We consider reflexive Bergman spaces Ap(Ω) on polygonal domains Ω of the complex plane. With some restrictions to the angles of the boundary of Ω, we show that the boundedness of the Toeplitz operator Tg:Ap(Ω)→Ap(Ω) with a positive symbol g is equivalent to the boundedness of the Berezin transform of g or to g times the area measure being a Carleson measure. The result is also formulated for more general

In this paper, we establish some sharp Liouville type theorems for nonnegative weak supersolutions in RN×R of porous medium equation ut−Δum=up, and of porous medium system {ut−Δum=vpvt−Δvm=uq, where p, q>m>1. More precisely, we show that the sharp conditions for nonexistence of supersolutions to the scalar equation and the system are respectively given by N≤2p−m and N≤max(2(p+1)p(q+1)−m(p+1),2(q+1)q(p+1)−m(q+1))

In the present paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system with logarithmic and critical nonlinearity: (a+b[u]s2)(−Δ)su+V(x)u+ϕu=λ|u|q−2uln⁡|u|2+|u|2s∗−2u,x∈Ω,(−Δ)tϕ=u2,x∈Ω,u=0,x∈R3∖Ω, where s∈34,1,t∈(0,1),λ,a,b>0,4

In this paper, we are interested in operators arising from the product of a weighted composition operator and a maximal differential operator on the Fock space. Let Tmax and Cu,φ denote a maximal differential operator and a weighted composition operator, respectively. We characterize some C-selfadjoint operators Cu,φTmax with respect to special conjugations C. Moreover, we obtain a partial characterization

Let C be the familiar class of normalized close-to-convex functions in the unit disk. In Koepf [On the Fekete-Szegö problem for close-to-convex functions. Proc Amer Math Soc. 1987;101:89–95], Koepf proved that for a function f(z)=z+∑k=2∞akzk in the class C, |a3−λa22|≤{3−4λ,λ∈[0,13],13+49λ,λ∈[13,23],1,λ∈[23,1]. As an important application, in the same paper, Koepf showed that ||a3|−|a2||≤1 for close-to-convex

In the hypercomplex analysis, it is well known that the quaternionic slice regular functions are defined on axially symmetric slice domains and that each quaternion is represented in the form x+iy, where x,y∈R and i∈S2 but the mapping ((x,y),i)↦x+iy allows to see that any axially symmetric slice domain is the base space of a trivial sphere bundle. Remember that a sphere bundle is a useful concept in

The aim of this work is to consider the bicomplex third-order Jacobsthal numbers and to present some properties involving this sequence, including the Binet-style formulae and the generating functions. Furthermore, Cassini's identity and d'Ocagne's identity for this type of bicomplex numbers are given, and a different way to find the nth term of this sequence is stated using the determinant of a four-diagonal

The aim of this paper is to study the existence of positive solutions to the fractional Kirchhoff-type problems involving steep potential well λV and sign-changing nonlinearity f(x)|u|p−2u(2

In this work, we consider a model of functional differential variational inequalities formulated by a delay differential inclusion and an elliptic variational inequality in Banach spaces. We prove the global solvability of the problem and we have given some sufficient conditions to ensure the existence of decay solutions. Moreover, the existence of a global attractor for the semiflow governed by our

We show existence of solution for a supercritical nonlinear Schrödinger equation on the whole RN by means of an approximation scheme. We prove a Sobolev embedding corresponding to the variable exponent of the equation.

We present results concerning the cyclic behaviour of weighted shifts and composition operators on the disc algebra. Most of our proofs use the representations of the dual space of the disc algebra as Banach spaces of analytic functions on the unit disc. Taking into account that no composition operator on the disc algebra can be hypercyclic, the results we obtain show that the composition operators

We study the compact embedding for Sobolev spaces W01,Φ(G) of two variable exponents, where Φ(x,t)=tp(x)(log⁡(e+t))q(x). Here p(⋅) and q(⋅) are variable exponents satisfying the log-Hölder and log log-Hölder conditions, respectively.

In Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. we have considered the problem about asymptotic formulas for all eigenvalues of Tn(a), as n goes to infinity, assuming that a is a specific

This paper is concerned with the Riemann problem for bianalytic functions on a closed Jordan curve with a certain degree of fractality. We derive a solvability condition under which an explicit solution of this problem is obtained.

ABSTRACT In this paper, we consider the following quasilinear equation: Δp,gu+a(x)|u|p−2u=K(x)|u|p∗(s)−2udg(x,x0)s+h(x)|u|r−2u,x∈M, where M is a compact Riemannian manifold with dimension n⩾3 without boundary, and x0∈M. Here a(x), K(x) and h(x) are continuous functions on M satisfying some further conditions. The operator Δp,g is the p-Laplace–Beltrami operator on M associated with the metric g, and

ABSTRACT In this paper, using a change of variables and variational method, positive solutions of the stationary relativistic nonlinear Schrödinger equation involving critical exponent and Hardy potential are studied when the potential function has positive lower bound and radial symmetry. We extend the result of Huang, Xiang (Soliton solutions for a quasilinear Schrödinger equation with critical exponent

This article considers an optimal control problem for the stationary Stokes system in a three-dimensional domain with a highly oscillating boundary. The controls are acting on the state through the Neumann data on the oscillating part of the boundary with appropriate scaling parameters ϵα with α≥1. The periodic unfolding operators are used to characterize the optimal controls. Using the unfolding operators

In this paper, we investigate the Fredholm solvability of the generalized Neumann problem for a high-order elliptic equation in the plane. The equivalence of the solvability conditions of the generalized Neumann problem to the complementarity condition (Shapiro–Lopatinsky condition) is proved. The formula for the index of the specified problem in the class C2l,μ(D¯) is calculated.

We give an overview of some concepts related to the approximation of solutions of a strongly elliptic operator. A definition of a complete system of classical solutions is given, and we show how using the Runge property, it is possible to extend the completeness of these systems onto strictly internal domains in the L2, H1 and H2 norms. This result is applied to Schrödinger equations with potentials

Asmar et al. [Note on norm convergence in the space of weak type multipliers. J Operator Theory. 1998;39(1):139–149] proved that the space of weak-type Fourier multipliers acting from Lp into Lp,∞ is continuously embedded into L∞. We obtain a sharper result in the setting of abstract Lorentz spaces Λq(X) with 0

ABSTRACT Based on the Runge theorem for generalized analytic vectors proved by Goldschmidt in 1979, we provide a Mergelyan-type and a Carleman-type approximation theorems for solutions of Pascali systems.

An exact formula for the conformal map from the exterior of two slits onto the doubly connected flow domain is obtained when a fluid flows in a wedge about a vortex. The map is employed to determine the potential flow outside the vortex and the vortex domain boundary provided the circulation around the vortex and constant speed on the vortex boundary are prescribed, and there are no stagnation points

There are over 100 recent articles in the literature dealing with first-, second- and third-order differential superordinations, containments and inequalities in the complex plane, none of which deals with systems of such topics. This article investigates systems of two second-order simultaneous differential containments, superordinations and inequalities in two analytic functions p and q defined in

In this paper, we prove a criterion for plurisubharmonic functions in terms of integral mean by complex ellipsoids. Moreover, by using the criterion, we prove an analogue of Blaschke–Privalov theorem for plurisubharmonic functions.

This paper deals with the problem of determining a radially dependent coefficient a(r) in the equation Δu−a(r)u=0, in the unit sphere Ω from the Dirichlet–Neumann data pair {u,∂u∂ν}|∂Ω. We discuss the uniqueness of this determination.

ABSTRACT In this article, we study the existence of infinitely many nontrivial solutions for the following doubly nonlocal problem involving fractional (p(x),p+)-Laplacian. (−Δ)p(x)su+(−Δ)p+su=|u|r(x)−2u+|u|ps∗(x)−2u+λf(x,u)inΩ,u=0inRN∖Ω. Here, Ω⊂RN is a bounded Lipschitz domain, λ>0, qs∈(0,1),p(⋅,⋅) is a continuous, bounded, symmetric function in RN×RN such that p+=sup(x,y)∈RN×RN{p(x,y)}

We review mathematical results on the interaction of time-harmonic electromagnetic waves with ‘geometric’ and ‘electromagnetic’ chiral objects and discuss the relation between these two notions.

We are interested in singular positive solutions of a quasilinear elliptic equation with a singular coefficient r−(γ−1)(rα|u′|β−1u′)′+krα−β−γuβ+up=0,0β, 0

This paper is concerned with a class of uniformly elliptic nonlocal Bellman systems in unbounded Lipschitz domains. There are two major ingredients. The first is a narrow region principle in unbounded domains. We can use this to carry out the direct method of moving planes. The second ingredient is to prove the monotonicity of positive bounded solutions for a cooperative system with uniformly elliptic

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations

In L2(R), consider a second-order elliptic differential operator Aϵ, ϵ>0, of the form Aϵ=−ddxg(x/ϵ)ddx+ϵ−2p(x/ϵ) with periodic coefficients. For small ε, we study the behavior of the semigroup e−Aϵt, t>0, cut by the spectral projection of the operator Aϵ for the interval [ϵ−2ν,+∞). Here ϵ−2ν is the right edge of a spectral gap for the operator Aϵ. We obtain approximation for the ‘cut semigroup’ in

In this paper, we investigate the following autonomous Choquard equation −Δu+u=(Iα∗F(u))F′(u)in RN, where N≥3, Iα denotes the Riesz potential of order α∈(0,N) and F satisfies general critical growth conditions. By using the variational methods and the Pohožaev manifold techniques, we prove the existence of radially symmetric positive ground state solution.

In this paper, we consider some three-dimensional noncommutative algebra A~2 over the field C, which contains the algebra of bicomplex numbers B(C) as a subalgebra. Locally bounded and Gâteaux-differentiable mappings defined in the domains of the three-dimensional subspace of the algebra B(C) and taking values in the algebra A~2 are considered. Such mappings generalize holomorphic functions of bicomplex

Existence of radially symmetric positive solutions for a general class of quasi linear elliptic equations in the zero mass situation is proven. Our results include ground-state solutions for both power-type and exponential-type critical growths in the whole space and the class concerned includes, as particular cases, the Laplace, p-Laplace and k-Hessian operators in radial form.

In this paper, we prove that ζ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in Γ(α),Γ(n),Γ(l) over the ring of polynomials in C, l>n>α≥0 are nonnegative integers. We extend the result that ζ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in Γ,Γ′,Γ″ over the field of complex numbers, which

In this paper, we continue the study of new weighted spaces of holomorphic functions on the unit disc with the mixed norm defined in terms of conditions on Fourier coefficients of a function. Here we present the case in which the corresponding conditions are related to grand and small Lebesgues spaces, i.e. the Fourier coefficients as functions of radial variable belong to either grand or small space

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the C∗-algebra B generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator ST and by the unitary weighted shift operators Ug:φ↦|g′|1/2(φ∘g), g∈G, acting on the space L2(T) over the unit circle T⊂C. Here G

In this paper, we consider the following quasilinear Schrödinger-Poisson system with critical growth: −Δu+V(x)u−12uΔ(u2)+ϕu=|u|4u+μg(u),x∈R3,−Δϕ=u2,x∈R3, where μ>0, V(x) is a smooth potential function and g is a appropriate nonlinear function. For the sake of overcoming the technical difficulties caused by the quasilinear term, we shall apply the perturbation method by adding a 4-Laplacian operator

This paper is concerned with the existence and multiplicity of sign-changing solutions for the following modified Schrödinger–Poisson system: −Δu+V(x)u+κϕu−12u△u2=f(u),x∈R3,−△ϕ=u2,x∈R3, where κ∈(0,1),V(x) is coercive, the nonlinear term f is 4-superlinear at infinity but does not need any increasing condition. By using the method of invariant sets of descending flow, the existence and multiplicity

In the paper, we study the following system of partial differential equations (PDEs) involving fractional Laplacian operators (1) (−△)α2u=vq,x∈R+n,(−△)β2v=up,x∈R+n,(1) under the boundary conditions u≡0, v≡0, x∈Rn∖R+n, where p∈(1,n+βn−α], q∈(1,n+αn−β], 0<α, β<2 and n≥3. In order to overcome the difficulty that there are no corresponding maximum principles for the operators (−△)α2 and (−△)β2 in R+n,

In this paper, we study the existence and regularity of a non-trivial weak solution to the following semilinear elliptic problem with singular weights and singular nonlinearity: {−div(|x|−2β∇w)−μw|x|2(β+1)=g(x)wθ1+f(x)wθ2inΩ,w>0inΩ,w=0on∂Ω, under assumptions θ1≥θ2>0,0<μ<(N−2(β+1)2)2 and 0≤g∈Lk(Ω), 0≤f∈Lk(Ω),1

We derive sufficient conditions for the surjectivity of the Cauchy–Riemann operator ∂¯ between weighted spaces of smooth Fréchet-valued functions. This is done by establishing an analog of Hörmander's theorem on the solvability of the inhomogeneous Cauchy–Riemann equation in a space of smooth C-valued functions whose topology is given by a whole family of weights. Our proof relies on a weakened variant

In this paper, we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative BC-linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand–Mazur theorem for

In this paper, we are mainly concerned with one certain type of non-linear complex differential equation fn(z)+Pd(z,f)=p1eα1z+p2eα2z, where p1,p2,α1,α2 are non-zero constants and Pd(z,f) is a differential polynomial in f of degree d at most n−1. For this kind of equation, if it admits a meromorphic solution such that N(r,f)=S(r,f), then firstly we give a positive answer to Li's question with a weaker

(2021). Publisher's note. Complex Variables and Elliptic Equations: Vol. 66, Boundary Value Problems for Partial Differential Equations and Function Spaces - Special Issue Dedicated to the 110th Anniversary of the Birthday of Sergei L. Sobolev. Guest Editors: Heinrich Begehr, Gennadii V. Demidenko, and Inessa I. Matveeva, pp. 1209-1210.

(2021). Correction. Complex Variables and Elliptic Equations: Vol. 66, Boundary Value Problems for Partial Differential Equations and Function Spaces - Special Issue Dedicated to the 110th Anniversary of the Birthday of Sergei L. Sobolev. Guest Editors: Heinrich Begehr, Gennadii V. Demidenko, and Inessa I. Matveeva, pp. 1211-1211.

(2021). Correction. Complex Variables and Elliptic Equations: Vol. 66, Boundary Value Problems for Partial Differential Equations and Function Spaces - Special Issue Dedicated to the 110th Anniversary of the Birthday of Sergei L. Sobolev. Guest Editors: Heinrich Begehr, Gennadii V. Demidenko, and Inessa I. Matveeva, pp. 1212-1212.

We consider the problem {−Δu=λK(x)f(u)in B1c,u(x)=0on ∂B1,u(x)→0as |x|→∞, where B1c={x∈Rn:|x|>1}, n>2, λ is a positive parameter, K:B1c→R+ belongs to a class of continuous functions which satisfy certain decay assumptions, and f:[0,∞)→R belongs to a class of continuous functions which are superlinear at ∞ with f(0)<0. Recently, several authors have studied positive radial solutions to this problem

Let F be a smooth Riemann surface foliation on M∖E, where M is a complex manifold and E⊂M is a closed set. Fix a hermitian metric g on M∖E and assume that all leaves of F are hyperbolic. For each leaf L⊂F, the ratio of g|L, the restriction of g to L, and the Poincaré metric λL on L defines a positive function η that is known to be continuous on M∖E under suitable conditions on M, E. For a domain U⊂M

In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions f∈S given by f(z)=z+∑n=2∞anzn for D={z∈C:|z|<1} has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form H2,2(f)=a2a4−a32. A non-sharp bound for H2,2(f) when f∈K(α), α∈[0,1) consisting of convex functions of order α was found

We discuss the Schwarz problem on Reuleaux triangle. By the technique of parqueting reflection principle, we translate the Schwarz problem into a Riemann boundary value problem on a system of arcs. We first state that the Riemann problem is solvable, while the solutions are represented by a Cauchy type integral. At last, as the main theorem, we prove that those two problems are equivalent to each other

In this paper, we give the nth derivertive criterion for functions belonging to function spaces HK2. Furthermore, some sufficient conditions for coefficients of the linear differential equation f(k)+Ak−1(z)f(k−1)+⋯+A1(z)f′+A0(z)f=Ak(z)are found such that all solutions belong to the HK2 spaces, where Aj(z) are analytic functions in D, j=0,…,k.

Let μ be a normal function on [0,1) and p>0. Suppose φ is a holomorphic self-map on the unit ball B of Cn. In this paper, the authors characterize the conditions such that the composition operator Cφ is bounded or compact from the normal weight Dirichlet type space Dμp(B) to the normal weight Bloch type space Bνp(B) when n>1, where νp(r)=(1−r2)npμ1p(r) ( 0≤r<1). Moreover, the authors give a simple

In this paper, we consider the existence of multiple solutions for a quasilinear Schrödinger equation with Robin boundary condition involving critical Hardy–Sobolev exponent as follows: −Δu−Δ(u2)u+u=(u2)2∗(s)−1|x|su+f(x,u)inΩ,∂u∂n+β(x)u=0on∂Ω, where f∈C(Ω¯×R,R) satisfies suitable condition. Using variable substitution and Mountain Pass Theorem, we prove that the above equation admits at least two nontrivial

We characterize wandering subspace property of the shift operator on zero-based invariant subspaces of reproducing kernel Hilbert spaces by using the kernels of Toeplitz operators. As applications, we prove that the wandering subspace property of the shift operator holds on the weighted Bergman space Aα2 ( α>1) for zero-based invariant subspaces when the zeros lie in a small disk centred at the origin

We prove a Pohozaev-type identity for both the problem (−Δ+m2)su=f(u) in RN and its harmonic extension to R+N+1 when 0

The present article is concerned with a class of corner-degenerate viscoelastic higher hyperbolic equations with nonlinear damping and boundary source terms. First, we prove some invariance results about the energy functional and the solution set of our problem by using potential wells methods on the manifolds with corner singularity. Then, we establish existence results of global weak solution on

Let M¯ be a smoothly bounded orientable pseudoconvex CR manifold of finite 1-type with comparable eigenvalues of the Levi-form. Then we extend the given CR structure on M¯ to an integrable almost complex structure on the concave side of M when dimR⁡M≥3. If dimR⁡M≥7, we extend the structure on the convex side. Therefore we may regard M as a real hypersurface in Cn when dimR⁡M≥7.

We study the existence of critical points of stable stationary solutions to reaction–diffusion problems on topological tori. Stable nonconstant stationary solutions are often called patterns. We construct topological tori and patterns with prescribed numbers of critical points whose locations are explicit.

We consider the generalized quasilinear Schrödinger equations (P) −div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=λf(x,u),x∈RN,(P) where N≥3, g:R→R+ is an even differentiable function, f:RN×R→R satisfies the Carathéodory condition, the potential V(x):RN→(0,∞) is continuous and λ is a parameter. The intention of the article is to determine the precise positive interval of λ when the problem possesses at least two