details to the comprehensive exposition [2].
Lemma 2.1
Proof
Let us first recall that if \(\Vert P_UP_V\Vert <1\) for \(U,V\in {\mathcal {G}}(H)\), then \(\dim U=\dim V\) and \(\dim U^\perp =\dim V^\perp \) (cf. [11, I.4.6]). Consequently, if U and V belong to the same component of \({\mathcal {G}}(H)\), then they must have both the same dimension and the same codimension.
Now, let us assume that \(U,V\in {\mathcal {G}}_{nk}(H)\) for some k, n such that \(k+n=+\infty \). Since H is separable, it is easy to construct an orthogonal operator \(O:H\rightarrow H\) such that \(O(U)=V\). Denoting by \(\mathcal {O}(H)\) the subspace of \({\mathcal {L}}(H)\) consisting of all orthogonal operators, it is easily seen from functional calculus that \(\mathcal {O}(H)\) is connected^{1}. Hence, there is a path \(M:[0,1]\rightarrow \mathcal {O}(H)\) joining the identity operator \(I_H\) to O. Finally, since \(P_{M_t(U)}=M_tP_UM^{1}_t\) for each \(t\in [0,1] \), we have that \(\{M_t(U)\}_{t\in [0,1]}\) is continuous and so a path in \({\mathcal {G}}(H)\) that joins U to V. \(\square \)
Remark 2.2
A computation of all homotopy groups \(\pi _i({\mathcal {G}}_{nk}(H))\), \(i\in \mathbb {N}\), can be found in [2, Section 2].
The following lemma is essentially well known (cf. e.g. [7, Appendix A]), but as we are not aware of a proof in the literature, we include it here for the sake of completeness. The reader may compare it with a related assertion on Banach bundles, which can be found e.g. in [25] and also [23], and on which our argument is based.
Lemma 2.3
Proof
Let us first fix some \(t_0\in [a,b]\). Since \(A_{t_0}\) is surjective, there exists \(M_0\in {\mathcal {L}}(X,H)\) such that \(A_{t_0}M_0=I_X\), with \(I_X\) the identity operator on X. From the fact that the invertible elements in \({\mathcal {L}}(X)\) are open, we see that \(A_t M_0\) is invertible for all t in a neighbourhood \({\mathcal {I}}_0\) of \(t_0\).
Now, if we set \(M_{0,t}:=M_0(A_t M_0)^{1}\) for \(t\in {\mathcal {I}}_0\), then \(A_t M_{0,t}=I_X\).
Note that if \(M_1,M_2\in {\mathcal {L}}(X,H)\) are such that \(A_tM_i=I_X\), then \(A_t(\alpha M_1+(1\alpha )M_2)=I_X\) for all \(0\le \alpha \le 1\). Consequently, by using a partition of unity, we may conclude that there exists a path \(M:[a,b]\rightarrow {\mathcal {L}}(X,H)\) such that \(A_t M_t=I_X\) for all \(t\in [a,b]\).
Finally, that \(\ker (A_t)\in {\mathcal {G}}_{nk}(H)\) with \(k=\dim X\) and \(n=\dim H\dim X\) is an immediate consequence of the rank–nullity theorem in linear algebra.
\(\square \)
2.2 The spectral flow
The group GL(H) of all invertible operators on H acts on \(\Phi _S(H)\) by mapping \(M\in GL(H)\) and \(L\in \Phi _S(H)\) to \(M^*LM\), which is called the cogredient action. One of the main theorems in [9] states that for any path \(L:[a,b]\rightarrow \Phi _S(H)\) there exist a path \(M:[a,b]\rightarrow GL(H)\) and a selfadjoint invertible operator \(J\in \Phi _S(H)\), such that \(M^*_tL_tM_t=J+K_t\) with \(K_t\) selfadjoint and compact for each \(t\in [a,b]\).
Definition 2.4
 (i)
If \(L_{t}\) is invertible for all \(t\in [a,b]\), then \({{\mathrm{sf}}}(L,[a,b])=0\).
 (ii)If \(H_1\) and \(H_2\) are separable Hilbert spaces and the paths \(L_1:[a,b]\rightarrow \Phi _S(H_1)\) and \(L_2:[a,b]\rightarrow \Phi _S(H_2)\) have invertible endpoints, then$$\begin{aligned} {{\mathrm{sf}}}(L_1\oplus L_2,[a,b])={{\mathrm{sf}}}(L_1,[a,b])+{{\mathrm{sf}}}(L_2,[a,b]). \end{aligned}$$
 (iii)Let \(h:[0,1]\times [a,b]\rightarrow \Phi _S(H)\) be a homotopy such that h(s, a) and h(s, b) are invertible for all \(s\in [0,1]\). Then,$$\begin{aligned} {{\mathrm{sf}}}(h(0,\cdot ),[a,b])={{\mathrm{sf}}}(h(1,\cdot ),[a,b]). \end{aligned}$$
 (iv)If \(L_t\in \Phi _S^+(H)\), \(t\in [a,b]\), and \(L_a\), \(L_b\) are invertible, then the spectral flow of L is the difference of the Morse indices at its endpoints:where$$\begin{aligned} {{\mathrm{sf}}}(L,[a,b])=\mu _{\mathrm{Morse}}(L_a)\mu _{\mathrm{Morse}}(L_b), \end{aligned}$$$$\begin{aligned} \mu _{\mathrm{Morse}}(L_t)=\sup \dim \{V\subset H:\, \langle L_t u,u\rangle _H<0\quad \text {for all } u\in V{\setminus }\{0\}\}. \end{aligned}$$(2.1)
3 Bifurcation along gap continuous paths of subspaces
Definition 3.1
 (i)
\(t_n\rightarrow t^*\) in [a, b] and \(u_n\rightarrow 0\) in H as \(n\rightarrow +\infty \);
 (ii)
\(u_n\in H_{t_n}\) and \(u_n\ne 0\) for all \(n\in \mathbb {N}\);
 (iii)
\(u_n\) is a critical point of \({\mathcal {J}}\mid _{H_{t_n}}\) for all \(n\in \mathbb {N}\).
Now, let us state our main theorems and a corollary, which we are proving in the next section.
Theorem 3.2
Let \(\{H_t\}_{t\in [a,b]}\) be an admissible path in \({\mathcal {G}}_{nk}(H)\) such that either \(n\ne +\infty \) or \(k\ne +\infty ,\) and let us assume that the operator T introduced in (3.1) is Fredholm.
Note that the case in which the path \(\{H_t\}_{t\in [a,b]}\) is in the connected component \({\mathcal {G}}_{\infty ,\infty }(H)\) of \({\mathcal {G}}(H)\) is excluded in Theorem 3.2. Our second theorem deals with this setting, but we have to impose a restriction on the form of the operator T.
Theorem 3.3
We assume that \(T=I_H+K\) for some compact operator K, and that \(\{H_t\}_{t\in [a,b]}\) is an admissible path in \({\mathcal {G}}_{\infty ,\infty }(H)\). Then the operators \(L_t\) in (3.2) are Fredholm, and if \({{\mathrm{sf}}}(L,[a,b])\ne 0,\) then there is a bifurcation point of \({\mathcal {J}}\) along \(\{H_t\}_{t\in [a,b]}.\)

if \(n\ne +\infty \) in Theorem 3.2, since each \(L_t\) is positive on the subspace \(H^\perp _t\) which is of finite codimension;

if \(T\in \Phi ^+_S(H)\) in Theorem 3.2, as \(\mu _{\mathrm{Morse}}(L_t)\le \mu _{\mathrm{Morse}}(T)\) for all \(t\in [a,b]\);

for all compact operators K in Theorem 3.3 by the same argument as in the previous item.
4 Proofs of the main theorems
Theorem 4.1
Lemma 4.2
The critical points of \(f_t\) are precisely the critical points of \({\mathcal {J}}\mid _{H_t},\) \(t\in [a,b].\)
Proof
Consequently, it follows from Definition 3.1 and Lemma 4.2 that \(t^*\in [a,b]\) is a bifurcation point of \({\mathcal {J}}\) along \(\{H_t\}_{t\in [a,b]}\) if and only if it is a bifurcation point for the family of functionals \(f_t\).
Now, we deduce Theorems 3.2 and 3.3 from Theorem 4.1 but before we note for later reference the following immediate consequence of the definition of Fredholm operators.
Lemma 4.3
In what follows, we will apply Lemma 4.3 to \(L_t\mid _{H_t}:H_t\rightarrow H_t\) and \(L_t\mid _{H^\perp _t}:H^\perp _t\rightarrow H^\perp _t\).
Proof of Theorem 3.2
Let us now assume that \(k\ne +\infty \). Since \(L_a\) and \(L_b\) are again invertible by assumption, in order to apply Theorem 4.1 it is enough to show that \(L_t\) is Fredholm for all \(t\in (a,b)\). However, by Lemma 4.3 we just need to prove that \(P_tTP_t\) is Fredholm on \(H_t\). Now the kernel and cokernel of the projection \(P_t\) are \(H^\perp _t\), which is of finite dimension \(k< +\infty \), and so \(P_t\) is a Fredholm operator. This shows that indeed \(P_t TP_t\) is Fredholm as the composition of Fredholm operators is again Fredholm (cf. [10, Theorem 3.2]). \(\square \)
Proof of Theorem 3.3
Proof of Corollary 3.4
We have already shown that a bifurcation point \(t^* \in (a,b)\) exists under the assumptions of Theorem 3.2 or Theorem 3.3, respectively. Now we assume for a contradiction that \({{\mathrm{im}}}(T\mid _{H_{t^*}})\cap H^\perp _{t^*}= \{0\}\). Then \(\ker (L_{t^*})=\{0\}\) by (4.2), and so \(L_{t^*}\) is invertible as it is Fredholm of index 0 by Theorems 3.2 and 3.3.
5 An example
Throughout this section, we set \(I:=[0,1]\) and we denote by \(H^1_0(I,{\mathbb {R}}^n)\) the Hilbert space of all absolutely continuous functions \(u:I\rightarrow {\mathbb {R}}^n\) such that the derivative \(u'\) is square integrable.
Definition 5.1
 (i)
\(t_k\rightarrow t^*\) in [a, b] and \(u_k\rightarrow 0\) in \(H^1_0(I,{\mathbb {R}}^n)\) as \(k \rightarrow +\infty \);
 (ii)
\(u_k\not \equiv 0\) for each \(k\in \mathbb {N}\);
 (iii)for every \(k\in \mathbb {N}\), the restriction \(u_{0,k}:=u_k\mid _{[0,t_k]}\) satisfies$$\begin{aligned} (A(x)u'_{0,k}(x))'+\nabla _\xi g(x,u_{0,k}(x))=0,\quad x\in [0,t_k]; \end{aligned}$$
 (iv)for every \(k\in \mathbb {N}\), the restriction \(u_{1,k}:=u_k\mid _{[t_k,1]}\) satisfies$$\begin{aligned} (A(x)u'_{1,k}(x))'+\nabla _\xi g(x,u_{1,k}(x))=0,\quad x\in [t_k,1], \end{aligned}$$
 (v)
\(u_{0,k}(t_k)=u_{1,k}(t_k)=0\) for each \(k\in \mathbb {N}\).
Let us note that the two restrictions \(u_{0,k}\) and \(u_{1,k}\) in Definition 5.1 define a global solution of (5.1) if and only if \(u'_{0,k}(t_k)=u'_{1,k}(t_k)\).
Lemma 5.2
There is a bifurcation point of (5.1) at \(t^*\in (a,b)\) if and only if \(t^*\) is a bifurcation point of \({\mathcal {J}}\) along \(\{H_t\}_{t\in [a,b]}\).
Proof
In summary, from Lemma 5.2 the existence of bifurcation points of (5.1) can be reduced to the study of bifurcation points of the functional \({\mathcal {J}}\) on \(\{H_t\}_{t\in [a,b]}\). One may wonder if our approach is really necessary to study bifurcation points as in Definition 5.1, or whether there is a simple transformation from \({\mathcal {J}}\mid _{H_t}\) to some functional \({\mathcal {J}}_t:H^1_0(I,{\mathbb {R}}^n)\rightarrow {\mathbb {R}}\) whose form is similar to that of \({\mathcal {J}}\) in (5.2). If so, Theorem 4.1 might be directly applied to obtain bifurcation points. However, this is not possible, as the functions \(u_k\in H^1_0(I,{\mathbb {R}}^n)\) in Definition 5.1 do not belong to \(H^2(I,{\mathbb {R}}^n)\) in general, whereas critical points of \({\mathcal {J}}_t\) would be in this space by elliptic regularity.
Proposition 5.3
 (i)
\(E(a_,0)\cap E(a_+,0)=E(b_,0)\cap E(b_+,0)=\{0\},\)
 (ii)
\(\mu (a)\ne \mu (b),\)
Proof
 1.
the restrictions of \(\mathrm{d}^2_0{\mathcal {J}}\) to \(H_a\) and \(H_b\) are nondegenerate,
 2.
\(\mu _{\mathrm{Morse}}(L_a) \ne \mu _{\mathrm{Morse}}(L_b)\).
Footnotes
Notes
Acknowledgements
This work was commenced when the first author stayed at the Institut für Mathematik, Humboldt Universität zu Berlin, and it was continued during a visit of the second author at the Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro. The authors would like to thank both departments for their kind hospitality.
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