On semi‐isogenous mixed surfaces

Let C be a smooth projective curve and G be a finite subgroup of whose action is mixed, i.e. there are elements in G exchanging the two isotrivial fibrations of . Let be the index two subgroup . If G⁰ acts freely, then is smooth and we call it semi‐isogenous mixed surface. In this paper we give an algorithm to determine semi‐isogenous mixed surfaces with given geometric genus, irregularity and self‐intersection of the canonical class. As an application we classify irregular semi‐isogenous mixed surfaces with and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with . We provide an example of a minimal surface of general type with and .     

Necessary and sufficient conditions for existence of blow‐up solutions for elliptic problems in Orlicz–Sobolev spaces

This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem where either with is a smooth bounded domain or . The function ϕ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. The proofs are based on comparison principle, variational methods and topological arguments on the Orlicz–Sobolev spaces.     

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.     

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.     

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 .     

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.     

Some characterizations of BLO space

For , a real‐valued function belongs to space if In this paper, we establish a version of John–Nirenberg inequality suitable for the space with . As a corollary, it is proved that spaces are independent of the scale in sense of norm. Also, we characterize the space through weighted Lebesgue spaces and variable Lebesgue spaces, respectively.     

The automorphism group of a rigid affine variety

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group of a rigid affine variety contains a unique maximal torus . If the grading on the algebra of regular functions defined by the action of  is pointed, the group  is a finite extension of . As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.     

Gonality of complete graphs with a small number of omitted edges

Let be the complete metric graph on d vertices. We compute the gonality of graphs obtained from by omitting edges forming a , or general configurations of at most edges. We also investigate if these graphs can be lifted to curves with the same gonality. We lift the former graphs and the ones obtained by removing up to edges not forming a K ₃ using models of plane curves with certain singularities. We also study the gonality when removing edges not forming a K ₃. We use harmonic morphism to lift these graphs to curves with the same gonality because in this case plane singular models can no longer be used due to a result of Coppens and Kato.     

Ground states for a nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents

This paper studies the following nonhomogeneous elliptic system involving Hardy–Sobolev critical exponents where , Ω is a C ¹ open bounded domain in containing the origin, and . The existence result of positive ground state solution is established.     

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds:

• 1. h is bounded in .
• 2. h satisfies the condition in with .
• 3. both h and g are bounded in .
• 4. h is bounded and .

We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in is strictly less than 1. In addition, we determine the Bohr radius for the space of analytic Bloch functions and the space of harmonic Bloch functions. The paper concludes with two conjectures.     

On the topological type of a set of plane valuations with symmetries

Let be a set of irreducible plane curve singularities. For an action of a finite group G , let be the Alexander polynomial in variables of the algebraic link and let with identical variables in each group. (If , is the monodromy zeta function of the function germ , where is an equation defining the curve C ₁.) We prove that determines the topological type of the link L . We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincaré series of a set of valuations.     

On the extremal Betti numbers of the binomial edge ideal of closed graphs

We study the equality of the extremal Betti numbers of the binomial edge ideal and those of its initial ideal for a closed graph G. We prove that in some cases there is a unique extremal Betti number for and as a consequence there is a unique extremal Betti number for and these extremal Betti numbers are equal.     

Characterisation of zero trace functions in variable exponent Sobolev spaces

It is well known that if u belongs to the Sobolev space , where Ω is an open subset of and , then if belongs to weak , where dist . Results of this type are given here for Sobolev spaces with a variable exponent p , under the conditions that Ω is bounded and satisfies a mild regularity condition, and p is a bounded, log‐Hölder continuous function that is bounded away from 1. The outcome includes theorems that are new even when p is constant. In particular it is shown that if and only if and .     

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 .     

Stability of integral operators on a space of homogeneous type

In this paper, we consider integral operators T on compact spaces of homogeneous type with finite diameter, whose kernels have certain Hölder regularity and mild singularity near the diagonal. We show that given any , the ‐stability of the operator is equivalent for different , where I stands for the identity operator.     

Properties of analytic Morrey spaces and applications

In this note, we aim to study analytic Morrey spaces . We first give the canonical factorization for . Then by applying p‐Carleson measure, we prove an atomic decomposition theorem of . As an application of the decomposition theorem, the interpolation problem of is solved. Finally, we show the boundedness and compactness of Toeplitz operators on .     

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 .     

Multiplicity results for a class of quasilinear equations with exponential critical growth

In this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations: where is the N‐Laplacian operator, , f is a function with exponential critical growth, μ and ε are positive parameters and A is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments and Lusternik–Schnirelman category theory.