Inhomogeneous Cauchy-Riemann Equation Research Articles

COMPLEX BOUNDARY VALUE PROBLEMS FOR THE CAUCHY–RIEMANN OPERATOR ON A TRIANGLE

There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

Schwarz problem in a ring domain*

In this article, we discuss the solutions of Schwarz problem in a ring domain. Firstly we introduce the solutions of inhomogeneous Cauchy-Riemann equation. Afterwards we extend the discussions to higher-order equations. The relevant solvability conditions are obtained. The last part is reserved for the solutions of higher-order linear equations in .

The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator overline{partial } on spaces {mathcal {E}}{mathcal {V}}(varOmega ,E) of {mathcal {C}}^{infty }-smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights {mathcal {V}}. Vector-valued means that these functions have values in a locally convex Hausdorff space E over {mathbb {C}}. We derive a counterpart of the Grothendieck-Köthe-Silva duality {mathcal {O}}({mathbb {C}}setminus K)/{mathcal {O}}({mathbb {C}})cong {mathscr {A}}(K) with non-empty compact Ksubset {mathbb {R}} for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of overline{partial }:{mathcal {E}} {mathcal {V}}(varOmega ,E)rightarrow {mathcal {E}}{mathcal {V}} (varOmega ,E) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on {mathcal {E}}{mathcal {V}}(varOmega ,{mathbb {C}}).

Surjectivity of the ̄∂-operator between weighted spaces of smooth vector-valued functions

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 -valued functions whose topology is given by a whole family of weights. Our proof relies on a weakened variant of weak reducibility of the corresponding subspace of holomorphic functions in combination with the Mittag–Leffler procedure. Using tensor products, we deduce the corresponding result on the solvability of the inhomogeneous Cauchy–Riemann equation for Fréchet-valued functions.

Robin Boundary Value Problem Depending on Parameters in a Ring Domain

This study is devoted to give solvability conditions and solutions of the Robin boundary problem with constant coefficients for the homogeneous and the inhomogeneous Cauchy-Riemann equation in an annular domain. In order to get results, known representations and theorems in the literature are used. The representations for the solutions and solvability conditions are given in explicit form and here only a special Robin problem is considered. At the end of the paper, it is concluded that with some choices, boundary value problems for the Cauchy-Riemann equation reduce to some basic boundary problems in the ring domain.

Riemann–Hilbert-type boundary value problems on a half hexagon

In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann–Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are obtained. The boundary behavior of these operators is discussed. Finally, we investigate the Schwarz problem and the Riemann–Hilbert problem for inhomogeneous Cauchy–Riemann equations.

Continuous solutions of nonlinear Cauchy-Riemann equations and pseudoholomorphic curves in normal coordinates

We establish elliptic regularity for nonlinear, inhomogeneous Cauchy-Riemann equations under weak assumptions, and give a counterexample in a borderline case. In some cases where the inhomogeneous term has a separable factorization, the solution set can be explicitly calculated. The methods also give local parametric formulas for pseudoholomorphic curves with respect to some continuous almost complex structures.

Solvability of the inhomogeneous Cauchy–Riemann equation in projective weighted spaces

We establish an analog of Hormander’s Theorem on solvability of the inhomogeneous Cauchy–Riemann equation for a space of measurable functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence defining the space. The same conditions guarantee the weak reducibility of the corresponding space of entire functions. Basing on these results, we solve the problem of describing the multipliers in weighted spaces of entire functions with the projective and inductive-projective topological structure. Applications are obtained to convolution operators in the spaces of ultradifferentiable functions of Roumieu type.

On the existence of solutions of one nonlinear boundary-value problem for shallow shells of Timoshenko type with simply supported edges

Solvability of one system of nonlinear second order partial differential equations with given initial conditions is considered in an arbitrary field. Reduction of the initial system of equations to one nonlinear operator equation is used to study the problem. The solvability is established with the use of the principle of contracting mappings. The method used in these studies is based on the integral representations for the displacements. These representations are constructed with the use of general solutions to the inhomogeneous Cauchy-Riemann equation.

Schwarz boundary value problem for the Cauchy-Riemann equation in a rectangle

In this paper, Schwarz problem for the inhomogeneous Cauchy-Riemann equation in a rectangle is investigated explicitly. By the parqueting reflection principle and the Cauchy-Pompeiu formula, a modified Schwarz-Poisson representation formula in a rectangle is constructed. In particular, the boundary behaviors for the related Schwarz-type and Pompeiu-type operators at the corner points are discussed in detail.

Inhomogeneous Cauchy-Riemann equation in Octonion space

The integral representation of differentiable functions in Octonion space is obtained and the explicit solution of the inhomogeneous Cauchy-Riemann equation is given by integral representation. As an application, the Cousin problem analogue of Mittag-Laffler problem is discussed.

Schwarz boundary value problem in half lens and half lune

In this article, firstly we obtain Cauchy–Schwarz representation in a half lens in . Then, we consider the Schwarz boundary value problem for the inhomogeneous Cauchy–Riemann equation and explicit expression of solution is obtained. Secondly, the Schwarz problem is explicitly solved in two half lunes which are complements of the half lens with respect to two orthogonal half circles.

Solvability of inhomogeneous Cauchy–Riemann equation in spaces of functions with a system of uniform weight estimates

We obtain an analog of the Hormander theoremon solvability of the \(\overline \partial \)-problemin spaces of functions satisfying a system of uniform estimates. The result is formulated in terms of the weight sequence determining the space. We apply the results for multipliers of projective and inductiveprojective weight spaces of entire functions and for convolution operators in the Roumieu spaces of ultradifferentiable functions.

A Riemann–Hilbert boundary value problem in a triangle

In this article a Riemann–Hilbert boundary value problem on an isosceles orthogonal triangle is considered. Using explicit Schwarz–Poisson-type formulae for the triangle, Schwarz-type and Pompeiu-type operators are obtained. Boundary behaviors of these operators are discussed in detail. Finally, we investigate the Riemann–Hilbert boundary value problem for both homogeneous and inhomogeneous Cauchy–Riemann equations. An explicit solvability of the problem is obtained.

A Riemann-Hilbert boundary value problem in a bounded sector

In this article a Riemann-Hilbert boundary value problem in a bounded sector domain with angle is considered. A plane parqueting-reflection technique is used. Boundary behaviours of resulting integral operators are discussed in detail. Then we study the Riemann-Hilbert boundary value problem, with an arbitrary index, for both homogeneous and inhomogeneous Cauchy–Riemann equations. Expressions of solutions and the condition of solvability are explicitly obtained.

On the existence of solutions to geometrically nonlinear problems for shallow Timoshenko-type shells with free edges

In the paper we investigate the solvability of the boundary-value problems for shallow isotropic elastic shells within the framework of Timoshenko’s shearmodel. The considered problems are nonlinear geometrically and linear physically. The method of studying consists in reducing the initial system of equilibrium equations to one nonlinear differential equation with respect to deflections. In doing so integral representations for the tangential displacements and angles of rotation play a significant role. The representations are deduced by making use of general solutions to the inhomogeneous Cauchy-Riemann equation. The solvability is established by the principle of contracting mappings.

Schwarz problem in lens and lune

The Schwarz problem is explicitly solved for the inhomogeneous Cauchy–Riemann equation in a particular circular lens and two related complementary lunes. The solutions are given by the same analytical formula restricted to the respective domain. The parqueting-reflection method is used to construct the Cauchy–Schwarz representation formula leading to the Schwarz and thus the Poisson kernels for the three domains.

Two basic boundary-value problems for the inhomogeneous Cauchy–Riemann equation in an infinite sector

The object of the present article is to investigate the Schwarz and Dirichlet boundary value problems for the inhomogeneous Cauchy–Riemann equation in an infinite sector. Firstly, we obtain the Schwarz–Poisson formula in a sector with angle (). Secondly, boundary behaviors of some linear integrals will be studied, especially at the corner point. Finally, the solutions and the conditions of solvability are explicitly obtained.

Pointwise Estimates for the Bergman Kernel of the Weighted Fock Space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\phi})$ where $\phi$ is a subharmonic function with $\Delta \phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta \phi$.

Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauchy–Riemann equation

The Riemann–Hilbert–Poincaré problem with general coefficient for the inhomogeneous Cauchy–Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coeffcient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.