Perturbation results for a class of singular Hamiltonian systems

Antonio Ambrosetti; Ivar Ekeland

Displaying similar documents to “Perturbation results for a class of singular Hamiltonian systems”

Perturbation results for a class of singular Hamiltonian systems

Antonio Ambrosetti, Ivar Ekeland (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The existence of solutions with prescribed period $T$ for a class of Hamiltonian systems with a Keplerian singularity is discussed.

Problems remaining NP-complete for sparse or dense graphs

Ingo Schiermeyer (1995)

Discussiones Mathematicae Graph Theory

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For each fixed pair α,c > 0 let INDEPENDENT SET ( $m \leq c n^{α}$ ) and INDEPENDENT SET ( $m \geq (ⁿ ₂) - c n^{α}$ ) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m \leq c n^{α}$ or $m \geq (ⁿ ₂) - c n^{α}$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ( $m \leq n + c n^{α}$ ) and HAMILTONIAN PATH ( $m \leq n + c n^{α}$ ) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m \leq n + c n^{α}$ edges. For each ϵ > 0 let HAMILTONIAN CIRCUIT (m ≥ (1 - ϵ)(ⁿ₂)) and HAMILTONIAN PATH (m ≥ (1 - ϵ)(ⁿ₂)) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH...

Rigidity and gluing for Morse and Novikov complexes

Octav Cornea, Andrew Ranicki (2003)

Journal of the European Mathematical Society

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We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M, ω)$ with $c_{1} {|_{π_{2} (M)} = [ω] |}_{π_{2} (M)} = 0$ . The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently $C^{0}$ close generic...

On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei, Chang-Mu Chu, Hong-Min Suo, Chun-Lei Tang (2015)

Annales Polonici Mathematici

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In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ $- (a + b \int_{Ω} | \nabla u | ² d x) Δ u = u ⁵ + λ u^{q - 1} / {| x |}^{β}$ in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

Factors and circuits in $K_{1, 3}$ -free graphs

Zdeněk Ryjáček (1989)

Banach Center Publications

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The cubic Szegő equation

Patrick Gérard, Sandrine Grellier (2010)

Annales scientifiques de l'École Normale Supérieure

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We consider the following Hamiltonian equation on the $L^{2}$ Hardy space on the circle, $i \partial_{t} u = {Π (| u |}^{2} u),$ where $Π$ is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several...

Approximative sequences and almost homoclinic solutions for a class of second order perturbed Hamiltonian systems

Marek Izydorek, Joanna Janczewska (2014)

Banach Center Publications

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In this work we will consider a class of second order perturbed Hamiltonian systems of the form $q ̈ + V_{q} (t, q) = f (t)$ , where t ∈ ℝ, q ∈ ℝⁿ, with a superquadratic growth condition on a time periodic potential V: ℝ × ℝⁿ → ℝ and a small aperiodic forcing term f: ℝ → ℝⁿ. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system...

Hofer’s metrics and boundary depth

Michael Usher (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that if $(M, ω)$ is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of $(M, ω)$ has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in $M \times M$ ...

Hamiltonian-colored powers of strong digraphs

Garry Johns, Ryan Jones, Kyle Kolasinski, Ping Zhang (2012)

Discussiones Mathematicae Graph Theory

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For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^{k}$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^{k}$ if the directed distance $^{\to} d_{D} (u, v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^{k}$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^{k}$ is distance-colored if each arc (u, v) of $D^{k}$ is assigned the color i where $i =^{\to} d_{D} (u, v)$ . The digraph...

Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang (2016)

Annales Polonici Mathematici

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We study the following singular elliptic equation with critical exponent ⎧ $- Δ u = Q (x) u^{2 * - 1} + λ u^{- γ}$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

On some completions of the space of hamiltonian maps

Vincent Humilière (2008)

Bulletin de la Société Mathématique de France

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In one of his papers, C. Viterbo defined a distance on the set of Hamiltonian diffeomorphisms of $ℝ^{2 n}$ endowed with the standard symplectic form $ω_{0} = d p \land d q$ . We study the completions of this space for the topology induced by Viterbo’s distance and some others derived from it, we study their different inclusions and give some of their properties. In particular, we give a convergence criterion for these distances that allows us to prove that the completions contain non-ordinary elements, as for example,...

A note on a new condition implying pancyclism

Evelyne Flandrin, Hao Li, Antoni Marczyk, Mariusz Woźniak (2001)

Discussiones Mathematicae Graph Theory

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We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to $K_{n / 2, n / 2}$ .

Potential forbidden triples implying hamiltonicity: for sufficiently large graphs

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson (2005)

Discussiones Mathematicae Graph Theory

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In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with $G_{i} = K_{1, 3}$ for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a $K_{1, s}$ , s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being $K_{1, 3}$ , such that all G₁G₂G₃-free...

Forbidden triples implying Hamiltonicity: for all graphs

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson (2004)

Discussiones Mathematicae Graph Theory

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In [2], Brousek characterizes all triples of graphs, G₁, G₂, G₃, with $G_{i} = K_{1, 3}$ for some i = 1, 2, or 3, such that all G₁G₂G₃-free graphs contain a hamiltonian cycle. In [6], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁, G₂, G₃, none of which is a $K_{1, s}$ , s ≥ 3 such that G₁, G₂, G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In this paper, a characterization will be given of all triples G₁, G₂, G₃ with none being $K_{1, 3}$ , such that all...

Extension of several sufficient conditions for Hamiltonian graphs

Ahmed Ainouche (2006)

Discussiones Mathematicae Graph Theory

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Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $(K_{r} \cup K ₛ \cup K ₜ) \lor K ₂$ , 1 ≤ r ≤ s ≤ t or the graph $((n + 1) / 2) K ₁ \lor K_{(n - 1) / 2}$ . It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity...

First steps in stable Hamiltonian topology

Kai Cieliebak, Evgeny Volkov (2015)

Journal of the European Mathematical Society

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In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on $S^{3}$ is homotopic to a stable Hamiltonian structure which cannot be embedded in $ℝ^{4}$ . Moreover, we derive a structure theorem...

Analytic invariants for the $1 : - 1$ resonance

José Pedro Gaivão (2013)

Annales de l’institut Fourier

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Associated to analytic Hamiltonian vector fields in $ℂ^{4}$ having an equilibrium point satisfying a non semisimple $1 : - 1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

On the differentiability of certain saltus functions

Gerald Kuba (2011)

Colloquium Mathematicae

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We investigate several natural questions on the differentiability of certain strictly increasing singular functions. Furthermore, motivated by the observation that for each famous singular function f investigated in the past, f’(ξ) = 0 if f’(ξ) exists and is finite, we show how, for example, an increasing real function g can be constructed so that $g^{'} (x) = 2^{x}$ for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.

Derivation of Hartree’s theory for mean-field Bose gases

Mathieu Lewin (2013)

Journées Équations aux dérivées partielles

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This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of $N$ bosons with an interaction of intensity $1 / N$ (mean-field regime). In the limit $N \to \infty$ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation...

The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

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If $G$ is a connected graph of order $n \geq 1$ , then by a hamiltonian coloring of $G$ we mean a mapping $c$ of $V (G)$ into the set of all positive integers such that $| c (x) - c (y) | \geq n - 1 - D_{G} (x, y)$ (where $D_{G} (x, y)$ denotes the length of a longest $x - y$ path in $G$ ) for all distinct $x, y \in V (G)$ . Let $G$ be a connected graph. By the hamiltonian chromatic number of $G$ we mean $min (max (c (z); z \in V (G))),$ where the minimum is taken over all hamiltonian colorings $c$ of $G$ . The main result of this paper can be formulated as follows: Let $G$ be a connected graph of order $n \geq 3$ . Assume that there exists...