The prime purpose of this article aims to establish connections between various sub-classes of harmonic univalent mappings by applying a convolution operator associated with the well-known Wright function. In particular, the inclusion relations among Goodman–Rønning-type harmonic univalent mappings, harmonic k-uniformly convex mappings and harmonic k-uniformly starlike mappings in the plane are established

The goal of this work is to extend the concepts of generalized Lelong number of positive currents investigated by Skoda, Demailly and Ghiloufi in complex analysis, to weakly positive supercurrents on the real superspaces. We generalize then a result of Lagerberg when the supercurrent is closed as well as a very recent result of Berndtsson for minimal supercurrents associated to submanifolds of R n

The paper is devoted to the proof of new estimations of harmonic measure of a boundary arc of a Jordan domain D. The main result allows to estimate a distortion of harmonic measure under locally quasiconformal or quasiconformal mappings of the unit disk of complex plane onto domain D. As an application of the main results, some estimations for quasiconformal mappings of the disk are obtained.

In this paper we show that continuous endomorphisms on the space of entire functions with a given order can be expressed as differential operators of infinite order satisfying suitable growth conditions.

For any Lipschitz domain we construct an arbitrarily small, localized perturbation which splits the spectrum of the Dirichlet, Neumann or Robin Laplacian into simple eigenvalues. We showcase two different approaches. The first one consists in the excision of a hole inside the domain and the perturbation of its boundary, and is based on a Hadamard's formula and sharp spectral stability estimates. The

This paper concerns univalent analytic functions f for which log⁡f′ belong to Dirichlet-Morrey spaces. We establish some characterizations of Dirichlet-Morrey domains. Our results involve Schwarzian derivatives, quasiconformal extensions and Grunsky kernels of univalent functions.

The n-dimensional generalized Hartogs triangles are domains defined by Hpn:={(z1,…,zn)∈Cn:|z1|p1<⋯<|zn|pn<1} with p:=(p1,…,pn)∈(Z+)n and n≥2. In this paper, we first obtain an estimate for the Bergman kernel of Hpn and then use it to establish the Lp boundedness of the associated Bergman projections. Our result generalizes the Lp boundedness result for two-dimensional generalized Hartogs triangles

In this paper, we consider a type of Kirchhoff problem as follows −(a+b∫RN|∇u|2)Δu=(1+εK(x))u2∗−1,u>0, in RN, where a, b>0 are given constants, ε>0 is a small parameter and 2∗=2N/(N−2),(N≥3). We show that if K(x) has k critical points near which K(x) satisfies some expansion assumption, then by Lyapunov–Schmidt reduction method, we construct multi-peak solutions for ε>0 small, which concentrate at

We completely solve several important problems, such as boundedness and compactness, topological structure, ergodic and dynamical properties, for composition operators on weighted Banach spaces of entire functions with sup-norm.

In this paper, we consider the Choquard equations with a local perturbation −Δu+u=∫RN|u(y)|2|x−y|N−αdyu+|u|q−2uin RN, where N≥3, α∈((N−4)+,N), q∈(2,2N/(N−2)). By using variational method and approximating approach, we prove that for any given positive integer k, the above equation has a least energy radial solution changing sign exactly k times. This solution is constructed as the limit of such solutions

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball Bm(R0) in Cm ( 02n+1+α+ρK) hyperplanes in general position regardless of multiplicity with certain positive constants K and α<1 (explicitly estimated), then f1=f2

In this paper, we study the existence of positive solutions to a class of singular quasilinear elliptic problems −div(|x|−ap|∇u|p−2∇u)+V(x)|u|p−2u=λh(x)|u|q−2u+H(x)|u|m−2u,x∈RNu(x)>0,x∈RN,u∈X≡Da1,p(RN,|x|−ap)∩Lp(RN,V), where 10,1

In this paper, after introducing the concepts of quaternionic dual group and the quaternionic valued character on locally compact abelian group G2, the inverse of the quaternionic Fourier transform (QFT) on locally compact abelian groups is investigated. Due to the non-commutativity of multiplication of quaternions, there are different types of QFTs right, left and two-sided quaternionic Fourier transform

We study polynomial interpolation of Hermite type of holomorphic functions based on Radon projections. We give two kinds of interpolation schemes and show that the interpolation polynomials are continuous with respect to the angles and the distances. When the chords are suitably distributed, we prove that the interpolation polynomials converge geometrically on the closed unit disk to the functions

In this paper, we characterize the kernels and finite rank properties of the Hankel operators and small Hankel operators with pluriharmonic symbols on Hardy-Sobolev spaces Hβ2 and pluriharmonic Hardy-Sobolev spaces Hβ,ph2. For β=12, the result was given by Y. J. Lee.

(2020). Professor Jinyuan Du: Citation for his 70th birthday. Complex Variables and Elliptic Equations. Ahead of Print.

In this paper, the authors study the boundedness of the Toeplitz type operator Tϕ and the Hankel type operator Vϕ on Hp,q,s(B). Further, the authors discuss the solvability of the Gleason's problem on Hp,q,s(B). Finally, the inclusion relations of the space Hp,q,s(B) and some typical function spaces are also considered.

We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with polyharmonic functions, the elements of the kernel of an iterated Laplacian. Here, we consider polynomials of the form P(x)=∑k=0dxkak, with Clifford coefficients ak∈Rn, and get

In this paper, we consider a class of fractional Laplacian problems of the form: (−Δ)p1(x,.)su+(−Δ)p2(x,.)su+|u|q(x)−2u=g(x)u−γ(x)+λf(x,u)in Ω,u=0,on ∂Ω, where Ω⊂RN, (N≥2), is a bounded domain and (−Δ)pi(x,.)s is the fractional pi(x,.)-Laplacian. We assume that λ is a nonnegative parameter and γ:Ω¯→(0,1) is a continuous function. By using variational methods and monotonicity arguments combined with

In this article, we study the existence/multiplicity results for the variable order nonlocal Choquard problem with variable exponents (−Δ)p(⋅)s(⋅)u(x)=λ|u(x)|α(x)−2u(x)+∫ΩF(y,u(y))|x−y|μ(x,y)dyf(x,u(x)),x∈Ω,u(x)=0,x∈Ωc:=RN∖Ω, where Ω⊂RN is a smooth and bounded domain, N≥2, p,s,μ and α are continuous functions on RN×RN and f(x,t) is a Carathéodory function with F(x,t):=∫0tf(x,s)ds. Under suitable assumption

The boundedness, invertibility and smoothness properties of multidimensional singular integral operators and solvability of the corresponding singular integral equations are investigated in Besov spaces.

A model of resonators for which boundary is composed of the Helmholtz resonators is constructed. It is based on the theory of self-adjoint extensions of symmetric operators. We consider a limiting procedure when the number of resonators tends to infinity and look after the spectrum of the Neumann Laplacian. It is shown that there is a sequence of the model systems and, correspondingly, a sequence of

Answering a question about triangle inequality suggested by R. Li, Barrlund [The p-relative distance is a metric. SIAM J Matrix Anal Appl. 1999;21:699.702] introduced a distance function which is a metric on a subdomain of Rn. We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under

We solve the Riemann boundary value problem for bi-analytic functions on a contour consisting of non-rectifiable closed Jordan curves. In the classical results on this problem, curves are piecewise-smooth. But the Riemann boundary value problem has a lot of applications in the theory of solid media and other fields, and some of these applications allow fractal (and, consequently, non-rectifiable) contours

The aim of the D-bar method is to solve inverse conductivity problems. For complex conductivity, Hamilton et al. presented a reconstruction method which consists of six steps. In this article, we present a new regularization denoted by Sm based on an iterative algorithm which allows us to reconstruct an approximation of the conductivity of the domain. The advantage of this new regularization is the

This paper is concerned on the fourth-order elliptic equation Δ2u−Δu+V(x)u−λΔ[ρ(u2)]ρ′(u2)u=f(u) in RN,u∈W2,2(RN),Pλ where Δ2=Δ(Δ) is the biharmonic operator, 3≤N≤ 6, the radially symmetric potential V may change sign and infRNV(x)=−∞ is allowed. If f satisfies a type of nonquadracity and monotonicity conditions and ρ is a suitable smooth function, we prove, via variational approach, the existence

In the present paper, by using a variational approach combined with the Nehari manifold method and fibreing maps, the existence of two non-trivial solutions for a class of p-biharmonic problems is obtained.

We make sharp estimates to obtain a Schwarz-type lemma for the symmetrized polydisc Gn and for the extended symmetrized polydisc G~n. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in [Young NJ. The automorphism group of the tetrablock. J Lond Math Soc. 2008;77(2):757–770]. Also we find a few geometric interplay between the members

In this article, we introduce a subclass B(α,μ,A,B) of Bazilevič functions of order α+iμ and estimate the logarithmic coefficients and the difference of moduli of successive coefficients for the function belonging to the above class. The results obtained generalize some known results.

In this paper, we study the existence of a positive solution to the elliptic problem: (−Δ)su=u−q+f(x)h(u)+μin Ω,u>0in Ω,u=0in (RN∖Ω). Here Ω⊂RN (N>2s) is an open bounded domain with smooth boundary, s∈(0,1) and q∈(0,1). For s∈(0,12), we take advantage of the convexity of Ω. The operator (−Δ)s indicates the restricted fractional Laplacian, and μ is a non-negative Radon measure as a source term. The

In this paper, we introduce a class of holomorphic functions Mg(U) on unit disk U. Let w(ξ) be a normalized holomorphic function on U such that w(ξ)/w′(ξ)∈Mg(U), and w(ξ)−ξ has a zero of order k + 1 at ξ=0. By using the formula of Faà di Bruno for the kth derivative of a composite function, we establish the Fekete and Szegö inequality for w(ξ), and then we extend this result to higher dimensions. The

This paper presents the global Borel-Pompieu- and the global Cauchy-type integral formulas for the quaternionic slice regular functions using the relationship between this function space and a non-constant coefficient differential operator given by G:=∥x→∥2∂0+x→∑k=13xk∂k, according to [González-Cervantes JO. On cauchy integral theorem for quaternionic slice regular functions. Complex Anal Oper Theory

In this paper, we are interested in the existence of solutions to the anisotropic nonlocal problem {(P)δ}−∑i=1N∂∂xi∂u∂xipi−2∂u∂xi=∫ΩF(x,u)rf(x,u)+δ|u|p∗−2uin Ω,u≥0in Ω,u=0on ∂Ω, where Ω is a smooth bounded domain of RN, N≥2, δ=0 or δ=1,pi and r≥0, 1

In this paper, we discuss several results regarding existence, multiplicity and stability of solutions of differential equation with discontinuous nonlinearities relevant in several applications. Our problem can be formulated as a free boundary problem with unknown boundaries to be characterized. We first show that the bifurcation diagram is a generalized ‘S-shaped’ curve. Then we study the stability

In this article, we study the quasilinear elliptic equation: −Δu+V(x)u−Δ[(1+u2)1/2]u2(1+u2)1/2=h(u),x∈RN, where N≥3, V:RN→R satisfies suitable assumptions. Unlike h∈C1(R,R), we only need to assume that h∈C(R,R). By using a change of variable, we obtain the existence of ground state solutions with general critical growth.

In this paper, 8×8 real matrix representations of elliptic biquaternions are obtained and by means of these representations, a general method is developed to solve the linear elliptic biquaternion equations. Then, this method is applied to the well-known quaternion equations X−QXR = S and QX−XR = S over the elliptic biquaternion algebra. Also, some illustrative numerical examples are given to show

The goal of this paper is to investigate a parametric Dirichlet problem with (p,q)-Laplacian and concave–convex nonlinearity. Denoting by Sλ the set of positive solutions of the problem corresponding to the parameter λ>0, we prove that the set Sλ is compact in the W01,p(Ω)-topolgy. We also establish an upper semicontinuity and a Mosco convergence result for the multivalued mapping λ↦Sλ.

In this article, we are concerned with the existence of solutions for the following critical fractional p&q-Laplacian system (1) (−Δ)ps1u+(−Δ)qs2u=λQ(x)|u|r−2u+2αα+β|u|α−2u|v|β,in Ω;(−Δ)ps1v+(−Δ)qs2v=μQ(x)|v|r−2v+2βα+β|u|α|v|β−2vin Ω;u=v=0,in Rn∖Ω,(1) where Ω is a smooth bounded set in Rn, λ,μ>0 are two parameters, 0ps1, and α>1,β>1 satisfy α+β=ps1∗ with ps1∗=npn−ps1 is the

In this paper, we develop a direct method of moving planes in Rn without any decay conditions at infinity for solutions for fractional Laplacian. We first prove a monotonicity result for semi-linear equations involving the fractional Laplacian equation in Rn, and we also derive a one-dimensional symmetry result, which indicates that fractional De Giorgi conjecture is valid under some conditions. During

Given holomorphic functions ψ0 and ψ1, we consider first-order differential operators acting on Hardy space, generated by the formal differential expression E(ψ0,ψ1)f(z)=ψ0(z)f(z)+ψ1(z)f′(z). We characterize these operators which are complex symmetric with respect to weighted composition conjugations. In parallel, as a basis of comparison, a characterization for differential operators which are hermitian

This paper is concerned with the existence of multiple non-radial sign-changing solutions for −Δu+u+αK(|x|)Φ(x)u=|u|p−2u,x∈R3,−ΔΦ=K(|x|)u2,x∈R3, where 2

In this paper, we are interested in the study of the following problem with Navier boundary conditions Δ(|x|p(x)|Δu|p(x)−2Δu)=λV(x)|u|q(x)−2uin Ω,u=Δu=0on ∂Ω, where Ω is a smooth bounded domain in Rn, λ>0, the potential V is in some generalized Sobolev space, and p,q:Ω→[1,∞) are continuous functions. The main tools used here are based on the variational method combined with the Mountain Pass theorem

In this paper, we study the multiplicity of weak solutions to the boundary value problem −Gαu=f(x,y,u)+g(x,y,u)in  Ω,u=0on  ∂Ω, where Ω is a bounded domain with smooth boundary in RN (N≥2), α∈N, f(x,y,ξ) is odd in ξ and g(x,y,ξ) is a perturbation term. Under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. Here we do not require that f satisfies

In this work, we study Schrödinger type equations with domain in a complex space and their solutions in the classical as well as in the sense of the space of currents. For such an equation with a (possibly) complex parameter and a (possibly) nonzero potential, conditions are given under which an analogue of the Weyl's lemma holds, meaning that each solution within the space of currents is distributionally

In this article, we prove modular and norm Pólya–Szegö inequalities in general fractional Orlicz–Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of these spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary, we prove a Rayleigh–Faber–Krahn type inequality

In this note, we prove the existence of a weak solution for nonlinear p-sub-Laplacian equations with the Dirichlet boundary condition on the Heisenberg group. In the proof, we use a version of the mountain pass theorem and the Folland–Stein embedding theorem as the main tools.

The extension of the Schwarz representation formula to simply connected domains with harmonic Green function and its polyanalytic generalization is not valid in general. They do hold only for certain domains.

In this paper, we investigate the quasilinear Schrödinger elliptic equations with infinitely many solutions: −Δu+V(x)u+τΔ(1+u2)u21+u2=W(x)f(x,u),x∈RN, where N≥3, τ≥2. It is assumed that the nonlinearity f(x,t) is sublinear in a neighbourhood of t = 0. Some techniques are used here, such as variational methods, Z2 genus and the Moser iteration.

The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its Minkowski function is plurisubharmonic. In this short note, we prove that under the additional assumption of smoothness of the boundary, the Minkowski function of

In this paper, we study the linear coupled elliptic equations: ({Pε})−ε2Δu+u=u3+λvin Ω,−ε2Δv+v=v3+λuin Ω,u>0,v>0in Ω,∂u∂n=∂v∂n=0on ∂Ω, where Ω is a smooth bounded domain in R3, ϵ is a small parameter and λ is a coupling constant. Due to Lyapunov–Schmidt reduction method, we prove that (Pε) has at least O1ε3|ln⁡ε|3 synchronized and O1ε6|ln⁡ε|6 segregated solutions for ϵ sufficiently small and some λ

In this paper, we investigate the following Chern–Simons–Schrödinger system −Δu+u+λ∫|x|∞h(s)su2(s)ds+h2(|x|)|x|2u=f(x,u)inR2, where λ>0, h(s)=∫0st2u2(t)dt and the nonlinearity f(x,s)∈C(R2×R,R) behaves like e4πs2 as |s|→+∞. By using the variational methods and the Trudinger–Moser inequality, we obtain the existence of positive solutions for this system. Moreover, we observe the concentrate behavior

We answer the question how arbitrarily small perturbations of a pair of one arbitrary and one symmetric 2×2 matrix can change a normal form with respect to a certain linear group action. This result is then applied to describe the quadratic part of normal forms of complex points of small C2-perturbations of real 4-manifolds embedded in a complex 3-manifold.

In this paper, we prove the existence and uniqueness of the admissible solution to complex Hessian equations with Neumann boundary condition in balls in Cn.

In this paper, we consider the following problem: −Δpu−ζ|u|p−2u|x|p=∑i=1kIαi∗|u|pαi∗|u|pαi∗−2u+|u|p∗−2u,in RN, where N=3,4,5, p∈(1,2], ζ∈0,Λ, Λ=N−ppp, Δp:=div(|∇u|p−2∇u) is the p-Laplacian operator, p∗=NpN−p is the critical Sobolev exponent, pαi∗=p2N+αiN−p are the Hardy–Littlewood–Sobolev critical upper exponents, the parameters αi satisfy some assumptions. First, we establish the refined Sobolev inequality

The usual definitions of fractional derivatives and integrals are very well-suited for a fractional generalisation of real analysis. But the basic building blocks of complex analysis are different: although fractional derivatives of complex-valued functions and to complex orders are well known, concepts such as the Cauchy–Riemann equations and d-bar derivatives have no analogues in the standard fractional

In this paper, we obtain conditions of Noetherian solvability and the Index formula for a singular integral equation with a Cauchy kernel and a Carleman shift in Besov space, which is embedded into the space of continuous functions on a closed Lyapunov contour but not into the class of functions satisfying Hölder condition.

We study some variational problems related to the controllability of the transition between two quantum states of the driven quantum harmonic oscillator. We focus on the existence of minimum points of the total energy functional which is obtained by adding to the cost of the control a term penalizing the distance from the maximum transition probability between given energy states.

We prove a Schwarz type lemma for harmonic mappings between the unit disk and a geodesic line in a Riemann surface. It presents a certain extension of the Schwarz lemma for the real harmonic mappings with respect to the Euclidean metric defined on the unit disk.

We consider an elliptic equation driven by a p-Laplacian-like operator, on an n-dimensional Riemannian manifold. The growth condition on the right-hand side of the equation depends on the geometry of the manifold. We produce a nontrivial solution by using a Palais–Smale compactness condition and a mountain pass geometry.

In this work, deBranges–Rovnyak spaces, H(b), on the unit ball of Cn are studied. An integral representation of the functions in H(b) through the Clark measure on Sn associated with b is given and a characterization of admissible boundary limits is given in relation with finite angular derivatives. Lastly, the interplay between Clark measures and angular derivatives is examined and it is obtained that