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

Abstract: Communications in Contemporary Mathematics, Ahead of Print. Let [math] be a compact, connected, almost complex manifold of dimension [math] endowed with a [math]-preserving circle action with isolated fixed points. In this paper, we analyze the “geography problem” for such manifolds, deriving equations relating the Chern numbers to the index [math] of [math]. We study the symmetries and zeros of the Hilbert polynomial, which imply many rigidity results for the Chern numbers when [math]. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a 6-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. Another consequence is that, if the action is Hamiltonian, the minimal Chern number coincides with the index and is at most [math]. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:12Z DOI: 10.1142/S0219199717500432

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

Abstract: Communications in Contemporary Mathematics, Ahead of Print. In this paper, we establish a min-max theory for minimal surfaces using sweepouts of surfaces of genus [math]. We develop a direct variational method similar to the proof of the famous Plateau problem by Douglas [Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931) 263–321] and Rado [On Plateau’s problem, Ann. Math. 31 (1930) 457–469]. As a result, we show that the min-max value for the area functional can be achieved by a bubble tree limit consisting of branched genus-[math] minimal surfaces with nodes, and possibly finitely many branched minimal spheres. We also prove a Colding–Minicozzi type strong convergence theorem similar to the classical mountain pass lemma. Our results extend the min-max theory by Colding–Minicozzi and the author to all genera. Citation: Communications in Contemporary Mathematics PubDate: 2017-04-25T10:15:09Z DOI: 10.1142/S0219199717500419

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Abstract: Communications in Contemporary Mathematics, Ahead of Print. The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for [math], where [math] is the base matrix and [math] is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of [math] in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of [math] and for the distribution of the norm of [math] applied to a fixed vector. The bounds are uniform in [math] and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated. Citation: Communications in Contemporary Mathematics PubDate: 2017-02-17T08:12:50Z DOI: 10.1142/S0219199717500286

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

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

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

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

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