Boundedness in chemotaxis systems with rotational flux terms

We consider the chemotaxis system with rotation under no‐flux boundary conditions in the bounded domain , . Here the matrix‐valued function fulfills () for all with some nondecreasing function S₀ and is a nonnegative function with for all . Moreover, f satisfies for all with nondecreasing function f₀. It is shown that for the nonnegative initial data and with , if at least one of the following assumptions holds:

• ,
• , and ,
• ,

then the corresponding initial‐boundary value problem possesses a unique global classical solution that is uniformly bounded.     

Positive solutions of nonlinear Schrödinger equation with peaks on a Clifford torus

We prove the existence of large energy positive solutions for a stationary nonlinear Schrödinger equation with peaks on a Clifford type torus. Here where with for all Each is a function and is defined by the generalized notion of spherical coordinates. The solutions are obtained by a or a process.     

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 .     

Two lower bounds for the Stanley depth of monomial ideals

Let be two monomial ideals of the polynomial ring . In this paper, we provide two lower bounds for the Stanley depth of . On the one hand, we introduce the notion of lcm number of , denoted by , and prove that the inequality holds. On the other hand, we show that , where denotes the order dimension of the lcm lattice of . We show that I and satisfy Stanley's conjecture, if either the lcm number of I or the order dimension of the lcm lattice of I is small enough. Among other results, we also prove that the Stanley–Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.     

On a ‐Kirchhoff equation with critical exponent and an additional nonlocal term via truncation argument

We study existence and multiplicity of solutions of the following nonlocal ‐Kirchhoff equation with critical exponent, via truncation argument on the Sobolev space with variable exponent, where Ω is a bounded smooth domain of , , M, f are continuous functions, , and are real parameter.     

Infinitely many homoclinic solutions for a second‐order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second‐order Hamiltonian system where , and . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, is not necessary to be uniformly positive definite or coercive; through is still assumed to be superquadratic near , it is not assumed to be superquadratic near .     

Sobolev estimates for (pseudo)‐differential operators and applications

In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space E to a complex vector space F, the Sobolev a priori estimate holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.     

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let be the ring of (continuous) semialgebraic functions on a semialgebraic set M and its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps and induced by the inclusion of a semialgebraic subset N of M. The ring can be understood as the localization of at the multiplicative subset of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion that reduces both problems above to an analysis of the fibers of the spectral map . If we denote , it holds that the restriction map is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of at the points of Z. The size of the fibers of prime ideals “close” to the complement provides valuable information concerning how N is immersed inside M. If N is dense in M, the map is surjective and the generic fiber of a prime ideal contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber is a finite set for . If such is the case, our procedure allows us to compute the size s of . If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in .     

Variational and Perron–Wiener solutions in stratified Lie groups

We consider sub‐Laplacians in stratified Lie groups and we compare Perron–Wiener and weak‐variational solutions of the Dirichlet problem , where Ω is a bounded open set in and φ is the restriction to the boundary of a function such that . The result we obtained extends a previous theorem by Arendt and Daners, related to the classial Laplacian in .     

Lifting Galois sections along torsors

The cuspidalization conjecture, which is a consequence of Grothendieck's section conjecture, asserts that for any smooth hyperbolic curve X over a finitely generated field k of characteristic 0 and any non empty Zariski open , every section of lifts to a section of . We consider in this article the problem of lifting Galois sections to the intermediate quotient introduced by Mochizuki 10 . We show that when and is an union of torsion sub‐packets every Galois section actually lifts to . One of the main tools in the proof is the construction of torus torsors and over X and the geometric interpretation .     

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, , .     

A Mehta–Ramanathan theorem for linear systems with basepoints

Let be a normal complex projective polarized variety and an H‐semistable sheaf on X. We prove that the restriction to a sufficiently positive general complete intersection curve passing through a prescribed finite set of points remains semistable, provided that at each , the variety X is smooth and the factors of a Jordan–Hölder filtration of are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.     

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.     

On some structural sets and a quaternionic ‐hyperholomorphic function theory

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy‐Riemann equations to the quaternion skew field . It relies heavily on results on functions defined on domains in or with values in . This theory is centred around the concept of ψ‐hyperholomorphic functions related to a so‐called structural set ψ of or respectively. The main goal of this paper is to develop the nucleus of the ‐hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy‐Riemann operators associated to a pair of structural sets of . Following a matrix approach, a generalized Borel‐Pompeiu formula and the corresponding Plemelj‐Sokhotzki formulae are established.     

Well‐posedness of second order degenerate integro‐differential equations with infinite delay in vector‐valued function spaces

We study the well‐posedness of the second order degenerate differential equations with infinite delay: with periodic boundary conditions , where and M are closed linear operators in a Banach space satisfying , . Using operator‐valued Fourier multiplier techniques, we give necessary and sufficient conditions for the well‐posedness of this problem in Lebesgue‐Bochner spaces , periodic Besov spaces and periodic Triebel‐Lizorkin spaces .     

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 .     

Solvable extensions of negative Ricci curvature of filiform Lie groups

We give necessary and sufficient conditions of the existence of a left‐invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra is a filiform Lie algebra . It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or , and is a one‐dimensional extension of , in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.     

‐bounded ‐calculus for sectorial operators via generalized Gaussian estimates

We show that, for negative generators of analytic semigroups, a bounded ‐calculus self‐improves to an ‐bounded ‐calculus in an appropriate scale of ‐spaces if the semigroup satisfies suitable generalized Gaussian estimates. As application of our result we obtain that large classes of differential operators have an ‐bounded ‐calculus.     

On the geometry of complex ‐spaces

It is known that applying an ‐homothetic deformation to a complex contact manifold whose vertical space is annihilated by the curvature yields a condition which is invariant under ‐homothetic deformations. A complex contact manifold satisfying this condition is said to be a complex ‐space. In this paper, we deal with the questions of Bochner, conformal and conharmonic flatness of complex ‐spaces when , and prove that such kind of spaces cannot be Bochner flat, conformally flat or conharmonically flat.