Showing papers on "Geometry and topology published in 2019"

Superlinear Schrödinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent

Abstract: Abstract This paper concerns the existence and multiplicity of solutions for the Schrődinger–Kirchhoff type problems involving the fractional p–Laplacian and critical exponent. As a particular case, we study the following degenerate Kirchhoff-type nonlocal problem: ‖ u ‖λ(θ−1)p[ λ(−Δ)psu+V(x)| u |p−2u ]=| u |ps⋆−2u+f(x,u) in ℝN,‖ u ‖λ=(λ∫ℝ∫2N| u(x)−u(y) |p| x−y |N+psdxdy+∫ℝNV(x)| u |pdx)1/p $$\\begin{align}& \\left\\| u \\right\\|_{\\lambda }^{\\left( \\theta -1 \\right)p}\\left[ \\lambda \\left( -\\Delta \\right)_{p}^{s}u+V\\left( x \\right){{\\left| u \\right|}^{p-2}}u \\right]={{\\left| u \\right|}^{p_{s}^{\\star }-2}}u+f\\left( x,u \\right)\\,in\\,{{\\mathbb{R}}^{N}}, \\\\ & {{\\left\\| u \\right\\|}_{\\lambda }}={{\\left( \\lambda \\int\\limits_{\\mathbb{R}}{\\int\\limits_{2N}{\\frac{{{\\left| u\\left( x \\right)-u\\left( y \\right) \\right|}^{p}}}{{{\\left| x-y \\right|}^{N+ps}}}}dxdy+\\int\\limits_{{{\\mathbb{R}}^{N}}}{V\\left( x \\right){{\\left| u \\right|}^{p}}dx}} \\right)}^{{1}/{p}\\;}} \\\\ \\end{align}$$ where (−Δ)ps $\\left( -\\Delta \\right)_{p}^{s}$is the fractional p–Laplacian with 0 < s < 1 < p < N/s, ps⋆=Np/(N−ps) $p_{s}^{\\star }={Np}/{\\left( N-ps \\right)}\\;$is the critical fractional Sobolev exponent, λ > 0 is a real parameter, 1<θ≤ps⋆/p, $1<\\theta \\le {p_{s}^{\\star }}/{p}\\;,$and f : ℝN × ℝ → ℝ is a Carathéodory function satisfying superlinear growth conditions. For θ∈(1,ps⋆/p), $\\theta \\in \\left( 1,{p_{s}^{\\star }}/{p}\\; \\right),$by using the concentration compactness principle in fractional Sobolev spaces, we show that if f(x, t) is odd with respect to t, for any m ∈ ℕ+ there exists a Λm > 0 such that the above problem has m pairs of solutions for all λ ∈ (0, Λm]. For θ=ps⋆/p, $\\theta ={p_{s}^{\\star }}/{p}\\;,$by using Krasnoselskii’s genus theory, we get the existence of infinitely many solutions for the above problem for λ large enough. The main features, as well as the main difficulties, of this paper are the facts that the Kirchhoff function is zero at zero and the potential function satisfies the critical frequency infx∈ℝ V(x) = 0. In particular, we also consider that the Kirchhoff term satisfies the critical assumption and the nonlinear term satisfies critical and superlinear growth conditions. To the best of our knowledge, our results are new even in p–Laplacian case.

Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

Abstract: Abstract The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.

Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian

Abstract: Abstract The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case ( a + b ⁢ ∥ u ∥ s p ⁢ ( θ - 1 ) ) ⁢ [ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ ( x , u ) + ( ∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u in ⁢ ℝ N , (a+b\\|u\\|_{s}^{p(\\theta-1)})[(-\\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\\lambda f(x,u)% +\\Bigg{(}\\int_{\\mathbb{R}^{N}}\\frac{|u|^{p_{\\mu,s}^{*}}}{|x-y|^{\\mu}}\\,dy% \\Biggr{)}|u|^{p_{\\mu,s}^{*}-2}u\\quad\\text{in }\\mathbb{R}^{N}, where ∥ u ∥ s = ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \\|u\\|_{s}=\\Bigg{(}\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \\,dx\\,dy+\\int_{\\mathbb{R}^{N}}V(x)|u|^{p}\\,dx\\Biggr{)}^{\\frac{1}{p}}, a , b ∈ ℝ 0 + {a,b\\in\\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\\lambda>0} is a parameter, s ∈ ( 0 , 1 ) {s\\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / ( N - p ⁢ s ) ) {\\theta\\in[1,N/(N-ps))} , ( - Δ ) p s {(-\\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\\mathbb{R}^{N}\\rightarrow\\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\\mu

Weaving geodesic foliations

Abstract: We study discrete geodesic foliations of surfaces---foliations whose leaves are all approximately geodesic curves---and develop several new variational algorithms for computing such foliations. Our key insight is a relaxation of vector field integrability in the discrete setting, which allows us to optimize for curl-free unit vector fields that remain well-defined near singularities and robustly recover a scalar function whose gradient is well aligned to these fields. We then connect the physics governing surfaces woven out of thin ribbons to the geometry of geodesic foliations, and present a design and fabrication pipeline for approximating surfaces of arbitrary geometry and topology by triaxially-woven structures, where the ribbon layout is determined by a geodesic foliation on a sixfold branched cover of the input surface. We validate the effectiveness of our pipeline on a variety of simulated and fabricated woven designs, including an example for readers to try at home.

Weil's Conjecture for Function Fields: Volume I (Ams-199)

Abstract: Let X be an algebraic curve defined over a finite field Fq and let G be a smooth affine group scheme over X with connected fibers whose generic fiber is semisimple and simply connected. In this paper, we affirm a conjecture of Weil by establishing that the Tamagawa number of G is equal to 1.

Constant sign and nodal solutions for parametric (p, 2)-equations

Abstract: When q = 2, we have the Laplace di erential operator denoted by ∆. In the right hand side (reaction) of the problem, we have a parametric term x 7→ λ|x|p−2xwith λ > 0 being a parameter and also a perturbation f (z, x)which is a Caratheodory function (that is, for all x ∈ R, z 7→ f (z, x) is measurable and for a.a. z ∈ Ω, x 7→ f (z, x) is continuous). We do not impose any sign condition on f (z, ·) and we assume that for a.a. z ∈ Ω, f (z, ·) is (p − 1)−superlinear near ±∞. However, we do not assume that it satis es the usual in such cases AmbrosettiRabinowitz condition (the AR-condition for short). Our aim is to prove multiplicity theorems providing sign information for all the solutions produced. To this end, rst we look for constant sign solutions andwe prove bifurcation-type results describing in a precise way the changes in the sets of positive and negative solutions respectively as the parameter λ moves in the positive semiaxis (0, +∞). We also show that there exist extremal constant sign solutions (that is, a smallest positive solution and a biggest negative solution). Then these extremal constant sign solutions are used to generate nodal (that is, sign changing) solutions. By strengthening the conditions on the perturbation f (z, ·) and using also tools from the theory of critical groups (Morse theory), we prove a multiplicity theorem for small values of the parameter λ > 0. So, we show that when the parameter λ > 0 is small, problem (Pλ) has at least seven nontrivial solutions all with sign information: two positive, two negative and three nodal. We mention that (p, 2)−equations (that is, equations driven by a p−Laplacian and a Laplacian), arise in problems of mathematical physics (see, for example, Benci-D’Avenia-Fortunato-Pisani [1]). We also mention

Statistical approximation properties of λ-Bernstein operators based on q-integers

Abstract: Abstract In this paper, we introduce a new generalization of λ-Bernstein operators based on q-integers, we obtain the moments and central moments of these operators, establish a statistical approximation theorem and give an example to show the convergence of these operators to f(x). It can be seen that in some cases the absolute error bounds are smaller than the case of classical q-Bernstein operators to f(x).

η-Ricci solitons on nearly Kenmotsu manifolds

Abstract: In this paper, we study the geometry and topology of η-Ricci solitons satisfying Ricci-semisymmetry condition, S ⋅ R = 0 condition and finally Einstein-semisymmetry condition on nearly Kenmotsu man...

Sign-changing multi-bump solutions for the Chern-Simons-Schrödinger equations in ℝ2

Abstract: Abstract In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity −Δu+λV(|x|)u+h2(|x|)|x|2+∫|x|∞h(s)su2(s)dsu=f(u),x∈R2, $$\\begin{array}{} \\displaystyle -{\\it\\Delta} u+\\lambda V(|x|)u+\\left(\\frac{h^2(|x|)}{|x|^2}+\\int\\limits^{\\infty}_{|x|}\\frac{h(s)}{s}u^2(s)ds\\right)u=f(u),\\,\\, x\\in\\mathbb R^2, \\end{array}$$ where λ > 0, V is an external potential and h(s)=12∫0sru2(r)dr=14π∫Bsu2(x)dx $$\\begin{array}{} \\displaystyle h(s)=\\frac{1}{2}\\int\\limits^s_0ru^2(r)dr=\\frac{1}{4\\pi}\\int\\limits_{B_s}u^2(x)dx \\end{array}$$ is the so-called Chern-Simons term. Assuming that the external potential V is nonnegative continuous function with a potential well Ω := int V–1(0) consisting of k + 1 disjoint components Ω0, Ω1, Ω2 ⋯, Ωk, and the nonlinearity f has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as λ → +∞ are also considered.

Continuity results for parametric nonlinear singular Dirichlet problems

Abstract: Abstract In this paper we study from a qualitative point of view the nonlinear singular Dirichlet problem depending on a parameter λ > 0 that was considered in [32]. Denoting by Sλ the set of positive solutions of the problem corresponding to the parameter λ, we establish the following essential properties of Sλ: there exists a smallest element uλ∗ $\\begin{array}{} u_\\lambda^* \\end{array}$ in Sλ, and the mapping λ ↦ uλ∗ $\\begin{array}{} u_\\lambda^* \\end{array}$ is (strictly) increasing and left continuous; the set-valued mapping λ ↦ Sλ is sequentially continuous.

Multiplicity and concentration results for magnetic relativistic Schrödinger equations

Abstract: Abstract In this paper, we consider the following magnetic pseudo-relativistic Schrödinger equation εi∇−A(x)2+m2u+V(x)u=f(|u|)uinRN, $$\\begin{array}{} \\displaystyle \\sqrt{\\left(\\frac{\\varepsilon}{i}\ abla-A(x)\\right)^2+m^2}u+V(x)u= f(|u|)u \\quad {\\rm in}\\,\\,\\mathbb{R}^N, \\end{array}$$ where ε > 0 is a parameter, m > 0, N ≥ 1, V : ℝN → ℝ is a continuous scalar potential satisfies V(x) ≥ − V0 > − m for any x ∈ ℝN and f : ℝN → ℝ is a continuous function. Under a local condition imposed on the potential V, we discuss the number of nontrivial solutions with the topology of the set where the potential attains its minimum. We proof our results via variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

Local regularity results for solutions of linear elliptic equations with drift term

Abstract: Abstract We study the local regularity of the solution u of the following nonlinear boundary value problem: { 𝒜 ⁢ u = - div ⁡ [ E ⁢ ( x ) ⁢ u + F ⁢ ( x ) ] in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle\mathcal{A}u&\displaystyle=-\operatorname{% div}{[E(x)u+F(x)]}&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded open subset of ℝ N {\mathbb{R}^{N}} , with N > 2 {N>2} , 𝒜 {\mathcal{A}} is a nonlinear Leray–Lions operator in divergence form, and E ⁢ ( x ) {E(x)} and F ⁢ ( x ) {F(x)} are vector fields satisfying suitable local summability properties.

On curvature tensors of Norden and metallic pseudo-Riemannian manifolds

Abstract: Abstract We study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.

The Brunn--Minkowski inequality and a Minkowski problem for -harmonic Green's function

Abstract: Abstract In this article we study two classical problems in convex geometry associated to 𝒜{\\mathcal{A}}-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p-Laplace equation. Let p be fixed with 2≤n≤p<∞{2\\leq n\\leq p<\\infty}. For a convex compact set E in ℝn{\\mathbb{R}^{n}}, we define and then prove the existence and uniqueness of the so-called 𝒜{\\mathcal{A}}-harmonic Green’s function for the complement of E with pole at infinity. We then define a quantity C𝒜⁢(E){\\mathrm{C}_{\\mathcal{A}}(E)} which can be seen as the behavior of this function near infinity. In the first part of this article, we prove that C𝒜⁢(⋅){\\mathrm{C}_{\\mathcal{A}}(\\,\\cdot\\,)} satisfies the following Brunn–Minkowski-type inequality: [C𝒜⁢(λ⁢E1+(1-λ)⁢E2)]1p-n≥λ⁢[C𝒜⁢(E1)]1p-n+(1-λ)⁢[C𝒜⁢(E2)]1p-n[\\mathrm{C}_{\\mathcal{A}}(\\lambda E_{1}+(1-\\lambda)E_{2})]^{\\frac{1}{p-n}}\\geq% \\lambda[\\mathrm{C}_{\\mathcal{A}}(E_{1})]^{\\frac{1}{p-n}}+(1-\\lambda)[\\mathrm{C% }_{\\mathcal{A}}(E_{2})]^{\\frac{1}{p-n}} when n

An investigation of fractional Bagley-Torvik equation

Abstract: Abstract In this paper the authors prove the existence as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our results rely on an appropriate fixed point theorem in partially ordered normed linear spaces. Illustrative examples are included to demonstrate the validity and applicability of our technique.

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case

Abstract: We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-manifold is leaf conjugate to the time one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz-Hertz-Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds.

Razumikhin-type theorem on time-changed stochastic functional differential equations with Markovian switching

Abstract: Abstract This work is mainly concerned with the exponential stability of time-changed stochastic functional differential equations with Markovian switching. By expanding the time-changed Itô formula and the Razumikhin theorem, we obtain the exponential stability results for the time-changed stochastic functional differential equations with Markovian switching. What’s more, we get many useful stability results by applying our new results to several important types of functional differential equations. Finally, an example is given to demonstrate the effectiveness of the main results.

Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential

Abstract: Abstract In this paper, we consider the nonlinear eigenvalue problem: Δ(|Δu|p(x)−2Δu)=λ|u|q(x)−2uδ(x)2q(x)inΩ,u∈W02,p(x)(Ω), $$\\begin{array}{} \\displaystyle \\begin{cases} {\\it\\Delta}(|{\\it\\Delta} u|^{p(x)-2}{\\it\\Delta} u)= \\lambda \\frac{|u|^{q(x)-2}u}{{\\delta(x)}^{2q(x)}} \\;\\; \\mbox{in}\\;\\; {\\it\\Omega}, \\\\ u\\in W_0^{2,p(x)}({\\it\\Omega}), \\end{cases} \\end{array}$$ where Ω is a regular bounded domain of ℝN, δ(x) = dist(x, ∂Ω) the distance function from the boundary ∂Ω, λ is a positive real number, and functions p(⋅), q(⋅) are supposed to be continuous on Ω satisfying 1 0 provided that Hardy-Rellich inequality holds.

A Survey of Riemannian Contact Geometry

Abstract: Abstract This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

Topological Indices of Para-line Graphs of V-Phenylenic Nanostructures

Abstract: Abstract The degree-based topological indices are numerical graph invariants which are used to correlate the physical and chemical properties of a molecule with its structure. Para-line graphs are used to represent the structures of molecules in another way and these representations are important in structural chemistry. In this article, we study certain well-known degree-based topological indices for the para-line graphs of V-Phenylenic 2D lattice, V-Phenylenic nanotube and nanotorus by using the symmetries of their molecular graphs.

Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization

Abstract: Abstract The existing results on the variational inequality problems for fuzzy mappings and their applications were based on Zadeh’s decomposition theorem and were formally characterized by the precise sets which are the fuzzy mappings’ cut sets directly. That is, the existence of the fuzzy variational inequality problems in essence has not been solved. In this paper, the fuzzy variational-like inequality problems is incorporated into the framework of n-dimensional fuzzy number space by means of the new ordering of two n-dimensional fuzzy-number-valued functions we proposed [Fuzzy Sets and Systems 295 (2016) 19-36]. As a theoretical basis, the existence and the basic properties of the fuzzy variational inequality problems are discussed. Furthermore, the relationship between the variational-like inequality problems and the fuzzy optimization problems is discussed. Finally, we investigate the optimality conditions for the fuzzy multiobjective optimization problems.

Geometry and Topology of the Space of Plurisubharmonic Functions

Abstract: Let $\Omega$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $\Omega$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local Kahler-Einstein metrics.

L-topological-convex spaces generated by L-convex bases

Abstract: Abstract In this paper, axiomatic definitions of both L-convex bases and L-convex subbases are introduced and their relations with L-convex spaces are studied. Based on this, the notion of L-topological-convex space is introduced as a triple (X, 𝓣, 𝓒), where X is a nonempty set, 𝓒 is an L-convex structure on X and 𝓣 is an L-cotopology on X compatible with 𝓒. It can be characterized by many means.

Spectral geometry of the Steklov problem on orbifolds

Abstract: We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two-dimensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components.