Authors:A. Ballester-Bolinches; S. Camp-Mora; M. R. Dixon; R. Ialenti; F. Spagnuolo Abstract: Abstract In this paper, we study the behavior of locally finite groups of infinite rank whose proper subgroups of infinite rank have one of the three following properties, which are generalizations of permutability: S-permutability, semipermutability and S-semipermutability. In particular, it is proved that if G is a locally finite group of infinite rank whose proper subgroups of infinite rank are S-permutable (resp. semipermutable), then G is locally nilpotent (resp. all subgroups are semipermutable). For locally finite groups whose proper subgroups of infinite rank are S-semipermutable, the same statement can be proved only for groups with min-p for every prime p. A counterexample is given for the general case. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0642-7

Authors:Hai-Ping Fu; Jian-Ke Peng Abstract: Abstract Let \((M^n, g)(n\ge 3)\) be an n-dimensional complete, simply connected, locally conformally flat Riemannian manifold with constant scalar curvature S. Denote by T the trace-free Ricci curvature tensor of M. The main result of this paper states that T goes to zero uniformly at infinity if for \(p\ge \frac{n}{2}\) , the \(L^{p}\) -norm of T is finite. As applications, we prove that \((M^n, g)\) is compact if the \(L^{p}\) -norm of T is finite and S is positive, and \((M^n, g)\) is scalar flat if \((M^n, g)\) is a noncompact manifold with nonnegative constant scalar curvature and the \(L^{p}\) -norm of T is finite. We prove that \((M^n, g)\) is isometric to a sphere if S is positive and the \(L^{p}\) -norm of T is pinched in [0, C), where C is an explicit positive constant depending only on n, p and S. Finally, we prove an \(L^{p}(p\ge \frac{n}{2})\) -norm of T pinching theorem for complete, simply connected, locally conformally flat Riemannian manifolds with negative constant scalar curvature. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0646-3

Authors:Chang-Yu Guo Abstract: Abstract In this short note, we prove that if F is a weak upper semicontinuous admissible Finsler structure on a domain in \(\mathbb {R}^n, n\ge 2\) , then the intrinsic distance and differential structures coincide. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0634-7

Authors:Mohammad Shahrouzi Abstract: Abstract In this paper we consider a nonlinear higher-order viscoelastic inverse problem with memory in the boundary. Under some suitable conditions on the coefficients, relaxation function and initial data, we proved a blow-up result for the solution with positive initial energy. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0644-5

Authors:M. A. Shan Abstract: Abstract We study a class of quasilinear parabolic equations with model representative $$\begin{aligned} \frac{\partial u}{\partial t}-\sum \limits _{i=1}^{n} \frac{\partial }{\partial x_i}\left( u ^{m_i-1} \frac{\partial u}{\partial x_i}\right) =0, m_i > 1,\, i=1,\ldots ,s,\,\,m_i < 1,\, i=s+1,\ldots ,n. \end{aligned}$$ We establish the pointwise condition for removability of singularity for solutions of such equations. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0647-2

Authors:Yuanze Wu; Tsung-fang Wu; Wenming Zou Abstract: Abstract In this paper, we study the following two-component systems of nonlinear Schrödinger equations $$\begin{aligned} \left\{ \begin{array}{ll} \Delta u-(\lambda a(x)+a_0(x))u+\mu _1u^3+\beta v^2u=0&{}\quad \text {in }\mathbb {R}^3,\\ \Delta v-(\lambda b(x)+b_0(x))v+\mu _2v^3+\beta u^2v=0&{}\quad \text {in }\mathbb {R}^3,\\ u,v\in H^1(\mathbb {R}^3), u,v>0&{}\quad \text {in }\mathbb {R}^3, \end{array}\right. \end{aligned}$$ where \(\lambda ,\mu _1,\mu _2>0\) and \(\beta <0\) are parameters; \(a(x), b(x)\ge 0\) are steep potentials and \(a_0(x),b_0(x)\) are sign-changing weight functions; a(x), b(x), \(a_0(x)\) and \(b_0(x)\) are not necessarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems with \(\lambda \) sufficiently large. The concentration behaviors of solutions as both \(\lambda \rightarrow +\infty \) and \(\beta \rightarrow -\infty \) are also considered. In particular, the phenomenon of phase separations is observed in the whole space \(\mathbb {R}^3\) . In the Hartree–Fock theory, this provides a theoretical enlightenment of phase separation in \(\mathbb {R}^3\) for the 2-mixtures of Bose–Einstein condensates. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0635-6

Authors:Vladimir I. Bogachev; Stanislav V. Shaposhnikov Abstract: Abstract We obtain sharp conditions for higher integrability and continuity of solutions to double divergence form second-order elliptic equations with coefficients of low regularity. In addition, we prove Harnack’s inequality in this case. PubDate: 2017-10-01 DOI: 10.1007/s10231-016-0631-2

Authors:Mauro Nacinovich; Egmont Porten Abstract: We introduce various notions of q-pseudo-concavity for abstract CR manifolds, and we apply these notions to the study of hypoellipticity, maximum modulus principle and Cauchy problems for CR functions. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0638-3

Authors:Marcelo F. Furtado; Ricardo Ruviaro; Edcarlos D. Silva Abstract: Abstract We prove the existence of two solutions for some elliptic equations with combined indefinite nonlinearities on the boundary. The main novelty is to consider variational methods together with a suitable split of the Sobolev space \(W^{1,2}(\Omega )\) . PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0645-4

Authors:David A. C. Mollinedo; Christian Olivera Abstract: Abstract In this article we study the existence and uniqueness of solutions of the stochastic continuity equation with irregular coefficients. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0633-8

Authors:Lucia Alessandrini Abstract: Abstract A product of Kähler manifolds also carries a Kähler metric. In this short note, we would like to study the product of generalized p-Kähler manifolds, compact or not. The results we get extend the known results (balanced, SKT, sG manifolds), and are optimal in the compact case. Hence we can give new non-trivial examples of generalized p-Kähler manifolds. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0640-9

Authors:G. Garrigós; S. Hartzstein; T. Signes; B. Viviani Abstract: Abstract We find optimal decay estimates for the Poisson kernels associated with various Laguerre-type operators L. From these, we solve two problems about the Poisson semigroup \(e^{-t\sqrt{L}}\) . First, we find the largest space of initial data f so that \(e^{-t\sqrt{L}}f(x)\rightarrow f(x)\) at \({\,a.e.\,}x\) . Secondly, we characterize the largest class of weights w which admit 2-weight inequalities of the form \(\Vert \sup _{0<t\le t_0} e^{-t\sqrt{L}}f \,\Vert _{L^p(v)}\lesssim \Vert f\Vert _{L^p(w)}\) , for some other weight v. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0648-1

Authors:P. Cianci; G. R. Cirmi; S. D’Asero; S. Leonardi Abstract: Abstract We consider the following prototype problem: $$\begin{aligned} \left\{ \begin{aligned}&-\Delta u + M \frac{ \nabla u ^2}{u^\theta }=f&\hbox { in}\ \varOmega \\&u=0&\hbox { on}\ \partial \varOmega \end{aligned}\right. \end{aligned}$$ and we study the regularity of the gradient of a solution both in Morrey spaces and in fractional Sobolev spaces in correspondence of the regularity of the right-hand side. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0636-5

Authors:Xavier Ros-Oton; Joaquim Serra Abstract: Abstract We establish sharp boundary regularity estimates in \(C^1\) and \(C^{1,\alpha }\) domains for nonlocal problems of the form \(Lu=f\) in \(\Omega \) , \(u=0\) in \(\Omega ^c\) . Here, L is a nonlocal elliptic operator of order 2s, with \(s\in (0,1)\) . First, in \(C^{1,\alpha }\) domains we show that all solutions u are \(C^s\) up to the boundary and that \(u/d^s\in C^\alpha (\overline{\Omega })\) , where d is the distance to \(\partial \Omega \) . In \(C^1\) domains, solutions are in general not comparable to \(d^s\) , and we prove a boundary Harnack principle in such domains. Namely, we show that if \(u_1\) and \(u_2\) are positive solutions, then \(u_1/u_2\) is bounded and Hölder continuous up to the boundary. Finally, we establish analogous results for nonlocal equations with bounded measurable coefficients in nondivergence form. All these regularity results will be essential tools in a forthcoming work on free boundary problems for nonlocal elliptic operators (Caffarelli et al., in Invent Math, to appear). PubDate: 2017-10-01 DOI: 10.1007/s10231-016-0632-1

Authors:Reiko Aiyama; Kazuo Akutagawa; Satoru Imagawa; Yu Kawakami Abstract: Abstract We perform a systematic study of the image of the Gauss map for complete minimal surfaces in Euclidean four-space. In particular, we give a geometric interpretation of the maximal number of exceptional values of the Gauss map of a complete orientable minimal surface in Euclidean four-space. We also provide optimal results for the maximal number of exceptional values of the Gauss map of a complete minimal Lagrangian surface in the complex two-space and the generalized Gauss map of a complete nonorientable minimal surface in Euclidean four-space. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0643-6

Authors:Claudia Garetto; Michael Ruzhansky Abstract: Abstract In this paper, we study first-order hyperbolic systems of any order with multiple characteristics (weakly hyperbolic) and time-dependent analytic coefficients. The main question is when the Cauchy problem for such systems is well-posed in \(C^{\infty }\) and in \({\mathcal {D}}'\) . We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantees the \(C^\infty \) well-posedness of the corresponding Cauchy problem. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0639-2

Authors:Alicia Dickenstein; Ragni Piene Abstract: Abstract The notion of higher order dual varieties of a projective variety, introduced in Piene [Singularities, part 2, (Arcata, Calif., 1981), Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, 1983], is a natural generalization of the classical notion of projective duality. In this paper, we present geometric and combinatorial characterizations of those equivariant projective toric embeddings that satisfy higher order selfduality. We also give several examples and general constructions. In particular, we highlight the relation with Cayley–Bacharach questions and with Cayley configurations. PubDate: 2017-10-01 DOI: 10.1007/s10231-017-0637-4

Authors:Yu Liu; Jie Xiao Abstract: Abstract Functional capacities on the Grushin space \({\mathbb {G}}^n_\alpha \) are introduced, developed, and subsequently applied to the theory of Sobolev embeddings. PubDate: 2017-09-18 DOI: 10.1007/s10231-017-0699-3

Authors:Francesco Geraci Abstract: Abstract We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the regularity of free-boundary points following the approaches by Caffarelli, Monneau and Weiss. PubDate: 2017-09-14 DOI: 10.1007/s10231-017-0692-x

Authors:Ilaria Del Corso; Roberto Dvornicich Abstract: 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: 2017-09-13 DOI: 10.1007/s10231-017-0697-5