Authors:Jifeng Chu Pages: 651 - 659 Abstract: We prove some results on the existence and uniqueness of solutions to a recently derived nonlinear model for the ocean flow in arctic gyres. We derive an equivalent formulation of the problem in the form of an integral equation on a semi-infinite interval and then use fixed point techniques. We also obtain some stability result. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0696-6 Issue No:Vol. 197, No. 3 (2018)

Authors:Ilaria Del Corso; Roberto Dvornicich Pages: 661 - 671 Abstract: Fuchs (Abelian groups, Pergamon, Oxford, 1960, Problem 72) asked the following question: which groups can be the group of units of a commutative ring' In the following years, some partial answers have been given to this question in particular cases. The aim of the present paper is to address Fuchs’ question when A is a finite characteristic ring. The result is a pretty good description of the groups which can occur as group of units in this case, equipped with examples showing that there are obstacles to a “short” complete classification. As a by-product, we are able to classify all possible cardinalities of the group of units of a finite characteristic ring, so to answer Ditor’s question (Ditor in Am Math Mon 78(5):522–523, 1971). PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0697-5 Issue No:Vol. 197, No. 3 (2018)

Authors:Yu Liu; Jie Xiao Pages: 673 - 702 Abstract: Functional capacities on the Grushin space \({\mathbb {G}}^n_\alpha \) are introduced, developed, and subsequently applied to the theory of Sobolev embeddings. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0699-3 Issue No:Vol. 197, No. 3 (2018)

Authors:Ming Xu Pages: 703 - 720 Abstract: In this paper, I study the isoparametric hypersurfaces in a Randers sphere \((S^n,F)\) of constant flag curvature, with the navigation datum (h, W). I prove that an isoparametric hypersurface M for the standard round sphere \((S^n,h)\) which is tangent to W remains isoparametric for \((S^n,F)\) after the navigation process. This observation provides a special class of isoparametric hypersurfaces in \((S^n,F)\) , which can be equivalently described as the regular level sets of isoparametric functions f satisfying \(-f\) is transnormal. I provide a classification for these special isoparametric hypersurfaces M, together with their ambient metric F on \(S^n\) , except the case that M is of the OT-FKM type with the multiplicities \((m_1,m_2)=(8,7)\) . I also give a complete classification for all homogeneous hypersurfaces in \((S^n,F)\) . They all belong to these special isoparametric hypersurfaces. Because of the extra W, the number of distinct principal curvature can only be 1, 2 or 4, i.e. there are less homogeneous hypersurfaces for \((S^n,F)\) than those for \((S^n,h)\) . PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0701-0 Issue No:Vol. 197, No. 3 (2018)

Authors:Corentin Audiard; Boris Haspot Pages: 721 - 760 Abstract: We consider the Euler equations with a capillary tensor in the case of the so-called quantum hydrodynamics, which is formally equivalent to the Gross–Pitaevskii equation. Our main result is the existence and uniqueness of global solutions without vortices for small data in dimension at least 3. The absence of vortices means that the density remains bounded away from 0. Previous results include existence of global solutions without uniqueness (Antonelli and Marcati in Commun Math Phys 287(2):657–686, 2009) and lower bounds on the first occurrence of vortices (Béthuel et al. in J Anal Math 110:297–338, 2010). Our proof uses in a crucial way some deep results on the scattering of the Gross–Pitaevskii equation due to Gustafson et al. (Commun Contemp Math 11(4):657–707, 2009). The optimality of our assumptions is discussed, in particular we show that for slightly less regular initial data the density does not even remain bounded. We also sketch in the appendix the key arguments for the scattering of solutions of the Gross–Pitaevskii equation. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0702-z Issue No:Vol. 197, No. 3 (2018)

Authors:Hans Weber Pages: 761 - 774 Abstract: We prove extension theorems for group-valued modular functions defined on orthomodular lattices or modular complemented lattices and for modular measures defined on (pseudo-)D-lattices generalizing results of Riečan (Mat Časopis Sloven Akad Vied 19:44–49, 1969; 20:239–244, 1970; Comment Math Univ Carolin 20:309–316, 1979), Avallone and De Simone (Ital J Pure Appl Math 9:109–122, 2001) and Avallone et al. (Bollettino UMI 9-B(8):423–444, 2006; Math Slovaca 66:421–438, 2016). As basic tool, we first prove an extension theorem for lattice uniformities. This also yields as immediate consequence a result on the extension of modular functions on arbitrary lattices similar to that of Fox and Morales (Fundam Math 78:99–106, 1973) and Kranz (Fundam Math 91:171–178, 1976). PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0703-y Issue No:Vol. 197, No. 3 (2018)

Authors:G. B. Maggiani; M. G. Mora Pages: 775 - 815 Abstract: In this paper, we rigorously deduce a quasistatic evolution model for shallow shells by means of \(\Gamma \) -convergence. The starting point of the analysis is the three-dimensional model of Prandtl–Reuss elasto-plasticity. We study the asymptotic behaviour of the solutions, as the thickness of the shell tends to zero. As in the case of plates, the limiting model is genuinely three-dimensional, limiting displacements are of Kirchhoff–Love type, and the stretching and bending components of the stress are coupled in the flow rule and in the stress constraint. However, in contrast with the case of plates, the equilibrium equations are not decoupled, because of the presence of curvature terms. An equivalent formulation of the limiting problem in rate form is also discussed. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0704-x Issue No:Vol. 197, No. 3 (2018)

Authors:Fumihiko Hirosawa; Wanderley Nunes do Nascimento Pages: 817 - 841 Abstract: The aim of this paper is to prove some energy estimates for Klein–Gordon equations with time-dependent potential. If the potential is “non-effective” and has “very slow oscillations” in the time-dependent coefficient, then energy estimates are proved in Ebert et al. (in: Dubatovskaya and Rogosin (eds) AMADE 2012, Cambridge Scientific Publishers, Cambridge, 2014). In contrast, the main goal of the present paper is to generalize the previous results to potentials with “very fast oscillations” in the time-dependent coefficient; consequently, the positivity of the potential is not required anymore. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0705-9 Issue No:Vol. 197, No. 3 (2018)

Authors:T. Tachim Medjo Pages: 843 - 868 Abstract: In this article, we study a globally modified Allen–Cahn–Navier–Stokes system in a three-dimensional domain. The model consists of the globally modified Navier–Stokes equations proposed in Caraballo et al. (Adv Nonlinear Stud 6(3):411–436, 2006) for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We prove the existence and uniqueness of strong solutions. Using the flattening property, we also prove the existence of global \({ \mathbb {V} }\) -attractors for the model. Using a limiting argument, we derive the existence of bounded entire weak solutions for the three-dimensional coupled Allen–Cahn–Navier–Stokes system with time-independent forcing. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0706-8 Issue No:Vol. 197, No. 3 (2018)

Authors:Alberto Boscaggin; Walter Dambrosio; Duccio Papini Pages: 869 - 882 Abstract: For the planar N-centre problem $$\begin{aligned} \ddot{x} = - \sum _{i=1}^N \frac{m_i (x-c_i)}{\vert x - c_i \vert ^{\alpha +2}}, \qquad x \in \mathbb {R}^2 {\setminus } \{ c_1,\ldots ,c_N \}, \end{aligned}$$ where \(m_i > 0\) for \(i=1,\ldots ,N\) and \(\alpha \in [1,2)\) , we prove the existence of entire parabolic trajectories, having prescribed asymptotic directions for \(t \rightarrow \pm \,\infty \) and prescribed topological characterization with respect to the set of the centres. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0707-7 Issue No:Vol. 197, No. 3 (2018)

Authors:Claudianor O. Alves; Jefferson A. dos Santos; Rodrigo C. M. Nemer Pages: 883 - 903 Abstract: In this work, we use the Lusternik–Schnirelmann category to estimate the number of nontrivial solutions for a problem with discontinuous nonlinearity and subcritical growth. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0708-6 Issue No:Vol. 197, No. 3 (2018)

Authors:Julià Cufí; Artur Nicolau; Andreas Seeger; Joan Verdera Pages: 905 - 940 Abstract: We prove a characterization of some \(L^p\) -Sobolev spaces involving the quadratic symmetrization of the Calderón commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type estimate is established for functions in homogeneous Hardy–Sobolev spaces \(\dot{H}^1_\alpha \) . We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0709-5 Issue No:Vol. 197, No. 3 (2018)

Authors:Luigi Montoro; Fabio Punzo; Berardino Sciunzi Pages: 941 - 964 Abstract: We consider positive weak solutions to \((-\Delta )^s u=f(x,u)\) in \(\Omega {\setminus } \Gamma \) under zero Dirichlet boundary condition. The domain \(\Omega \) is bounded or is the whole space, and the solution has a singularity on the singular set \(\Gamma \) . Under suitable assumptions on f we prove symmetry and monotonicity properties of the solutions when the singular set \(\Gamma \) has zero s-capacity. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0710-z Issue No:Vol. 197, No. 3 (2018)

Authors:Jacek Dziubański; Marcin Preisner Pages: 965 - 987 Abstract: Let \(T_t=e^{-tL}\) be a semigroup of self-adjoint linear operators acting on \(L^2(X,\mu )\) , where \((X,d,\mu )\) is a space of homogeneous type. We assume that \(T_t\) has an integral kernel \(T_t(x,y)\) which satisfies the upper and lower Gaussian bounds: $$\begin{aligned} \frac{C_1}{\mu (B(x,\sqrt{t}))} \exp \left( {-\,c_1d(x,y)^2/ t} \right) \le T_t(x,y)\le \frac{C_2}{\mu (B(x,\sqrt{t}))} \exp \left( {-\,c_2 d(x,y)^2/ t} \right) . \end{aligned}$$ By definition, f belongs to \(H^1(L)\) if \(\Vert f\Vert _{{H^1(L)}}=\Vert \sup _{t>0} T_tf(x) \Vert _{L^1(X,\mu )} <\infty \) . We prove that there is a function \(\omega (x)\) , \(0<c\le \omega (x)\le C\) , such that \(H^1(L)\) admits an atomic decomposition with atoms satisfying: \({\mathrm{supp}}\, a\subset B\) , \(\Vert a\Vert _{L^\infty } \le \mu (B)^{-1}\) , and the weighted cancelation condition \(\int a(x)\omega (x)\hbox {d}\mu (x)=0\) . PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0711-y Issue No:Vol. 197, No. 3 (2018)

Authors:Michele Rossi; Lea Terracini Pages: 989 - 998 Abstract: After the publication of [2], we realized that Proposition 3.1, in that paper, contains an error, whose consequences are rather pervasive along the whole section 3 and for some aspects of examples 5.1 and 5.2. Here we give a complete account of needed corrections. PubDate: 2018-06-01 DOI: 10.1007/s10231-017-0698-4 Issue No:Vol. 197, No. 3 (2018)

Authors:Weisheng Niu; Yao Xu Abstract: This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators \(\mathcal {L}_\varepsilon \) , with rapidly oscillating periodic coefficients. We derive uniform boundary \(C^{m-1,\lambda } (0\!<\!\lambda \!<\!1)\) and \( W^{m,p}\) estimates in \(C^1\) domains, as well as uniform boundary \(C^{m-1,1}\) estimate in \(C^{1,\theta } (0\!<\!\theta \!<\!1)\) domains without the symmetry assumption on the operator. The proof, motivated by the works “Armstrong and Smart in Ann Sci Éc Norm Supér (4) 49(2):423–481 (2016) and Shen in Anal PDE 8(7):1565–1601 (2015),” is based on a suboptimal convergence rate in \(H^{m-1}(\Omega )\) . Compared to “Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036 (2012) and Shen (2015),” the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of \(u_0\) (the solution to the homogenized problem), and thus might be of some independent interests even for second-order elliptic systems. PubDate: 2018-06-02 DOI: 10.1007/s10231-018-0764-6

Authors:Jesús Suárez de la Fuente Abstract: We give a simple proof that there is no strictly singular bicentralizer on a super-reflexive Schatten ideal. This result applies, in particular, to the p-Schatten class for \(1<p<\infty \) . PubDate: 2018-05-29 DOI: 10.1007/s10231-018-0760-x

Authors:Peter Pflug; Włodzimierz Zwonek Abstract: In this paper, we show the existence of different types of peak functions in classes of \(\mathbb {C}\) -convex domains. As one of the tools used in this context is a result on preserving the regularity of \(\mathbb {C}\) -convex domains under projection. PubDate: 2018-05-29 DOI: 10.1007/s10231-018-0759-3

Authors:Yimei Li; Jiguang Bao Abstract: In this paper, we study the local behaviors of positive solutions of $$\begin{aligned} (-\Delta )^\sigma u= x ^{\tau }u^p \end{aligned}$$ with an isolated singularity at the origin, where \((-\Delta )^\sigma \) is the fractional Laplacian, \(0<\sigma <1\) , \(\tau >-2\sigma \) and \(p>1\) . Our first results provide a blowup rate estimate near an isolated singularity, and show that the solution is asymptotically radially symmetric. PubDate: 2018-05-28 DOI: 10.1007/s10231-018-0761-9

Authors:Hiroaki Ishida; Hisashi Kasuya Abstract: For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bi-graded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan’s theory of mixed Hodge structures as similar to the study on smooth algebraic varieties. PubDate: 2018-05-26 DOI: 10.1007/s10231-018-0762-8