Authors:Carlos E. Arreche Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation [math] where the coefficients [math] are rational functions in [math] with coefficients in [math]. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:41Z DOI: 10.1142/S0219199716500565
Authors:Fabio Cavalletti, Andrea Mondino Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. In this paper we prove that in a metric measure space [math] verifying the measure contraction property with parameters [math] and [math], any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to [math] and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:39Z DOI: 10.1142/S0219199717500079
Authors:Andrzej Weber, Michał Wojciechowski Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. We consider Auerbach bases in Banach spaces of dimension [math]. We show that there exist at least [math] such bases. This estimate follows from the calculation of the Lusternik–Schnirelmann category of the flag variety. A better estimate is obtained for generic smooth Banach spaces using Morse theory. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:37Z DOI: 10.1142/S021919971750016X
Authors:The Anh Bui, Xuan Truong Le Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:35Z DOI: 10.1142/S0219199716500462
Authors:Debora Impera, Michele Rimoldi Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. In this paper, we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of [math]-minimal hypersurfaces. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:33Z DOI: 10.1142/S021919971750002X
Authors:César R. de Oliveira, Alessandra A. Verri Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. We study the operator [math] restricted to a waveguide [math], with heterogeneous function [math] constant in the longitudinal direction. The purpose is to obtain an effective operator, in the norm resolvent sense, when the diameter of [math] tends to zero with heterogeneity approaching a homogeneous situation (i.e. a constant function [math]). The effective operator presents a potential that, besides the traditional dependence on waveguide geometric properties, there is also a contribution from [math] which results, when combined with the curvature, for example, in the possibility of a repulsive interaction. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:31Z DOI: 10.1142/S0219199716500607
Authors:M. L. M. Carvalho, Edcarlos D. da Silva, C. Goulart Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. In this paper, the existence and multiplicity of solutions for a quasilinear elliptic problem driven by the [math]-Laplacian operator is established. These solutions are also built as ground state solutions using the Nehari method. The main difficulty arises from the fact that the [math]-Laplacian operator is not homogeneous and the nonlinear term is indefinite. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:29Z DOI: 10.1142/S0219199716500504
Authors:Tingzhi Cheng, Genggeng Huang, Congming Li Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: (−Δ)α 2u = f(x,u,∇u)in Ω, for α ∈ (0, 2).u> 0,in Ω; u ≡ 0,in ℝn\Ω, Here [math] is a domain (bounded or unbounded) in [math] which is convex in [math]-direction. [math] is the nonlocal fractional Laplacian operator which is defined as (−Δ)α 2u(x) = Cn,αP.V.∫ℝnu(x) − u(y) x − y n+α , 0 < α < 2. Under various conditions on [math] and on a solution [math] it is shown that [math] is strictly increasing in [math] in the left half of [math], or in [math]. Symmetry (in [math]) of some solutions is proved. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:26Z DOI: 10.1142/S0219199717500183
Authors:Kōdai Fujimoto, Naoto Yamaoka Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. This paper deals with an equivalent system to the nonlinear differential equation of Liénard type [math], where the range of the function [math] is bounded. Sufficient conditions are obtained for the system to have at least one limit cycle. The proofs of our results are based on phase plane analysis of the system with the Poincaré–Bendixon theorem. Moreover, to show that these sufficient conditions are suitable in some sense, we also establish the results that the system has no limit cycles. Finally, some examples are given to illustrate our results. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:24Z DOI: 10.1142/S0219199716500577
Authors:Weiren Zhao Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. In this paper, we prove the local existence and uniqueness of the solution with discontinuous density for the inhomogeneous non-resistive magnetohydrodynamics (MHD) equations on a [math] bounded domain [math] or [math], if the initial data [math] with [math] satisfies [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:22Z DOI: 10.1142/S0219199716500553
Authors:Michael Ruzhansky, Durvudkhan Suragan Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. We establish sharp remainder terms of the [math]-Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli–Kohn–Nirenberg-type inequalities in [math] with arbitrary quasi-norms. We also present explicit examples to illustrate our results for different weights and in abelian cases. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:21Z DOI: 10.1142/S0219199717500146
Authors:N. V. Krylov Abstract: Communications in Contemporary Mathematics, Volume 19, Issue 06, December 2017. We establish the existence of solutions of fully nonlinear elliptic second-order equations like [math] in smooth domains without requiring [math] to be convex or concave with respect to the second-order derivatives. Apart from ellipticity nothing is required of [math] at points at which [math], where [math] is any given constant. For large [math] some kind of relaxed convexity assumption with respect to [math] mixed with a VMO condition with respect to [math] are still imposed. The solutions are sought in Sobolev classes. We also establish the solvability without almost any conditions on [math], apart from ellipticity, but of a “cut-off” version of the equation [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-30T09:31:18Z DOI: 10.1142/S0219199717500092
Authors:Jacques Giacomoni, Vicenţiu Rădulescu, Guillaume Warnault Abstract: Communications in Contemporary Mathematics, Ahead of Print. We discuss the existence and uniqueness of the weak solution of the following nonlinear parabolic problem: (PT) ut −∇⋅a(x,∇u) = f(x,u)in QT=def(0,T) × Ω,u = 0 on ΣT=def(0,T) × ∂Ω,u(0,x) = u0(x) in Ω, which involves a quasilinear elliptic operator of Leray–Lions type with variable exponents. Next, we discuss the global behavior of solutions and in particular the convergence to a stationary solution as [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:48Z DOI: 10.1142/S0219199717500651
Authors:Taísa Junges Miotto, Márcio Luís Miotto Abstract: Communications in Contemporary Mathematics, Ahead of Print. This work has objective to obtain results of existence and multiplicity of solutions for an Ambrosetti–Prodi-type problem for the [math] operator. Moreover, it was proved a continuity result for the parameter which limits the existence of solutions in relation of the parameter [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:48Z DOI: 10.1142/S0219199717500675
Authors:Nader Masmoudi, Federica Sani Abstract: Communications in Contemporary Mathematics, Ahead of Print. Adams' inequality is the complete generalization of the Trudinger–Moser inequality to the case of Sobolev spaces involving higher order derivatives. The failure of the original form of the sharp inequality when the problem is considered on the whole space [math] served as a motivation to investigate in the direction of a refined sharp inequality, the so-called Adams' inequality with the exact growth condition. Due to the difficulties arising in the higher order case from the lack of direct symmetrization techniques, this refined result is known to hold on first- and second-order Sobolev spaces only. We extend the validity of Adams' inequality with the exact growth to higher order Sobolev spaces. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:47Z DOI: 10.1142/S0219199717500729
Authors:Binru Li Abstract: Communications in Contemporary Mathematics, Ahead of Print. The aim of this paper is to investigate the closed subschemes of moduli spaces corresponding to projective varieties which admit an effective action by a given finite group [math]. To achieve this, we introduce the moduli functor [math] of [math]-marked Gorenstein canonical models with Hilbert polynomial [math], and prove the existence of [math], the coarse moduli scheme for [math]. Then we show that [math] has a proper and finite morphism onto [math] so that its image [math] is a closed subscheme. In the end we obtain the canonical representation type decomposition [math] of [math] and use [math] to study the structure of [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:47Z DOI: 10.1142/S0219199717500614
Authors:Haidong Liu, Zhaoli Liu Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, existence and multiplicity of positive solutions of the elliptic system − Δu + V1(x)u = μ1(x)u3 + β(x)uv2in Ω ðœ€, − Δv + V2(x)v = β(x)u2v + μ 2(x)v3 in Ω ðœ€,u = v = 0 on ∂Ω𜀠is proved, where [math] is an exterior domain in [math] such that [math] is far away from the origin and contains a sufficiently large ball, [math], and the coefficients [math] are continuous functions on [math] which tend to positive constants at infinity. We do not assume [math] to be positive functions. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:47Z DOI: 10.1142/S0219199717500638
Authors:Toke Meier Carlsen, James Rout Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct (graded) groupoids from (graded) Steinberg algebras and use this to characterize when there is a diagonal-preserving (graded) isomorphism between two (graded) Steinberg algebras. We apply this characterization to groupoids of directed graphs in order to study diagonal-preserving (graded) isomorphisms of Leavitt path algebras and ∗-isomorphisms of graph [math]-algebras. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:47Z DOI: 10.1142/S021919971750064X
Authors:Debabrata Karmakar Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space [math] ∫ℍ4 e32π2u2 − 1 (1 + u )2 dvg ≤ C∥u∥L2(ℍ4)2, (0.1) [math]. We will also establish an Adachi–Tanaka-type inequality in this setting. Another aspect of this paper is the Lions lemma in the hyperbolic space. We prove Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:46Z DOI: 10.1142/S0219199717500663
Authors:Erez Buchweitz Abstract: Communications in Contemporary Mathematics, Ahead of Print. Given a suitably normalized random vector [math], we observe that the function [math], defined for [math], admits surprisingly strong concentration far surpassing what is expected on account of Lévy’s isoperimetric inequality. Among the measures to which the above holds are all log-concave measures, for which a solution of the similar problem concerning the third marginal moments [math] would imply the hyperplane conjecture. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:46Z DOI: 10.1142/S0219199717500687
Authors:Ugo Bruzzo, Antonella Grassi Abstract: Communications in Contemporary Mathematics, Ahead of Print. The Noether–Lefschetz theorem asserts that any curve in a very general surface [math] in [math] of degree [math] is a restriction of a surface in the ambient space, that is, the Picard number of [math] is [math]. We proved previously that under some conditions, which replace the condition [math], a very general surface in a simplicial toric threefold [math] (with orbifold singularities) has the same Picard number as [math]. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in [math] in a linear system of a Cartier ample divisor with respect to a [math]-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:46Z DOI: 10.1142/S0219199717500705
Authors:Marian Bocea, Mihai Mihăilescu Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, the minimization problem Λ1(p) :=infu∈X0∖{0}∫Ωsinh( ∇u p)dx ∫Ωsinh( u p)dx , where [math] is studied when [math] ([math]) is an open, bounded, convex domain with smooth boundary and [math]. We show that [math] is either zero, when the maximum of the distance function to the boundary of [math] is greater than [math], or it is a positive real number, when the maximum of the distance function to the boundary of [math] belongs to the interval [math]. In the latter case, we provide estimates for [math] and show that for [math] sufficiently large [math] coincides with the principal frequency of the [math]-Laplacian in [math]. Some particular cases and related problems are also discussed. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:46Z DOI: 10.1142/S0219199717500742
Authors:Ping Lin Abstract: Communications in Contemporary Mathematics, Ahead of Print. This paper concerns a global controllability problem for heat equations. In the absence of control, the solution to the linear heat system globally exists. While for each initial data, we can find a feedback control acting on an internal subset of the space domain such that the corresponding solution to the system blows up at given time. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:45Z DOI: 10.1142/S0219199717500626
Authors:Hairong Liu, Tian Long, Xiaoping Yang Abstract: Communications in Contemporary Mathematics, Ahead of Print. We give an explicit description of polynomial growth solutions to some sub-elliptic operators of divergence form with [math]-periodic coefficients on the Heisenberg group, where the periodicity has to be meant with respect to the Heisenberg geometry. We show that the polynomial growth solutions are necessarily polynomials with [math]-periodic coefficients. We also prove the Liouville-type theorem for the Dirichlet problem to these sub-elliptic equations on an unbounded domain on the Heisenberg group, show that any bounded solution to the Dirichlet problem must be constant. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:45Z DOI: 10.1142/S0219199717500699
Authors:Abhishek Banerjee Abstract: Communications in Contemporary Mathematics, Ahead of Print. For a small abelian category [math], Auslander’s formula allows us to express [math] as a quotient of the category [math] of coherent functors on [math]. We consider an abelian category with the added structure of a cohereditary torsion pair [math]. We prove versions of Auslander’s formula for the torsion-free class [math] of [math], for the derived torsion-free class [math] of the triangulated category [math] as well as the induced torsion-free class in the ind-category [math] of [math]. Further, for a given regular cardinal [math], we also consider the category [math] of [math]-presentable objects in the functor category [math]. Then, under certain conditions, we show that the torsion-free class [math] can be recovered as a subquotient of [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:45Z DOI: 10.1142/S0219199717500717
Authors:Slaven Kožić, Mirko Primc Abstract: Communications in Contemporary Mathematics, Ahead of Print. In their seminal work Lepowsky and Wilson gave a vertex-operator theoretic interpretation of Gordon–Andrews–Bressoud’s generalization of Rogers–Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the principal Heisenberg subalgebra of standard [math]-modules, parametrized with partitions satisfying certain difference 2 conditions. In this paper, we define quasi-particles in the principal picture of [math] and construct quasi-particle monomial bases of standard [math]-modules for which principally specialized characters are given as products of sum sides of the corresponding analytic Rogers–Ramanujan-type identities with the character of the Fock space for the principal Heisenberg subalgebra. Citation: Communications in Contemporary Mathematics PubDate: 2017-08-17T06:24:44Z DOI: 10.1142/S0219199717500730
Authors:Carolina Araujo, Mauricio Corrêa, Alex Massarenti Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. As a consequence, we obtain a classification of codimension one del Pezzo distributions on Fano manifolds with Picard number one. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-21T09:19:21Z DOI: 10.1142/S0219199717500584
Authors:Man Chun Leung, Feng Zhou Abstract: Communications in Contemporary Mathematics, Ahead of Print. By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on [math] ([math]) when the prescribed function (after being projected to [math]) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness [math]), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:17Z DOI: 10.1142/S0219199717500511
Authors:Valentino Magnani, Dario Trevisan Abstract: Communications in Contemporary Mathematics, Ahead of Print. We introduce a class of “Lipschitz horizontal” vector fields in homogeneous groups, for which we show equivalent descriptions, e.g., in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness result both at equilibrium points and for vector fields of an involutive submodule of Lipschitz horizontal vector fields. We finally exhibit a counterexample to the general well-posedness theory for Lipschitz horizontal vector fields, in contrast with the Euclidean theory. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:17Z DOI: 10.1142/S0219199717500572
Authors:Luis J. Alías, Verónica L. Cánovas, Marco Rigoli Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study codimension two trapped submanifolds contained into one of the two following null hypersurfaces of de Sitter spacetime: (i) the future component of the light cone, and (ii) the past infinite of the steady state space. For codimension two compact spacelike submanifolds in the light cone we show that they are conformally diffeomorphic to the round sphere. This fact enables us to deduce that the problem of characterizing compact marginally trapped submanifolds into the light cone is equivalent to solving the Yamabe problem on the round sphere, allowing us to obtain our main classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space and characterize those having parallel mean curvature field. Finally, we consider the more general case of codimension two complete, non-compact, weakly trapped spacelike submanifolds contained into the light cone. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:17Z DOI: 10.1142/S0219199717500596
Authors:Alvaro Martínez-Pérez, José M. Rodríguez Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov boundary. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:16Z DOI: 10.1142/S021919971750050X
Authors:Sérgio H. Monari Soares, Yony R. Santaria Leuyacc Abstract: Communications in Contemporary Mathematics, Ahead of Print. We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system − Δu + V (x)u = g(v),x ∈ ℝ2, − Δv + V (x)v = f(u), x ∈ ℝ2, where [math] is a positive function which can vanish at infinity and be unbounded from above and [math] and [math] have exponential growth range. The proof involves a truncation argument combined with the linking theorem and a finite-dimensional approximation. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:16Z DOI: 10.1142/S0219199717500535
Authors:Marco Falconi Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we aim to characterize the cylindrical Wigner measures associated to regular quantum states in the Weyl C*-algebra of canonical commutation relations. In particular, we provide conditions at the quantum level sufficient to prove the concentration of all the corresponding cylindrical Wigner measures as Radon measures on suitable topological vector spaces. The analysis is motivated by variational and dynamical problems in the semiclassical study of bosonic quantum field theories. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:16Z DOI: 10.1142/S0219199717500559
Authors:Bendehiba Senoussi, Mohammed Bekkar Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we study translation surfaces of some new types with non-lightlike axis in 3-dimensional Minkowski space [math] satisfying the condition [math], where [math] and [math] denotes the Laplace operator and we obtain the complete classification for those ones. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:15Z DOI: 10.1142/S0219199717500523
Authors:Wenhua Zhao Abstract: Communications in Contemporary Mathematics, Ahead of Print. Let [math] be a commutative ring and [math] an [math]-algebra. An [math]-[math]-derivation of [math] is an [math]-linear map of the form [math] for some [math]-algebra endomorphism [math] of [math], where [math] denotes the identity map of [math]. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) [math]-derivation or [math]-[math]-derivation of [math] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [math], and whether or not a locally nilpotent (LN) [math]-derivation or [math]-[math]-derivation of [math] maps every ideal of [math] to a Mathieu subspace of [math]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [math] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [math]-[math]-derivations of integral domains of characteristic zero. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:15Z DOI: 10.1142/S0219199717500560
Authors:Vincenzo Ambrosio, Teresa Isernia Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we deal with the following fractional Kirchhoff equation p + q(1 − s)∬ℝ2N u(x) − u(y) 2 x − y N+2s dxdy (−Δ)su = g(u)inℝN, where [math], [math], [math], [math] is a small positive parameter and [math] is an odd function satisfying Berestycki–Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that [math] is sufficiently small. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:14Z DOI: 10.1142/S0219199717500547
Authors:Shicheng Xu Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold [math]: (1)the convexity radius of [math], [math], where [math] is the injectivity radius of [math] and [math] is the focal radius of open ball centered at [math] with radius [math]; (2)for any two points [math] in [math], [math] where [math] is the conjugate radius of [math]; (3)for any [math], any (not necessarily minimizing) geodesic in [math] has length [math]. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous. Citation: Communications in Contemporary Mathematics PubDate: 2017-06-13T09:36:14Z DOI: 10.1142/S0219199717500602
Authors:Changxing Miao, Xingdong Tang, Guixiang Xu Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we characterize a family of solitary waves for nonlinear Schrödinger equation (NLS) with derivative (DNLS) by the structure analysis and the variational argument. Since DNLS does not enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters [math] and the critical parameters [math], we show the existence and uniqueness of the solitary waves for DNLS, up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters [math], [math] and the supercritical parameters [math], there is no nontrivial solitary wave for DNLS. At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for DNLS with initial data in the invariant set [math], with [math], [math] or [math]. On the one hand, different with the scattering result for the [math]-critical NLS in [B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285(5) (2015) 1589–1618], the scattering result of DNLS does not hold for initial data in [math] because of the existence of infinity many small solitary/traveling waves in [math] with [math], [math] or [math]. On the other hand, our global result improves the global result in [Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. Partial Differential Equations 6(8) (2013) 1989–2002; Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. Partial Differential Equations 8(5) (2015) 1101–1112] (see Corollary 1.6). Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:14Z DOI: 10.1142/S0219199717500493
Authors:Jaume Llibre, Clàudia Valls Abstract: Communications in Contemporary Mathematics, Ahead of Print. We consider Hamiltonian systems with [math] degrees of freedom and a Hamiltonian of the form H = 1 2∑i=1dp 12 + V (q 1,…,qd), where [math] is a homogenous polynomial of degree [math]. We prove that such Hamiltonian systems with [math] odd or [math], have a Darboux first integral if and only if they have a polynomial first integral. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:11Z DOI: 10.1142/S0219199717500456
Authors:Guillaume Lévy Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we extend our previous result from [On uniqueness for a rough transport-diffusion equation, C. R. Acad. Sci. Sér. I[math] Math. 354(8) (2016) 804–807]. We prove that transport equations with rough coefficients do possess a uniqueness property, even in the presence of viscosity. Our method relies strongly on duality and bears a strong resemblance with the well-known DiPerna–Lions theory first developed in [Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511–547]. As an application, we show that the zero solution is the unique solution at the Leray regularity scale of the Euler and Navier–Stokes equations for zero initial datum. This uniqueness result allows us to reprove the celebrated theorem of Serrin [On the interior regularity of weak solutions of the Navier–Stokes equations, Arch. Ration. Mech. Anal. 9(1) (1962) 187–195] in a novel way. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:07Z DOI: 10.1142/S0219199717500481
Authors:Michael Brandenbursky, Jarek Kędra, Egor Shelukhin Abstract: Communications in Contemporary Mathematics, Ahead of Print. We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct explicit quasimorphisms on [math], some of them are [math]-continuous and vanish on all autonomous diffeomorphisms, and some of them are Calabi. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:06Z DOI: 10.1142/S0219199717500420
Authors:Chiara Camere Abstract: Communications in Contemporary Mathematics, Ahead of Print. We construct quasi-projective moduli spaces of [math]-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily–Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:04Z DOI: 10.1142/S0219199717500444
Authors:Miaomiao Niu, Zhongwei Tang, Lushun Wang Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: Δ2u + (λV (x) − δ(x))u = u p−2uinℝN, (P λ) where [math], [math], [math] is a parameter, [math] is a nonnegative potential function with nonempty zero set [math], [math] is a positive function such that the operator [math] is indefinite and non-degenerate for [math] large. We show that both in subcritical and critical cases, equation [math] admits a least energy solution which for [math] large localized near the zero set [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:01Z DOI: 10.1142/S021919971750047X
Authors:M. Schonbek, G. Seregin Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this note, we study the behavior of Lebesgue norms [math] of solutions [math] to the Cauchy problem for the Stokes system with drift [math], which is supposed to be a divergence free smooth vector valued function satisfying a scale invariant condition. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:14:58Z DOI: 10.1142/S0219199717500468
Authors:Andrei Minchenko, Alexey Ovchinnikov Abstract: Communications in Contemporary Mathematics, Ahead of Print. Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-21T03:46:53Z DOI: 10.1142/S0219199717500389
Authors:Fashun Gao, Minbo Yang Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we are concerned with the following nonlinear Choquard equation −Δu + V (x)u = ∫ℝN G(y,u) x − y μdyg(x,u)in ℝN, where [math], [math] and [math]. If [math] lies in a gap of the spectrum of [math] and [math] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579]. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-21T03:46:53Z DOI: 10.1142/S0219199717500377
Authors:Huyuan Chen, Feng Zhou Abstract: Communications in Contemporary Mathematics, Ahead of Print. Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case [math] −Δu + u = Iα[up]uqinℝN ∖{0},lim x →+∞u(x) = 0, where [math] and [math] is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent [math]. Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-21T03:46:52Z DOI: 10.1142/S0219199717500407
Authors:Fanyun Meng, Junchao Wei Abstract: Communications in Contemporary Mathematics, Ahead of Print. Let [math] be a ring and [math] an idempotent of [math], [math] is called an [math]-symmetric ring if [math] implies [math] for all [math]. Obviously, [math] is a symmetric ring if and only if [math] is a [math]-symmetric ring. In this paper, we show that a ring [math] is [math]-symmetric if and only if [math] is left semicentral and [math] is symmetric. As an application, we show that a ring [math] is left min-abel if and only if [math] is [math]-symmetric for each left minimal idempotent [math] of [math]. We also introduce the definition of strongly [math]-symmetric ring and prove that [math] is a strongly [math]-symmetric ring if and only if [math] and [math] is a symmetric ring. Finally, we introduce [math]-reduced ring and study some properties of it. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-21T03:46:52Z DOI: 10.1142/S0219199717500390
Authors:Lysianne Hari, Nicola Visciglia Abstract: Communications in Contemporary Mathematics, Ahead of Print. We study small data scattering in the energy space of solutions to the [math]-critical NLKG posed on product spaces [math] with [math] and [math] is a compact Riemannian manifold. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-19T09:35:33Z DOI: 10.1142/S0219199717500365
Authors:Luis Barreira, Claudia Valls Abstract: Communications in Contemporary Mathematics, Ahead of Print. We give a complete characterization of the existence of Lyapunov coordinate changes bringing an invertible sequence of matrices to one in block form. In other words, we give a criterion for the block-trivialization of a nonautonomous dynamics with discrete time while preserving the asymptotic properties of the dynamics. We provide two nontrivial applications of this criterion: we show that any Lyapunov regular sequence of invertible matrices can be transformed by a Lyapunov coordinate change into a constant diagonal sequence; and we show that the family of all coordinate changes preserving simultaneously the Lyapunov exponents of all sequences of invertible matrices (with finite exponent) coincides with the family of Lyapunov coordinate changes. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:57Z DOI: 10.1142/S0219199717500274
Authors:Daniel Pellegrino, Eduardo V. Teixeira Abstract: Communications in Contemporary Mathematics, Ahead of Print. We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [math]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [math]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [math], then the optimal constants of the [math]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [math]. It is likely that indeed the entropy grows as [math], and in this scenario, we show that the optimal constants are precisely [math]. In the bilinear case, [math], we show that any extremum of the Littlewood’s [math] inequality has entropy [math] and complexity [math], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [math]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [math]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [math]-linear forms, which has further consequences to the theory, as we explain herein. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:56Z DOI: 10.1142/S0219199717500298
Authors:Amal Attouchi, Mikko Parviainen Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we study an evolution equation involving the normalized [math]-Laplacian and a bounded continuous source term. The normalized [math]-Laplacian is in non-divergence form and arises for example from stochastic tug-of-war games with noise. We prove local [math] regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:56Z DOI: 10.1142/S0219199717500353
Authors:Young Jin Suh Abstract: Communications in Contemporary Mathematics, Ahead of Print. We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics [math], [math]. We show that [math] is even, say [math], and any such hypersurface becomes an open part of a tube around a [math]-dimensional complex hyperbolic space [math] which is embedded canonically in [math] as a totally geodesic complex submanifold or a horosphere whose center at infinity is [math]-isotropic singular. As a consequence of the result, we get the nonexistence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics [math], [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:54Z DOI: 10.1142/S0219199717500316
Authors:Arka Mallick, Cyril Tintarev Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we derive the following Leray–Trudinger type inequality on a bounded domain [math] in [math] containing the origin. supu∈W01,n(Ω),In[u,Ω,R]≤1∫Ωecn u(x) E2β x R n n−1dx < +∞, for some cn > 0 depending only on n. Here, [math], [math], [math] and [math], [math] for [math] This improves an earlier result by Psaradakis and Spector. Also, we prove that for any [math] in the place of [math], the above inequality is false if we take [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:52Z DOI: 10.1142/S0219199717500341
Authors:Ali Hyder, Stefano Iula, Luca Martinazzi Abstract: Communications in Contemporary Mathematics, Ahead of Print. Let [math] be an integer. For any open domain [math], non-positive function [math] such that [math], and bounded sequence [math] we prove the existence of a sequence of functions [math] solving the Liouville equation of order [math] (−Δ)mu k = Vke2mukin Ω,limsupk→∞∫Ωe2mukdx < ∞, and blowing up exactly on the set [math], i.e. limk→∞uk(x) = +∞ for x ∈ Sφandlimk→∞uk(x) = −∞ for x ∈ Ω ∖ Sφ, thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of [math] and to the case [math]. Several related problems remain open. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:52Z DOI: 10.1142/S0219199717500262
Authors:Lorenzo Brasco, Filippo Santambrogio Abstract: Communications in Contemporary Mathematics, Ahead of Print. We consider local weak solutions of the Poisson equation for the [math]-Laplace operator. We prove a higher differentiability result, under an essentially sharp condition on the right-hand side. The result comes with a local scaling invariant a priori estimate. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:49Z DOI: 10.1142/S0219199717500304
Authors:Jin Hong Kim Abstract: Communications in Contemporary Mathematics, Ahead of Print. It has been recently shown by Meng and Zhang that the full automorphism group [math] is a Jordan group for all projective varieties in arbitrary dimensions. The aim of this paper is to show that the full automorphism group [math] is, in fact, a Jordan group even for all normal compact Kähler varieties in arbitrary dimensions. The meromorphic structure of the identity component of the automorphism group and its Rosenlicht-type decomposition play crucial roles in the proof. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:48Z DOI: 10.1142/S0219199717500249
Authors:Jaume Llibre, Regilene Oliveira Abstract: Communications in Contemporary Mathematics, Ahead of Print. The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic [math]: [math], and a Darboux invariant of the form [math] with [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:47Z DOI: 10.1142/S021919971750033X
Authors:Alexander Quaas, Aliang Xia Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: (−Δ)αu = f(x,u,v,∇u,∇v)in Ω,(−Δ)αv = g(x,u,v,∇u,∇v)in Ω,u = v = 0 in ℝN∖Ω, where [math] denotes the fractional Laplacian and [math] is a smooth bounded domain in [math]. It shown that under some assumptions on [math] and [math], the problem has at least one positive solution [math]. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:45Z DOI: 10.1142/S0219199717500328
Authors:Jun Cao, Luong Dang Ky, Dachun Yang Abstract: Communications in Contemporary Mathematics, Ahead of Print. Let [math] and [math] be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of [math] and their dual spaces. More precisely, the authors prove that [math] and, for any [math], [math], where [math] denotes the local BMO space, [math], for any [math] and [math], the inhomogeneous Lipschitz space and [math] a variant of the local Orlicz–Hardy space related to the Orlicz function [math] for any [math] which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:44Z DOI: 10.1142/S0219199717500250
Authors:Kanishka Perera, Marco Squassina Abstract: Communications in Contemporary Mathematics, Ahead of Print. We obtain nontrivial solutions for a class of double-phase problems using Morse theory. In the absence of a direct sum decomposition, we use a cohomological local splitting to get an estimate of the critical groups at zero. Citation: Communications in Contemporary Mathematics PubDate: 2017-01-17T11:17:52Z DOI: 10.1142/S0219199717500237
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