Raffaele Chiappinelli

NORMALIZED EIGENVECTORS OF A PERTURBED LINEAR OPERATOR VIA GENERAL BIFURCATION

Let X be a real Banach space, A : X → X a bounded linear operator, and B : X → X a (possibly nonlinear) continuous operator. Assume that λ = 0 is an eigenvalue of A and consider the family of perturbed operators A + eB, where e is a real parameter. Denote by S the unit sphere of X and let SA = S ∩ KerA be the set of unit 0-eigenvectors of A. We say that a vector x0 ∈ SA is a bifurcation point for the unit eigenvectors of A+ eB if any neighborhood of (0, 0, x0) ∈ R × R × X contains a triple (e, λ, x) with e 6= 0 and x a unit λ-eigenvector of A+ eB, i.e. x ∈ S and (A+ eB)x = λx. We give necessary as well as sufficient conditions for a unit 0-eigenvector of A to be a bifurcation point for the unit eigenvectors of A + eB. These conditions turn out to be particularly meaningful when the perturbing operator B is linear. Moreover, since our sufficient condition is trivially satisfied when KerA is one-dimensional, we extend a result of the first author, under the additional assumption that B is of class C2.

Topological persistence of the normalized eigenvectors of a perturbed self-adjoint operator

Abstract Let T be a self-adjoint bounded operator acting in a real Hilbert space H , and denote by S the unit sphere of H . Assume that λ 0 is an isolated eigenvalue of T of odd multiplicity greater than 1 . Given an arbitrary operator B : H → H of class C 1 , we prove that for any e ≠ 0 sufficiently small there exists x e ∈ S and λ e near λ 0 , such that T x e + e B ( x e ) = λ e x e . This result was conjectured, but not proved, in a previous article by the authors. We provide an example showing that the assumption that the multiplicity of λ 0 is odd cannot be removed.

An estimate on the eigenvalues in bifurcation for gradient mappings

Let H be a real Hilbert space and let A: H→H be a nonlinear operator such that A (0) = 0. We consider the eigenvalue problem Recall that λ 0 e ℝ is said to be a bifurcation point for (1.1) if every neighbourhood of (λ 0 , 0) in ℝ × H contains solutions of (1.1).

Topological Persistence of the Unit Eigenvectors of a Perturbed Fredholm Operator of Index Zero

Let A,C : E → F be two bounded linear operators between real Banach spaces, and denote by S the unit sphere of E (or, more generally, let S = g−1(1), where g is any continuous norm in E). Assume that μ0 is an eigenvalue of the problem Ax = μCx, that the operator L = A − μ0C is Fredholm of index zero, and that C satisfies the transversality condition ImgL + C(KerL) = F , which implies that the eigenvalue μ0 is isolated (and when F = E and C is the identity implies that the geometric and the algebraic multiplicities of μ0 coincide). We prove the following result about the persistence of the unit eigenvectors: Given an arbitrary C1 map M : E → F , if the (geometric) multiplicity of μ0 is odd, then for any real ε sufficiently small there exists xε ∈ S and με near μ0 such that Axε + εM(xε) = μεCxε. This result extends a previous one by the authors in which E is a real Hilbert space, F = E, A is selfadjoint and C is the identity. We provide an example showing that the assumption that the multiplicity of μ0 is odd cannot be removed.

PERSISTENCE OF THE NORMALIZED EIGENVECTORS OF A PERTURBED OPERATOR IN THE VARIATIONAL CASE

Let H be a real Hilbert space and denote by S its unit sphere. Consider the nonlinear eigenvalue problem Ax+eB(x) = δx, where A : H → H is a bounded self-adjoint (linear) operator with nontrivial kernel KerA, and B : H → H is a (possibly) nonlinear perturbation term. A unit eigenvector x0 ∈ S ∩ KerA of A (thus corresponding to the eigenvalue δ = 0, that we assume to be isolated) is said to be persistent, or a bifurcation point (from the sphere S ∩ KerA), if it is close to solutions x ∈ S of the above equation for small values of the parameters δ ∈ R and e 6= 0. In this paper we prove that if B is a C1 gradient mapping and the eigenvalue δ = 0 has finite multiplicity, then the sphere S ∩KerA contains at least one bifurcation point, and at least two provided that a supplementary condition on the potential of B is satisfied. These results add to those already proved in the non-variational case, where the multiplicity of the eigenvalue is required to be odd.

Variational Methods for NLEV Approximation Near a Bifurcation Point

We review some more and less recent results concerning bounds on nonlinear eigenvalues (NLEV) for gradient operators. In particular, we discuss the asymptotic behaviour of NLEV (as the norm of the eigenvector tends to zero) in bifurcation problems from the line of trivial solutions, considering perturbations of linear self-adjoint operators in a Hilbert space. The proofs are based on the Lusternik-Schnirelmann theory of critical points on one side and on the Lyapounov-Schmidt reduction to the relevant finite-dimensional kernel on the other side. The results are applied to some semilinear elliptic operators in bounded domains of . A section reviewing some general facts about eigenvalues of linear and nonlinear operators is included.

Nonlinear Stability of Eigenvalues of Compact Self-Adjoint Operators

Variational methods are used to study the effect of suitably restricted nonlinear perturbations upon the eigenvalues of a compact selfadjoint operator.

ON THE NUMBER OF CRITICAL POINTS OF A C 1 FUNCTION ON THE SPHERE

For a C 1 function f:R^n →R\;(n \ge 2) , we consider the least number k of distinct critical points that f must possess when restricted to the sphere S=\{x\in R^n: \Vert x\Vert =1\} . Clearly k \ge 2 (for f attains its absolute minimum and maximum on S ), and a result of Lusternik and Schnirelmann establishes that k=n if f is even. Here we prove that k=n if, for a given orthonormal system ( e_i ), \max\limits_{S \cap V_i}\,f , for all i=1, …n-1 , where V_i is the subspace spanned by e_1, …, e_i and V_i^\bot its orthogonal complement. It is shown that this criterion is satisfied by suitably restricted perturbations of quadratic forms having n distinct eigenvalues.