Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let be a bounded open domain of . Let denote the outward unit normal of . We assume that the Steklov problem Δu = 0 in and on has a simple eigenvalue of . Then we consider an annular domain obtained by removing from a small‐cavity size of ε > 0, and we show that under proper assumptions there exists a real valued and real analytic function defined in an open neighborhood of (0,0) in and such that is a simple eigenvalue for the Steklov problem Δu = 0 in and on for all ε > 0 small enough, and such that . Here denotes the outward unit normal of , and δ_2,2 ≡ 1 and δ_2,n ≡ 0 if n ≥ 3. Then related statements have been proved for corresponding eigenfunctions. Copyright © 2013 John Wiley & Sons, Ltd.     

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 < p < ∞ and further assume that with 1 < q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.     

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector‐valued) monogenic, an anti‐monogenic and a ψ‐hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems where , , and . The novelty of this paper is that, relaxing the conditions on the potential function W(t,x), we obtain infinitely many solutions via critical point theory. Our results generalize and improve some existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear statistics of matrix ensembles in classical background

Given a joint probability density function of N real random variables, , obtained from the eigenvector–eigenvalue decomposition of N × N random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely, . For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment‐generating function , where denotes expectation value over the orthogonal (β = 1) and symplectic (β = 4) ensembles, in the form one plus a Schwartz function, none vanishing over for the Gaussian ensembles and for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large N asymptotic of the linear statistics from suitably scaled F(·). Copyright © 2016 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to a class of nonlocal non‐autonomous diffusion equations

In this paper, we consider the nonlocal non‐autonomous evolution problems where Ω is a bounded smooth domain in , N≥1, β is a positive constant, the coefficient a is a continuous bounded function on , and K is an integral operator with symmetric kernel , being J a non‐negative function continuously differentiable on and . We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter β is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.     

Bilinear estimate on tent‐type spaces with application to the well‐posedness of fluid equations

We introduce a class of tent‐type spaces and establish a Poisson extension result of Triebel–Lizorkin spaces . As an application, we get the well‐posedness of Navier–Stokes equations and magnetohydrodynamic equations with initial data in critical Triebel–Lizorkin spaces , . Copyright © 2016 John Wiley & Sons, Ltd.     

On Martin's Axiom and Forms of Choice

Martin's Axiom is the statement that for every well‐ordered cardinal , the statement holds, where is “if is a c.c.c. quasi order and is a family of dense sets in P, then there is a ‐generic filter of P”. In , the fragment is provable, but not in general in . In this paper, we investigate the interrelation between and various choice principles. In the choiceless context, it makes sense to drop the requirement that the cardinal κ be well‐ordered, and we can define for any (not necessarily well‐ordered) cardinal the statement to be “if is a c.c.c. quasi order with , and is a family of dense sets in P, then there is a ‐generic filter of P”. We then define to be the statement that for every (not necessarily well‐ordered) cardinal , we have that holds. We then investigate the set‐theoretic strength of the principle .     

The threshold for combs in random graphs

For let denote the tree consisting of an ‐vertex path with disjoint ‐vertex paths beginning at each of its vertices. An old conjecture says that for any the threshold for the random graph to contain is at . Here we verify this for with any fixed . In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for (with ). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 794–802, 2016     

Clairvoyant embedding in one dimension

Let v, w be infinite 0‐1 sequences, and a positive integer. We say that is ‐embeddable in , if there exists an increasing sequence of integers with , such that , for all . Let and be coin‐tossing sequences. We will show that there is an with the property that is ‐embeddable into with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 520–560, 2015     

On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example

Using the general formalism of 12 , a study of index theory for non‐Fredholm operators was initiated in 9 . Natural examples arise from (1 + 1)‐dimensional differential operators using the model operator in of the type , where , and the family of self‐adjoint operators in studied here is explicitly given by Here has to be integrable on and tends to zero as and to 1 as (both functions are subject to additional hypotheses). In particular, , , has asymptotes (in the norm resolvent sense) as , respectively. The interesting feature is that violates the relative trace class condition introduced in 9 , Hypothesis 2.1 ]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of 9 enabling the following results to be obtained. Introducing , , we recall that the resolvent regularized Witten index of , denoted by , is defined by whenever this limit exists. In the concrete example at hand, we prove Here denotes the spectral shift operator for the pair of self‐adjoint operators , and we employ the normalization, , .     

Circular symmetrization,subordination and arclength problems on convex functions

We study the class of univalent analytic functions f in the unit disk of the form satisfying where Ω will be a proper subdomain of which is starlike with respect to . Let be the unique conformal mapping of onto Ω with and and . Let denote the arclength of the image of the circle , . The first result in this paper is an inequality for , which solves the general extremal problem , and contains many other well‐known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class .     

Semiframes with Block Size Three and Odd Group Size

A ‐semiframe of type is a ‐GDD of type , , in which the collection of blocks can be written as a disjoint union where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . A ‐SF is a ‐semiframe of type in which there are p parallel classes in and d holey parallel classes with respect to . In this paper, we shall show that there exists a (3, 1)‐SF for any if and only if , , , and .     

On the set‐theoretic strength of the existence of disjoint cofinal sets in posets without maximal elements

In set theory without the Axiom of Choice , we study the deductive strength of the statements (“Every partially ordered set without a maximal element has two disjoint cofinal subsets”), (“Every partially ordered set without a maximal element has a countably infinite disjoint family of cofinal subsets”), (“Every linearly ordered set without a maximum element has two disjoint cofinal subsets”), and (“Every linearly ordered set without a maximum element has a countably infinite disjoint family of cofinal subsets”). Among various results, we prove that none of the above statements is provable without using some form of choice, is equivalent to , + (Dependent Choices) implies , does not imply in (Zermelo‐Fraenkel set theory with the Axiom of Extensionality modified in order to allow the existence of atoms), does not imply in (Zermelo‐Fraenkel set theory minus ) and (hence, ) is strictly weaker than in .     

Riemann boundary value problems on half space in Clifford analysis

In this paper, we discuss some properties of the Cauchy type integral operator defined over the half space of . As applications, we study a type of Riemann boundary value problem for solutions to polynomially generalized Cauchy–Riemann equations including with and as special cases over the half space of . Making use of Fischer‐type decomposition and the Clifford calculus for solutions to these equations, we will obtain explicit expressions of solutions to the kind of boundary value problems over the half space of . Copyright © 2012 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

The Kidder Equation:

The Kidder problem is with and where . This looks challenging because of the square root singularity. We prove, however, that for all . Other very simple but very accurate curve fits and bounds are given in the text; . Maple code for a rational Chebyshev pseudospectral method is given as a table. Convergence is geometric until the coefficients are when the coefficients . An initial‐value problem is obtained if is known; the slope Chebyshev series has only a fourth‐order rate of convergence until a simple change‐of‐coordinate restores a geometric rate of convergence, empirically proportional to . Kidder's perturbation theory (in powers of α) is much inferior to a delta‐expansion given here for the first time. A quadratic‐over‐quadratic Padé approximant in the exponentially mapped coordinate predicts the slope at the origin very accurately up to about . Finally, it is shown that the singular case can be expressed in terms of the solution to the Blasius equation.     

Asymptotics for the best Sobolev constants and their extremal functions

Let , where Ω is a bounded domain of , , and . We prove that , where ρ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of converge (as ) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured domain , for some .