A new morse index theory for strongly indefinite functionals

In this paper, a new Morse index theory for strongly indefinite functionals was developed via Gălerkin approximation. In particular, the abstract theory is valid for those kinds of strongly indefinite functionals corresponding to wave equation and beam equation.     

Morse theory for indefinite nonlinear elliptic problems

Using the heat flow as a deformation, a Morse theory for the solutions of the nonlinear elliptic equation:

−Δu−λu=a₊(x)|u|^q−1u−a₋(x)|u|^p−1u+h(x,u)$-\mathrm{\Delta }u-\lambda u=a_{+}(x){|u|}^{q-1}u-a_{-}(x){|u|}^{p-1}u+h(x,u)$

On local Morse theory for p-area functionals,p > 2

This survey article deals with some Morse theoretic aspects for functionals defined in Sobolev Banach spaces, associated with quasilinear elliptic equations or systems, involving the p-Laplace operator, p > 2.We discuss the notion of nondegeneracy in a Banach (not Hilbert) variational framework and we present some developments concerning the critical groups estimates and the interpretation of the multiplicity of a critical point.     

Critical point theorems for indefinite functionals

A variational principle of a minimax nature is developed and used to prove the existence of critical points for certain variational problems which are indefinite. The proofs are carried out directly in an infinite dimensional Hilbert space. Special cases of these problems previously had been tractable only by an elaborate finite dimensional approximation procedure. The main applications given here are to Hamiltonian systems of ordinary differential equations where the existence of time periodic solutions is established for several classes of Hamiltonians.Supported in part by the U.S. Army under Contract No. DAAG-29-75-C-0024 and by the Conseglio Nazionale delle Ricerche-Gruppo Nazionale Analisi Funzionale e ApplicazioneSupported in part by the J.S. Guggenheim Memorial Foundation, and by the Office of Naval Research under Contract No. N00014-76-C-0300. Reproduction in whole or in part is permitted for any purpose of the U.S. Government     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

Morse lemma for functionals of variational calculus

Institute for Control Problems, Academy of the Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 1–11, July–September, 1991.     

Discrete Morse Theory for Computing Cellular Sheaf Cohomology

Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.     

Critical points theorems concerning strongly indefinite functionals and infinite many periodic solutions for a class of Hamiltonian systems

Based on new deformation theorems concerning strongly indefinite functionals, we give some new min-max theorems which are useful in looking for critical points of functionals which are strongly indefinite and satisfy Cerami condition instead of Palais-Smale condition. As one application of abstract results, we study existence of multiple periodic solutions for a class of non-autonomous first order Hamiltonian system

     

Multiplicity for strongly indefinite semilinear elliptic system

In this paper we study a multiplicity result for a strongly indefinite semilinear elliptic system

     

On the Morse critical groups for indefinite sublinear elliptic problems

We consider the Dirichlet problem for the equation −Δu=αu+m(x)u|u|^q−2+g(x,u), where q∈(1,2) and m changes sign. We prove that the Morse critical groups at zero of the energy functional of the problem are trivial. As a consequence, existence and bifurcation of nontrivial solutions of the problem are established.     

Multiple solutions for indefinite functionals and applications to asymptotically linear problems

In this paper, by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann, Conley, Zehnder and K.C. Chang. As applications, the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated. And our results generalize some recent ones due to Coti-Zelati, J.Q.Liu, S.Li, etc.This research was supported in part by the National Postdoctoral Science Fund.     

Applications of two critical point results for non-differentiable indefinite functionals

In this paper we study the existence of solution for two kinds of hemivariational inequalities: the first of them is of elliptic type, the second one of hamiltonian type. In those problems the energy functional is indefinite, so the classical variational principles can’t be used in a direct way. The results are an application of two theorems of existence of critical points for non-differentiable functionals recently obtained.     

An index theory for paths that are solutions of a class of strongly indefinite variational problems

We generalize the Morse index theorem of [12,15] and we apply the new result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of semi-Riemannian manifolds. More specifically, we consider semi-Riemannian manifolds admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for . In particular we obtain Morse relations for stationary semi-Riemannian manifolds (see [7]) and for the G?del-type manifolds (see [3]). Received: 4 April 2001 / Accepted: 27 September 2001 / Published online: 23 May 2002 The authors are partially sponsored by CNPq (Brazil) Proc. N. 301410/95 and N. 300254/01-6. Parts of this work were done during the visit of the two authors to the IMPA, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, in January and February 2001. The authors wish to express their gratitude to all Faculty and Staff of the IMPA for their kind hospitality.