Existence of geodesics in static Lorentzian manifolds with convex boundary

We study some global geometric properties of a static Lorentzian manifold Λ embedded in a differentiable manifold M, with possibly non-smooth boundary ∂Λ. We prove a variational principle for geodesics in static manifolds, and using this principle we establish the existence of geodesics that do not touch ∂Λ and that join two fixed points of Λ. The results are obtained under a suitable completeness assumption for Λ that generalizes the property of global hyperbolicity, and a weak convexity assumption on ∂Λ. Moreover, under a non-triviality assumption on the topology of Λ, we also get a multiplicity result for geodesics in Λ joining two fixed points.

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Reconstruction of Lorentzian Manifolds from Boundary Light Observation Sets

International Mathematics Research Notices ◽

10.1093/imrn/rnx320 ◽

2017 ◽

Vol 2019 (22) ◽

pp. 6949-6987

Author(s):

Peter Hintz ◽

Gunther Uhlmann

Keyword(s):

Open Subset ◽

Lorentzian Manifold ◽

Conformal Structure ◽

Conjugate Points ◽

Lorentzian Manifolds ◽

Convexity Assumption ◽

Nonempty Boundary ◽

Light Rays ◽

Snell’S Law ◽

Snell's Law

Abstract On a time-oriented Lorentzian manifold (M, g) with nonempty boundary satisfying a convexity assumption, we show that the topological, differentiable, and conformal structure of suitable subsets S ⊂ M of sources is uniquely determined by measurements of the intersection of future light cones from points in S with a fixed open subset of the boundary of M; here, light rays are reflected at ∂M according to Snell’s law. Our proof is constructive, and allows for interior conjugate points as well as multiply reflected and self-intersecting light cones.

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HETEROCLINIC ORBITS FOR MONOTONE TWIST MAPS THROUGH A VARIATIONAL PRINCIPLE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401003541 ◽

2001 ◽

Vol 11 (09) ◽

pp. 2451-2461

Author(s):

TIFEI QIAN

Keyword(s):

Variational Principle ◽

Variational Method ◽

Fixed Points ◽

Heteroclinic Orbit ◽

Geometric Method ◽

Heteroclinic Orbits ◽

Constructive Approach ◽

Connecting Orbits ◽

Relative Ease ◽

Twist Maps

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

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Geodesics in Static Lorentzian Manifolds with Critical Quadratic Behavior

Advanced Nonlinear Studies ◽

10.1515/ans-2003-0405 ◽

2003 ◽

Vol 3 (4) ◽

Cited By ~ 9

Author(s):

R. Bartolo ◽

A.M. Candela ◽

J.L. Flores ◽

M. Sánchez

Keyword(s):

Lorentzian Manifold ◽

Lorentzian Manifolds ◽

Geodesic Connectedness

AbstractThe aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉

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RECOVERY OF ZEROTH ORDER COEFFICIENTS IN NON-LINEAR WAVE EQUATIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748020000122 ◽

2020 ◽

pp. 1-27

Author(s):

Ali Feizmohammadi ◽

Lauri Oksanen

Keyword(s):

Wave Equations ◽

Model Problem ◽

Wave Interaction ◽

Lorentzian Manifold ◽

Gaussian Beams ◽

Linear Wave ◽

Lorentzian Manifolds ◽

Solution Map ◽

Globally Hyperbolic ◽

Linear Wave Equations

This paper is concerned with the resolution of an inverse problem related to the recovery of a function $V$ from the source to solution map of the semi-linear equation $(\Box _{g}+V)u+u^{3}=0$ on a globally hyperbolic Lorentzian manifold $({\mathcal{M}},g)$ . We first study the simpler model problem, where $({\mathcal{M}},g)$ is the Minkowski space, and prove the unique recovery of $V$ through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams.

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Supercritical problems with concave and convex nonlinearities in ℝN

Communications in Contemporary Mathematics ◽

10.1142/s0219199720500522 ◽

2020 ◽

pp. 2050052

Author(s):

João Marcos do Ó ◽

Pawan Kumar Mishra ◽

Abbas Moameni

Keyword(s):

Variational Principle ◽

Convex Sets ◽

Nonlinear Term ◽

Growth Restriction ◽

Multiplicity Result ◽

Sobolev Embeddings ◽

Existence And Multiplicity ◽

Semilinear Elliptic ◽

Semilinear Elliptic Problems ◽

Concave And Convex Nonlinearities

In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole [Formula: see text] with nonlinearities involving linear and superlinear terms. We shall impose no growth restriction on the nonlinear term, and consequently, our problem can be supercritical in the sense of the Sobolev embeddings.

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A variational principle and coupled fixed points

Journal of Fixed Point Theory and Applications ◽

10.1007/s11784-019-0706-y ◽

2019 ◽

Vol 21 (2) ◽

Cited By ~ 1

Author(s):

Boyan Zlatanov

Keyword(s):

Variational Principle ◽

Fixed Points ◽

Coupled Fixed Points

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Nehari manifold and multiplicity result for elliptic equation involving p-laplacian problems

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v36i4.34078 ◽

2018 ◽

Vol 36 (4) ◽

pp. 197-208

Author(s):

Khaled Ben Ali ◽

Abdeljabbar Ghanmi

Keyword(s):

Elliptic Equation ◽

Positive Solutions ◽

Smooth Function ◽

Smooth Boundary ◽

Nehari Manifold ◽

Multiplicity Result ◽

Existence And Multiplicity ◽

Open Set ◽

Multiplicity Of Positive Solutions

This article shows the existence and multiplicity of positive solutions of the $p$-Laplacien problem $$\displaystyle -\Delta_{p} u=\frac{1}{p^{\ast}}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u \quad \mbox{for } x\in\Omega;\quad \quad u=0,\quad \mbox{for } x\in\partial\Omega$$ where $\Omega$ is a bounded open set in $\mathbb{R}^n$ with smooth boundary, $1<q<p<n$, $p^{\ast}=\frac{np}{n-p}$, $\lambda \in \mathbb{R}\backslash \{0\}$ and $a$ is a smooth function which may change sign in $\overline{\Omega}$. The method is based on Nehari results on three sub-manifolds of the space $W_{0}^{1,p}$.

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A MULTIPLICITY THEOREM FOR A PERTURBED SECOND-ORDER NON-AUTONOMOUS SYSTEM

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150400149x ◽

2006 ◽

Vol 49 (2) ◽

pp. 267-275 ◽

Cited By ~ 22

Author(s):

Francesca Faraci ◽

Antonio Iannizzotto

Keyword(s):

Variational Principle ◽

Periodic Solutions ◽

Autonomous System ◽

Second Order ◽

Mountain Pass ◽

Connected Components ◽

Multiplicity Result ◽

Global Minima ◽

Multiplicity Theorem

AbstractIn this paper we establish a multiplicity result for a second-order non-autonomous system. Using a variational principle of Ricceri we prove that if the set of global minima of a certain function has at least $k$ connected components, then our problem has at least $k$ periodic solutions. Moreover, the existence of one more solution is investigated through a mountain-pass-like argument.

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