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Abstract The famous Weierstrass theorem asserts that every continuous function on a compact set in R d can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varjú which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials. [ABSTRACT FROM AUTHOR]

Abstract Minimizers in the least gradient problem with discontinuous boundary data need not be unique. However, all of them have a similar structure of level sets. Here, we give a full characterization of the set of minimizers in terms of any one of them and discuss stability properties of an approximate problem. [ABSTRACT FROM AUTHOR]

Let d ∈ N and let γ i ∈ [ 0 , ∞ ) , x i ∈ ( 0 , 1 ) be such that ∑ i = 1 d + 1 γ i = M ∈ ( 0 , ∞ ) and ∑ i = 1 d + 1 x i = 1 . We prove that a ↦ Γ ( a M + 1 ) ∏ i = 1 d + 1 Γ ( a γ i + 1 ) ∏ i = 1 d + 1 x i a γ i is completely monotonic on ( 0 , ∞ ) . This result generalizes the one found by Alzer [2] for binomial probabilities ( d = 1 ). As a consequence of the log-convexity, we obtain some combinatorial inequalities for multinomial coefficients. We also show how the main result can be used to derive asymptotic formulas for quantities of interest in the context of statistical density estimation based on Bernstein polynomials on the d -dimensional simplex. [ABSTRACT FROM AUTHOR]

Abstract In this paper we provide sufficient conditions for Carathéodory integrands φ (x , p) on which the formula ∫ Ω φ (x , D u) : = sup ϕ ∈ V ⁡ { − ∫ Ω u d i v ϕ + φ ⁎ (x , ϕ (x)) d x } = ∫ Ω φ (x , ∇ u) d x + ∫ Ω ψ (x) | D s u | holds for each u ∈ B V (Ω). Here φ : Ω × R N → R , Ω ⊂ R N open and bounded, φ convex in p for a.e. x , φ satisfies the linear growth assumption φ (x , p) ≤ ψ (x) | p | + c for each | p | ≥ β for a.e. x , V is defined as V = { ϕ ∈ C 0 1 (Ω , R N) : | ϕ (x) | ≤ ψ (x) for all x ∈ Ω } , and φ ⁎ is the conjugate function of φ. Lower semicontinuity of ∫ Ω φ (x , D u) in L 1 (Ω) then easily follows, whereas in contrast to earlier work by other authors we do not assume continuity in the x variable for φ. [ABSTRACT FROM AUTHOR]

In this paper we give a generalization of the problem that was posed by I. Raşa in [10] and was proved based on a probabilistic approach by Mrowiec, Rajba and Wąsowicz in [8] . We end this paper by proposing two open problems related to Raşa's proposed inequality. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a notion of ( ω , t ) -convexity as a natural generalization of the notion of usual t -convexity, t -strongly convexity, approximate t -convexity, delta t -convexity and many other. The main result of this paper establishes the necessary and sufficient conditions under which an ( ω , t ) -convex map can be supported at a given point by an ( ω , t ) -affine support function. Several applications of this support theorem are presented. For instance, new characterizations of inner product spaces among normed spaces involving the notion of ( ω , t ) -convexity are given. [ABSTRACT FROM AUTHOR]

In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic nonlinear operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the backward–forward algorithm as a splitting method to solve structured monotone inclusions, and convex minimization problems in Hilbert spaces. It has a natural link with the forward–backward algorithm and has the same computational complexity, since it involves the same basic blocks, but organized differently. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators. First, we show that these two methods enjoy remarkable involutive relations, which go far beyond the evident inversion of the order in which the forward and backward steps are applied. Next, we establish several convergence properties for both methods, some of which were unknown even for the forward–backward algorithm. This brings further insight into this well-known scheme. Finally, we specialize our results to structured convex minimization problems, the gradient-projection algorithms, and give a numerical illustration of theoretical interest. [ABSTRACT FROM AUTHOR]

In a Hilbert space setting H , we study the convergence properties as t → + ∞ of the trajectories of the second-order differential equation (AVD) α , ϵ x ¨ ( t ) + α t x ˙ ( t ) + ∇ Φ ( x ( t ) ) + ϵ ( t ) x ( t ) = 0 , where ∇Φ is the gradient of a convex continuously differentiable function Φ : H → R , α is a positive parameter, and ϵ ( t ) x ( t ) is a Tikhonov regularization term, with ϵ ( t ) positive, and lim t → ∞ ⁡ ϵ ( t ) = 0 . In this damped inertial system, the damping coefficient α t vanishes asymptotically, but not too quickly, a key property to obtain rapid convergence of the values. In the case ϵ ( ⋅ ) ≡ 0 , this dynamic has been highlighted recently by Su, Boyd, and Candès as a continuous version of the Nesterov accelerated gradient method. Depending on the speed of convergence of ϵ ( t ) to zero, we analyze the convergence properties of the trajectories of (AVD) α , ϵ . We obtain results ranging from the rapid convergence of Φ ( x ( t ) ) to min ⁡ Φ when ϵ ( t ) decreases rapidly to zero, up to the strong convergence of the trajectories to the element of minimum norm of the set of minimizers of Φ, when ϵ ( t ) tends slowly to zero. When ϵ ( t ) = 1 t r , the critical value of r separating the two above cases is r = 2 . [ABSTRACT FROM AUTHOR]

The objective of this work is to study a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that this system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions, by addressing a more general problem. In addition, we conclude, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-component test case. [ABSTRACT FROM AUTHOR]

In this paper we consider a nonlinear viscoelastic equation with minimal conditions on the L 1 ( 0 , ∞ ) relaxation function g namely g ′ ( t ) ≤ − ξ ( t ) H ( g ( t ) ) , where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of g at infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H ( s ) = s p and p covers the full admissible range [ 1 , 2 ) . We get the best decay rates expected under this level of generality and our new results substantially improve several earlier related results in the literature. [ABSTRACT FROM AUTHOR]

For automorphism groups, we prove that the orbit accumulation points are either hyperbolic or parabolic. This answers the question pertaining to the non-compactness of the inverses of a non-compact sequence of automorphisms. We also partially extend a previous theorem of the author on global pseudoconvexity to higher dimensions, provided it admits a non-compact automorphism group. In general, a full extension does not exist. We provide an example to elaborate on the nonexistence. This example is carefully constructed and an exact sequence of automorphisms is explicitly found. [ABSTRACT FROM AUTHOR]

The aim of this paper is to state a sharp version of the König supremum theorem, an equivalent reformulation of the Hahn–Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush–Kuhn–Tucker and Fritz John types, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results. [ABSTRACT FROM AUTHOR]

New upper bounds on the pointwise behavior of Christoffel function on convex domains in R d are obtained. These estimates are established by explicitly constructing the corresponding “needle”-like algebraic polynomials having small integral norm on the domain, and are stated in terms of few easy-to-measure geometric characteristics of the location of the point of interest in the domain. Sharpness of the results is shown and examples of applications are given. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study two scaling methods developed respectively by Pinchuk and Frankel. We introduce first a continuously-varying global coordinate system, and give an alternative proof to the convergence of Pinchuk's scaling sequence (and of our modification) on bounded domains with finite type boundaries in C 2 . Using this, we discuss the modification of the Frankel scaling sequence on nonconvex domains. We also observe that two modified scalings are equivalent. [ABSTRACT FROM AUTHOR]

Takeuchi's generalized complete elliptic integrals related to generalized trigonometric functions are of importance in the computation of the generalized pi and in the elementary proof of Ramanujan's cubic transformation. In this paper we generalize some well-known results of the classical complete elliptic integrals to the case of Takeuchi's generalized complete elliptic integrals. We obtain sharp monotonicity and convexity results for combinations of these functions, as well as functional inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we determine the radius of uniform convexity for three kinds of normalized Bessel functions of the first kind. In the cases considered the normalized Bessel functions are uniformly convex on the determined disks. Moreover, necessary and sufficient conditions are given for the parameters of the three normalized functions such that they to be uniformly convex in the open unit disk. The basic tool of this study is Bessel functions in series. [ABSTRACT FROM AUTHOR]

We establish a local Lipschitz regularity near the boundary of weak solutions to a general class of homogeneous quasilinear elliptic equations with Neumann boundary condition in bounded convex domains. [ABSTRACT FROM AUTHOR]

We adapt modulus of continuity estimates to the study of spectra of combinatorial graph Laplacians, as well as the Dirichlet spectra of certain weighted Laplacians. The latter case is equivalent to stoquastic Hamiltonians and is of current interest in both condensed matter physics and quantum computing. In particular, we introduce a new technique which bounds the spectral gap of such Laplacians (Hamiltonians) by studying the limiting behavior of the oscillations of their solutions when introduced into the heat equation. Our approach is based on recent advances in the PDE literature, which include a proof of the fundamental gap theorem by Andrews and Clutterbuck. [ABSTRACT FROM AUTHOR]

Locally L 0 -convex modules were introduced in Filipovic et al. (2009) [10] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in Drapeau et al. (2016) [8] . In this paper we study locally L 0 -convex modules, and find exactly which subclass of locally L 0 -convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we establish a version of the so-called Jouini–Schachermayer–Touzi theorem for robust representation of conditional convex risk measures defined on a L ∞ -type module. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the X θ ( ⋅ ) -valued function spaces. Based on the notions of bounded topological lattice, Banach space net, continuous modular net and the dual space net, we divide X θ ( ⋅ ) -valued function spaces into two classes: norm-modular spaces and modular-modular spaces. For the first class, we study the separability of L + p ( ⋅ ) ( I , X θ ( ⋅ ) ) , give a representation of the dual space L + p ( ⋅ ) ( I , X θ ( ⋅ ) ) ⁎ , find sufficient conditions of the equality L p ( ⋅ ) ( I , X θ ( ⋅ ) ) = L + p ( ⋅ ) ( I , X θ ( ⋅ ) ) , and prove the reflexivity of L p ( ⋅ ) ( I , X θ ( ⋅ ) ) under some reasonable conditions. And for the second class, we prove the completeness and uniform convexity of L ρ θ ( ⋅ ) ( I , X θ ( ⋅ ) ) in suitable situations. To show the naturality and rationality of these results, some concrete function spaces such as L p t ± ( I , L p ( x , t ) ( Ω ) ) , L p t ( I , W 1 , p ( x , t ) ( Ω ) ) and L p t ( I , W 0 1 , p ( x , t ) ( Ω ) ) together with L P p ( t ) ( I , L p ( x , t ) ( Ω ) ) are taken into account. [ABSTRACT FROM AUTHOR]

We explore the small-time behavior of solutions to the Yang–Mills heat equation with rough initial data. We consider solutions A ( t ) with initial value A 0 ∈ H 1 / 2 ( M ) , where M is a bounded convex region in R 3 or all of R 3 . The behavior, as t ↓ 0 , of the L p ( M ) norms of the time derivatives of A ( t ) and its curvature B ( t ) will be determined for p = 2 and 6, along with the H 1 ( M ) norm of these derivatives. [ABSTRACT FROM AUTHOR]

A sandwich theorem for m -convex functions with m ∈ [ 0 , 1 ] is proved. The counterexamples to a sandwich type result for m -convex functions with 0 < m < 1 , presented in [2] , are given. [ABSTRACT FROM AUTHOR]

Suppose that K and S are locally compact Hausdorff spaces, X is a Banach space and λ is a real number satisfying 2 λ ( 1 − δ X ⁎ ( 2 λ ) ) < r , where δ X ⁎ denotes the modulus of convexity of the dual space X ⁎ and r is the unique real root of the polynomial p ( x ) = 2 x 3 − x 2 + 8 x + 20 . We prove that if T is an isomorphism from C 0 ( K , X ) onto C 0 ( S , X ) with ‖ T ‖ ‖ T − 1 ‖ ≤ λ , then K is homeomorphic to S . This improves an earlier generalization of the Banach–Stone theorem via geometric properties of X ⁎ due to Jarosz, the case where K and S are compact and r is exchanged by the smaller number 4/3. In the process of the proof we establish a connection between the moduli of convexity of X and X ⁎ which has independent interest. Namely, put C = r / 4 . Then C is approximately 0.41883 and for all Banach space X and ε ∈ ( 0 , 2 ] , we have 1 − C ε < δ X ( ε ) ⟹ 1 − 1 2 ε < δ X ⁎ ( ε ) . [ABSTRACT FROM AUTHOR]

The subject of the paper concerns the existence of different inclusions between families S t , S c of C n – biholomorphic starlike or convex mappings and a family S ( k ) , k ≥ 2 , of C n – locally biholomorphic mappings. In the definition of S ( k ) we used a combination of a Kikuchi–Matsuno–Suffridge's starlikeness condition and a property of ( j , k ) -symmetrical mappings in C n . This definition generalizes a notion of planar Sakaguchi's functions onto n -dimensional case. A motivation for the family S ( k ) comes from the paper [11] of the second author and the paper [6] of Hamada and Kohr. The family S ( k ) is a superset of a family which is connected with a problem posed in the first paper and solved in the second one. [ABSTRACT FROM AUTHOR]

For a certain class of elementary functions, which comprise a generalization of the class of min-type functions, we apply techniques from abstract convex analysis to obtain various characterizations of the maximal elements of the support sets of sub-topical functions. As an application, using this class of elementary functions, we characterize the global maximizers of the difference of two strictly sub-topical functions. These results have many applications in various areas of applied mathematics, mathematical economics, and game theory. [ABSTRACT FROM AUTHOR]

We provide a-priori L ∞ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions v p and u q for any ( p , q ) lying on the critical Sobolev hyperbola. Our proof combines moving planes method and Rellich–Pohozaev type identities for systems. Our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded. Using these a priori bounds, and local and global bifurcation techniques, we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system. [ABSTRACT FROM AUTHOR]

In this paper, we study two properties ( P 1 ) and ( P 2 ) introduced in the literature for studying the existence and stability of relative Chebyshev centers. We reformulate these properties in a way that leads naturally to our results. We mainly relate these properties with some geometric properties of Banach spaces. In particular, reflexive spaces having the Kadec–Klee property are characterized in terms of property ( P 1 ) and uniform convexity is characterized in terms of property ( P 2 ) . Some continuity results for the center map are also presented. [ABSTRACT FROM AUTHOR]

Let K be a compact subset of a totally-real manifold M , where M is either a C 2 -smooth graph in C 2 n over C n , or M = u − 1 { 0 } for a C 2 -smooth submersion u from C n to R 2 n − k , k ≤ n . In this case we show that K is polynomially convex if and only if for a fixed neighbourhood U , defined in terms of the defining functions of M , there exists a plurisubharmonic function Ψ on C n such that K ⊂ { Ψ < 0 } ⊂ U . [ABSTRACT FROM AUTHOR]

This paper studies the attraction–repulsion chemotaxis system with logistic source u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) + f ( u ) , v t = Δ v − α 1 v + β 1 u , w t = Δ w − α 2 w + β 2 u in a smooth bounded convex domain Ω ⊂ R 3 , subject to nonnegative initial data and homogeneous Neumann boundary conditions, where χ , ξ , α i and β i ( i = 1 , 2 ) are positive parameters and the logistic source function f fulfills f ( s ) = s − μ s γ + 1 , s ≥ 0 , μ > 0 and γ ≥ 1 . It is shown that this system possesses a unique global bounded classical solution under the conditions α i ≥ 1 2 and μ ≥ max ⁡ { ( 41 2 χ β 1 + 9 ξ β 2 ) γ , ( 9 χ β 1 + 41 2 ξ β 2 ) γ } . Furthermore, whenever u 0 ≢ 0 and for any γ ∈ N , the solution of the system approaches to the steady state ( ( 1 μ ) 1 γ , ( 1 μ ) 1 γ β 1 α 1 , ( 1 μ ) 1 γ β 2 α 2 ) in the norm of L ∞ ( Ω ) as t → ∞ . [ABSTRACT FROM AUTHOR]

The Hölder convexity of the composition of inverse hyperbolic tangent function and Jacobian sine function is investigated in this paper. [ABSTRACT FROM AUTHOR]

In this paper we obtain the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for fractional integrals which generalize the two familiar fractional integrals namely, the Riemann–Liouville and the Hadamard fractional integrals into a single form. We prove that, in most cases, we obtain the Riemann–Liouville and the Hadamard equivalence just by taking limits when a parameter ρ → 1 and ρ → 0 + , respectively. [ABSTRACT FROM AUTHOR]

Let Ω be a smooth domain in R 2 , we prove that if g : [ 0 , + ∞ ) → [ 0 , + ∞ ] is convex with g ( 0 ) < g ( t ) whenever t > 0 then there exists an unique minimizer u ∈ C 0 , 1 ( Ω ) of the functional u ↦ ∫ Ω g ( | ∇ u | ) d x d y among all Lipschitz-continuous functions that assume the same value of u on ∂Ω. [ABSTRACT FROM AUTHOR]

Property UC, introduced in 2009, plays a crucial role in several existence and convergence results for best proximity points and fixed points. In this paper, we present two approximation theoretic characterizations of uniform convexity and as consequences of these results, we characterize the uniform convexity in terms of property UC. Characterizations of strong convexity in terms of property UC are also presented. [ABSTRACT FROM AUTHOR]

We estimate the Jacobian of harmonic mapping of the unit disk onto a smooth and convex Jordan domain via the boundary function and the boundary curvature of the image domain. By using this result we make some asymptotically sharp estimates of constant of quasiconformality for harmonic diffeomorphisms between the unit disk and the convex domains via their boundary mappings. [ABSTRACT FROM AUTHOR]

In this study, we first provide a necessary and sufficient condition for a strongly convex weakly Kähler-Finsler metric to be of constant flag curvature, and we then prove that: (i) a strongly convex weakly Kähler-Finsler metric of constant flag curvature is necessarily of constant holomorphic curvature; and (ii) a strongly convex Kähler-Berwald metric on a complex manifold of complex dimension n≥; 2 has constant flag curvature if and only if it comes from a strongly convex locally complex Minkowski metric. We also give two examples of nontrivial strongly convex Kähler-Finsler metrics. [ABSTRACT FROM AUTHOR]

We study the conormal derivative problem for an elliptic equation of p-Laplacian type with discontinuous coefficients in a non-smooth domain in order to look for the minimal assumptions necessary to have the nonlinear Calderón-Zygmund theory for such problem. Under the assumptions that the nonlinear operator is sufficiently close to the p-Laplacian operator in BMO semi-norm and the boundary of the domain can be locally approximated by the convex boundary, we prove that both the gradient and the associated nonhomogeneous term belong to the same Lq space for every q⊂ [p, ∞). As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction. [ABSTRACT FROM AUTHOR]

For test functions supported in a domain of the Euclidean space we consider the Hardy–Rellich inequality: ∫ | Δ f | 2 d x ≥ C 2 ∫ | f | 2 δ − 4 ( x ) d x , where C 2 = const ≥ 0 and δ ( x ) is the distance from x to the boundary of the domain. M.P. Owen proved that this inequality is valid in any convex domain with C 2 = 9 / 16 (M.P. Owen (1999) [21] ). We examine the inequality in non-convex domains. It is proved that a positive constant C 2 for a plane domain exists if and only if its boundary is a uniformly perfect set. For a domain of dimension d ≥ 2 we prove that the inequality holds with the sharp constant C 2 = 9 / 16 , if the domain satisfies an exterior sphere condition with certain restriction on the radius of the sphere. In addition, we obtain similar results for the inequality ∫ δ 2 ( x ) | Δ f | 2 d x ≥ C 2 ⁎ ∫ | f | 2 δ − 2 ( x ) d x . [ABSTRACT FROM AUTHOR]

Let G be a compact group acting on a Polish group X by means of automorphisms. It is proved that the orbit space X / G is an ℓ 2 -manifold (resp., homeomorphic to ℓ 2 ) provided X is a G -ANR (resp., G -AR) and the fixed point set X G is not locally compact. It is also proved that if a compact group G acts affinely on a separable closed convex subset K of a Fréchet space with a non-locally compact fixed point set K G , then the orbit space K / G is homeomorphic to ℓ 2 . In particular, (1) if C ( Y , X ) denotes the space of all maps from a compact metric G -space Y to a non-locally compact Polish ANR (resp., AR) group X , endowed with the compact-open topology and the induced action of G , then the orbit space C ( Y , X ) / G is an ℓ 2 -manifold (resp., homeomorphic to ℓ 2 ), and (2) if X is an infinite-dimensional separable Fréchet G -space and c c ( X ) denotes the hyperspace of all non-empty compact convex subsets of X , endowed with the Hausdorff metric topology and the induced action of G , then the orbit space c c ( X ) / G is homeomorphic to ℓ 2 , whenever the fixed point set c c ( X ) G is not locally compact. [ABSTRACT FROM AUTHOR]

It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent points { p 0 , … , p d } in R d , then the Euclidean distances { | x − p j | } j = 0 d determine the point x in R d uniquely. In this paper we investigate a similar problem in general normed spaces which is motivated by this known fact. Namely, we characterize those, at least d -dimensional, real normed spaces ( X , ‖ ⋅ ‖ ) for which every set of d + 1 affine independent points { p 0 , … , p d } ⊂ X , the distances { ‖ x − p j ‖ } j = 0 d determine the point x lying in the simplex Conv ( { p 0 , … , p d } ) uniquely. If d = 2 , then this condition is equivalent to strict convexity, but if d > 2 , then surprisingly this holds only in inner product spaces. The core of our proof is some previously known geometric properties of bisectors. The most important of these ( Theorem 1 ) is re-proven using the fundamental theorem of projective geometry. [ABSTRACT FROM AUTHOR]

We prove that, for a Tychonoff space X , the space C p ( X ) is barrelled if and only if it is a Mackey group. [ABSTRACT FROM AUTHOR]

In this paper we obtain the following global a priori Calderón–Zygmund estimates in a convex domain Ω | f | p 1 + | f | p 2 ∈ L γ ( Ω ) ⇒ | ∇ u | p 1 + | ∇ u | p 2 ∈ L γ ( Ω ) for any γ ≥ 1 of weak solutions for a class of non-uniformly nonlinear elliptic equations with vanishing Dirichlet data ∑ i = 1 2 div ( ( A i ∇ u ⋅ ∇ u ) p i − 2 2 A i ∇ u ) = ∑ i = 1 2 div ( | f | p i − 2 f ) , where 1 < p 1 < p 2 < ∞ . [ABSTRACT FROM AUTHOR]

In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space X is said to have the weak Lebesgue property if every Riemann integrable function from [ 0 , 1 ] into X is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under ℓ 1 -sums and obtain new examples of Banach spaces with and without this property. Furthermore, we characterize Dunford–Pettis operators in terms of Riemann integrability and provide a quantitative result about the size of the set of τ -continuous nonRiemann integrable functions, with τ a locally convex topology weaker than the norm topology. [ABSTRACT FROM AUTHOR]

In this note, we study a semilinear system involving the curl operator in a bounded and convex domain in R 3 , which is a limiting form of the Ginzburg–Landau model for superconductors in three dimensions for a large value of the Ginzburg–Landau parameter. We show the existence and establish the L ∞ estimate associated with the boundary data for the weak solution of this system. As an application, our result gives a lower bound of the strength of the applied field such that the superconducting state is locally stable. [ABSTRACT FROM AUTHOR]

We say that a symmetric matrix A is copositive if v T A v ≥ 0 for all nonnegative vectors v . The main result of this paper is a characterization of the cone of feasible directions at a copositive matrix A , i.e., the convex cone of symmetric matrices B such that there exists δ > 0 satisfying A + δ B being copositive. This cone is described by a set of linear inequalities on the elements of B constructed from the so called set of (minimal) zeros of A . This characterization is used to furnish descriptions of the minimal (exposed) face of the copositive cone containing A in a similar manner. In particular, we can check whether A lies on an extreme ray of the copositive cone by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition for the irreducibility of A with respect to a copositive matrix C . [ABSTRACT FROM AUTHOR]

We introduce a new isometric geometric property, namely asymptotic midpoint uniform convexity and investigate its connection to similar asymptotic properties. [ABSTRACT FROM AUTHOR]

Geometric properties of the Jackson and Hahn–Exton q -Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q -generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre–Pólya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn–Exton q -Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q -Bessel function and its derivatives. Some open problems are proposed at the end of the paper. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is twofold. First, the three-quarter theorem is established for a large class of polyharmonic mappings. Second, a class of polyharmonic mappings with positive Jacobian and being univalent in the unit disk D are constructed. [ABSTRACT FROM AUTHOR]

Starting from a 3 D non-convex and nonlocal micromagnetic energy for ferromagnetic materials, we determine, via an asymptotic analysis, the free energy of two joined ferromagnetic thin films. We distinguish different regimes depending on the limit of the ratio between the small thickness of the two films. [ABSTRACT FROM AUTHOR]