Infinite Dimensional Morse Theory Research Articles

Existence and multiplicity of periodic solutions for a second-order ODE at resonance with an Ahmad–Lazer–Paul condition

In this work, we study the following resonant boundary value problem (1)−u′′(x)=g(x,u(x))+f(x);x∈(0,2π)u(0)=u(2π)u′(0)=u′(2π),where g:[0,2π]×R→R is continuous and bounded, and f∈L2(0,2π) is assumed to have mean-value zero. Using a variational approach and infinite-dimensional Morse theory, we prove existence and multiplicity of periodic solutions of the boundary problem (1), where the nonlinearity g satisfies an Ahmad–Lazer–Paul condition.

Existence and multiplicity for a superlinear elliptic problem under a non-quadradicity condition at infinity

In this article, we study the existence and multiplicity of solutions of the boundary-value problem $$\displaylines{ -\Delta u = f(x,u), \quad \text{in } \Omega, \cr u = 0, \quad \text{on } \partial\Omega, }$$ where \(\Delta\) denotes the N-dimensional Laplacian, \(\Omega\) is a bounded domain with smooth boundary, \(\partial\Omega\), in \(\mathbb{R}^N\) \((N\geq 3)\), and f is a continuous function having subcritical growth in the second variable. Using infinite-dimensional Morse theory, we extended the results of Furtado and Silva [9] by proving the existence of a second nontrivial solution under a non-quadradicity condition at infinity on the non-linearity. Assuming more regularity on the non-linearity f, we are able to prove the existence of at least three nontrivial solutions.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/60/abstr.html

Multiple nontrivial solutions of a semilinear elliptic problem with asymmetric nonlinearity

In this paper we study the existence and multiplicity of solutions of the boundary–value problem(1){−Δu=−λ|u|q−2u+au+b(u+)p−1, in Ω;u=0, on ∂Ω, where Δ denotes the N-dimensional Laplacian, Ω is a bounded domain with smooth boundary, ∂Ω, in RN(N⩾3), u+ denotes the positive part of u:Ω→R, 1<q<2<p<2⁎=2N/(N−2), λ>0, a∈R and b>0. Using infinite-dimensional Morse Theory, we extend the results of Paiva and Presoto [14] and establish some conditions for the existence of at least four nontrivial solutions of (1).

The existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness

In this paper, we show the existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness, by using the technique of penalized functionals and an infinite dimensional Morse theory developed by Kryszewski and Szulkin. Two applications are given to Hamiltonian systems and elliptic systems.

Some remarks on the critical point theory

In this paper, we give some remarks on the development of critical point theory. One is mainly concerned with Mountain Pass theorem, which is well-known in variational theory. For this reason, we make some remarks on some aspects relating to MPT. Another is related to topological methods of nonlinear problems, which have a great progress during the past thirty years and refer to many problems. We merely present remarks on the aspect of infinite dimensional Morse theory and linking approaches.

Existence of solutions for a class of operator equations

In this paper we deal with the existence and multiplicity of nontrivial solutions to a class of operator equation. By using infinite dimensional Morse theory, we establish some conditions which guarantee that the equation has many nontrivial solutions.

Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds

We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of H1-curves, and two splitting lemmas for the functionals on Banach manifolds of C1-curves. Two results on critical groups of iterated closed geodesics are also proved; their corresponding versions on Riemannian manifolds are based on the usual splitting lemma by Gromoll and Meyer (1969)). Our approach consists in deforming the square of the Finsler metric in a Lagrangian which is smooth also on the zero section and then in using the splitting lemma for nonsmooth functionals that the author recently developed in Lu (2011, 0000, 2013). The argument does not involve finite-dimensional approximations and any Palais’ result in Palais (1966). As an application, we extend to Finsler manifolds a result by V. Bangert and W. Klingenberg (1983) about the existence of infinitely many, geometrically distinct, closed geodesics on a compact Riemannian manifold.

An asymmetric superlinear elliptic problem at resonance

In this article, we study the existence and multiplicity of solutions to the problem {−Δu=g(x,u), in Ω;u=0, on ∂Ω, where Ω is a bounded domain in RN(N⩾2) with smooth boundary, and g:Ω¯×R→R is a differentiable function. We will assume that g(x,s) has a resonant behavior for large negative values of s and that a Landesman–Lazer type condition is satisfied. We also assume that g(x,s) is superlinear, but subcritical, for large positive values of s. We prove the existence and multiplicity of solutions for problem (1.1) by using minimax methods and infinite-dimensional Morse theory.

The splitting lemmas for nonsmooth functionals on Hilbert spaces I

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_\Omega f(x, u,\cdots, D^mu)dx$ as in (1.1). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincare-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [19] with some techniques from [27,43,46]. Our theory is applicable to the Lagrangian systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.

Multiple solutions to fourth-order boundary value problems

In this paper, we study the existence and multiplicity of solutions to the fourth-order boundary value problem u ( 4 ) ( t ) + β u ″ ( t ) − α u ( t ) = f ( t , u ( t ) ) for all t ∈ [ 0 , 1 ] subject to Dirichlet boundary value condition, where f ∈ C 1 ( [ 0 , 1 ] × R 1 , R 1 ) , α , β ∈ R 1 . By using the critical point theory and the infinite dimensional Morse theory, we establish some conditions on f which are able to guarantee that this boundary value problem has at least one nontrivial, two nontrivial, m distinct pairs of solutions, and infinitely many solutions, respectively. Our results improve some recent works.

Curve Space: Classifying Curves On Surfaces

We design signatures for curves defined on genus zero surfaces. The signature classifies curves according to the conformal geometry of the given curves and their embedded surface. Based on Teichmuller theory, our signature describes not only the curve shape but also the intrinsic relationship between the curve and its embedded surface. Furthermore, the signature metric is stable, it is close to identity between surfaces sharing similar Riemannian geometry metrics. Based on this, we propose a surface matching framework: first, with curve signatures, we match the partitioning of two surfaces defined by simple closed curves on them; second, the segmented subregions are pairwisely matched and then compared on canonical planar domains. 1. Introduction. Shape analysis and shape comparison are fundamental prob- lems in computer vision, graphics and modeling fields with many important appli- cations. Lots of 2D and 3D shape analysis techniques have been developed in the past couple of decades, most of which are based on comparing curvature or spacial positions of the points on the curve. A complete different way is to consider all the closed curves on the surface. The curve space on surface conveys rich geometric information of the surface itself and is easy to process. The philosophy of analyzing shapes by their associated curve spaces has deep root in algebraic topology (8), infinite dimensional Morse theory (18) and Teichmuller space theory in complex geometry (31). Suppose M is a surface (a 2-manifold), a closed curve on M is a map

On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold ( M, g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p, q∈ M that are non-conjugate in a suitable sense, under the assumption that ( M, g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.

An infinite dimensional Morse theory with applications

In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.

On a Fermat principle in general relativity. A Morse theory for light rays

In this paper an infinite dimensional Morse theory for lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds is developed under intrinsic assumptions. It yields applications to the gravitational lens effect. In particular we show that the number of images in the gravitational lens effect is infinite or odd.

Critical point theorems and applications to a semilinear elliptic problem

The objective of this article is to establish the existence of critical points for functionals of classC2defined on real Hilbert spaces. The argument is based on the infinite dimensional Morse theory introduced by Gromoll-Meyer [13]. The abstract results are applied to study the existence of nonzero solutions for a class of semilinear elliptic problems where the nonlinearity possesses a superlinear growth on a direction of the real line.

Infinite dimensional Morse theory and Fermat’s principle in general relativity. I

The following theorem may be viewed as the general relativistic version of Fermat’s principle. Among all lightlike curves connecting a given point p to a given timelike curve γ in a Lorentzian manifold, the geodesics are characterized by stationary arrival time. Here ‘‘arrival time’’ refers to a smooth parametrization of γ. In this article the first steps are taken to make infinite dimensional Morse theory applicable to this variational problem. The space of trial curves is made into a Hilbert manifold by imposing an H2 Sobolev condition and Fermat’s principle is reformulated in this infinite dimensional setting. Moreover, a Morse index theorem is presented. The mathematical formalism developed here aims at applications to the gravitational lens effect.

Morse theory for periodic solutions of hamiltonian systems and the maslov index

AbstractIn this paper we prove Morse type inequalities for the contractible 1‐periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π2 (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1‐periodic solution has at least one Floquet multiplier which is not equal to 1.

Witten's complex and infinite-dimensional Morse theory

We investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space. The results are applied to an infinite dimensional problem involving the symplectίc action function.