The Brezis–Nirenberg problem for fractional elliptic operators

Let be a uniformly elliptic operator in divergence form in a bounded open subset Ω of . We study the effect of the operator on the existence and nonexistence of positive solutions of the nonlocal Brezis–Nirenberg problem where denotes the fractional power of with zero Dirichlet boundary values on , , and λ is a real parameter. By assuming for all and near some point , we prove existence theorems for any , where denotes the first Dirichlet eigenvalue of . Our existence result holds true for and in the interior case () and for and in the boundary case (). Nonexistence for star‐shaped domains is obtained for any .     

Riesz transforms of the Hodge–de Rham Laplacian on Riemannian manifolds

Let M be a complete non‐compact Riemannian manifold satisfying the volume doubling property. Let be the Hodge–de Rham Laplacian acting on 1‐differential forms. According to the Bochner formula, where and are respectively the positive and negative part of the Ricci curvature and ? is the Levi–Civita connection. We study the boundedness of the Riesz transform from to and of the Riesz transform from to . We prove that, if the heat kernel on functions satisfies a Gaussian upper bound and if the negative part of the Ricci curvature is ε‐sub‐critical for some , then is bounded from to and is bounded from to for where depends on ε and on a constant appearing in the volume doubling property. A duality argument gives the boundedness of the Riesz transform from to for where Δ is the non‐negative Laplace–Beltrami operator. We also give a condition on to be ε‐sub‐critical under both analytic and geometric assumptions.     

Existence of solutions to a non‐variational singular elliptic system with unbounded weights

In this paper we prove an existence result for the following singular elliptic system where Ω is a bounded open set in (), is the p‐laplacian operator, and are suitable Lebesgue functions and , , are positive parameters satisfying suitable assumptions.     

Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on , namely We assume that and is a nonnegative function. By definition, a function belongs to the Hardy space if Under certain assumptions on V we characterize the space in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to for with no additional assumptions on the potential V.     

A note on a generalization of the Schwarz theorem about the equality of mixed partial derivatives

We provide a generalization of the classical Schwarz theorem about the equality of mixed partial derivatives. More precisely we extend it to a Riemannian manifold , by proving the following statement: if is a couple of commuting vector fields on M and are such that the set is superdense at a certain point , then .     

A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

Let be a polyharmonic function of order N defined on the strip satisfying the growth condition (0.1) for and any compact subinterval K of , and suppose that vanishes on equidistant hyperplanes of the form for and Then it is shown that is odd at t ₀, i.e. that for . The second main result states that u is identically zero provided that u satisfies 0.1 and vanishes on 2N equidistant hyperplanes with distance c .     

Almost minimizers for semilinear free boundary problems with variable coefficients

We study regularity results for almost minimizers of the functional where is a matrix with Hölder continuous coefficients. In the case we show that an almost minimizer belongs to , where the exponent η is related with the competition between the Hölder continuity of the matrix , the parameter of almost minimization and γ. In some sense, this regularity is optimal. As far as the case is concerned, our results show that an almost minimizer is locally in each phase and , improving in some sense a recent result of David & Toro.     

Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

We prove a sharp Liouville type theorem for stable solutions of the equation on the entire Euclidean space , where and f is a continuous and nonnegative function in such that as , where and . Our theorem holds true for and is sharp in the case .     

Eigenvalues of Robin Laplacians in infinite sectors

For , let denote the infinite planar sector of opening 2α, and be the Laplacian in , , with the Robin boundary condition , where stands for the outer normal derivative and . The essential spectrum of does not depend on the angle α and equals , and the discrete spectrum is non‐empty if and only if . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and the nth eigenvalue of behaves as and admits a full asymptotic expansion in powers of α². The eigenfunctions are exponentially localized near the origin. The results are also applied to δ‐interactions on star graphs.     

Bifurcation properties for a class of fractional Laplacian equations in

This paper concerns with the study of some bifurcation properties for the following class of nonlocal problems (P) where , , , is a positive continuous function, is a bounded continuous function and is the fractional Laplacian. The main tools used are the Leray–Shauder degree theory and the global bifurcation result due to Rabinowitz.     

On Fejér's inequalities for the Legendre polynomials

We present various inequalities for the sum where denotes the Legendre polynomial of degree k . Among others we prove that the inequalities hold for all and . The constant factors 2/5 and are sharp. This refines a classical result of Fejér, who proved in 1908 that is nonnegative for all and .     

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

We study the well‐posedness of the fractional degenerate differential equations with finite delay on Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where A and M are closed linear operators on a Banach space X satisfying , F is a bounded linear operator from (resp. and ) into X, where is given by when and . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of in the above three function spaces.     

Microlocal analysis on wonderful varieties. Regularized traces and global characters

Let be a connected reductive complex algebraic group with split real form . Consider a strict wonderful ‐variety X equipped with its σ‐equivariant real structure, and let X be the corresponding real locus. Further, let E be a real differentiable G‐vector bundle over X . In this paper, we introduce a distribution character for the regular representation of G on the space of smooth sections of E given in terms of the spherical roots of , and show that on a certain open subset of G of transversal elements it is locally integrable and given by a sum over fixed points.     

Concave univalent functions and Dirichlet finite integral

The article deals with the class consisting of non‐vanishing functions f that are analytic and univalent in such that the complement is a convex set, and the angle at ∞ is less than or equal to for some . Related to this class is the class of concave univalent mappings in , but this differs from with the standard normalization A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for settled by Avkhadiev et al. 3 . Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for is also presented.     

Korovkin‐type results for multivariate functions which are periodic with respect to one variable

Let K be a compact metric space and let denote the real Banach space of all continuous functions which are 2π‐periodic with respect to the second variable. We prove the following Korovkin‐type result: Let be a continuous algebraic separating function such that for all , and let be a sequence of positive linear operators. If uniformly with respect to and uniformly on for all , then uniformly on for every . As a corollary we deduce: If , then uniformly on for every if and only if uniformly on for every , where and .     

Boundedness and compactness of Hardy operators on Lorentz‐type spaces

Given non‐negative measurable functions on we study the high dimensional Hardy operator between Orlicz–Lorentz spaces , where f is a measurable function of and is the ball of radius t in . We give sufficient conditions of boundedness of and . We investigate also boundedness and compactness of between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.     

Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1]d

The aim of this paper is to establish the isomorphic classification of Besov spaces over [0, 1]^d . Using the identification of the Besov space with the ‐infinite direct sum of finite‐dimensional spaces (which holds independently of the dimension and of the smoothness degree of the space ) we show that , , is a family of mutually non‐isomorphic spaces. The only exception is the isomorphism between the spaces and , which follows from Pełczyński's isomorphism between and . We also tell apart the isomorphic classes of spaces from the isomorphic classes of Besov spaces over the Euclidean space .     

Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity,and logistic source

In this paper we study the zero‐flux chemotaxis‐system Ω being a convex smooth and bounded domain of , , and where , and . For any the chemotactic sensitivity function is assumed to behave as the prototype , with and . We prove that for nonnegative and sufficiently regular initial data and , the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.     

A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

This paper deals with the Keller–Segel system where Ω is a bounded domain in with smooth boundary , ; χ is a nonnegative function satisfying for some and . In the case that and , Fujie 2 established global existence of bounded solutions under the condition . On the other hand, when , Winkler 14 asserted global existence of bounded solutions for arbitrary . However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary . Moreover, the condition for K when cannot connect with the condition when . The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for χ and to build a mathematical bridge between the cases and .     

Fredholm theory connected with a Douglis–Nirenberg system of differential equations over an exterior region in

We consider a boundary problem over an exterior subregion of for a Douglis–Nirenberg system of differential operators under limited smoothness asumptions and under the assumption of parameter‐ellipticity in a closed sector in the complex plane with vertex at the origin. We pose the problem in an Sobolev–Bessel potential space setting, , and denote by the operator induced in this setting by the boundary problem under null boundary conditions. We then derive results pertaining to the Fredholm theory for for values of the spectral parameter λ lying in as well as results pertaining to the invariance of the Fredholm domain of with p .