Generalized Cauchy-Riemann System Research Articles

Hilbert formulas for 𝑟-analytic functions and the Stokes flow about a biconvex lens

The so-called r r -analytic functions are a subclass of p p -analytic functions and are defined by the generalized Cauchy-Riemann system with p ( r , z ) = r p(r,z)=r . In the system of toroidal coordinates, the real and imaginary parts of an r r -analytic function are represented by Mehler-Fock integrals with densities, which are assumed to be meromorphic functions. Hilbert formulas, establishing relationships between those functions, are derived for the domain exterior to the contour of a biconvex lens in the meridional cross-section plane. The derivation extends the framework of the theory of Riemann boundary-value problems, suggested in our previous work, to solving the three-contour problem for the case of meromorphic functions with a finite number of simple poles. For numerical calculations, Mehler-Fock integrals with Hilbert formulas reduce to the form of regular integrals. The 3D problem of the axially symmetric steady motion of a rigid biconvex lens-shaped body in a Stokes fluid is solved, and the Hilbert formula for the real part of an r r -analytic function is used to express the pressure in the fluid via the vorticity analytically. As an illustration, streamlines and isobars about the body, the vorticity and pressure at the contour of the body and the drag force exerted on the body by the fluid are calculated.

On characteristics of solutions to a model generalized Cauchy–Riemann system degenerating on a part of the boundary†

The equation , given in the domain G={z: r0<|z|< R}, is considered. We formally construct a full solution and prove an analogy of the Liouville theorem when R=∞ . Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

The Riemann problem on a semicircle for a generalized Cauchy-Riemann system with a singular line

The Riemann problem on a semicircle for a generalized Cauchy-Riemann system with a singular line

L2-Estimates and Existence Theorems for Generalized Analytic Functions in Several Variables

We consider the generalized Cauchy–Riemann system with nonlinear terms in an arbitrary domain of the complex space. Under some natural conditions on the coefficients and compatibility conditions, we prove solvability of this system in the space of locally square integrable functions.

PROPAGATION OF SMALLNESS FOR SOLUTIONS OF GENERALIZED CAUCHY–RIEMANN SYSTEMS

AbstractLet $u$ be a solution of a generalized Cauchy–Riemann system in $\mathbb{R}^n$. Suppose that $|u|\le1$ in the unit ball and $|u|\le\varepsilon$ on some closed set $E$. Classical results say that if $E$ is a set of positive Lebesgue measure, then $|u|\le C\varepsilon^\alpha$ on any compact subset of the unit ball. In the present work the same estimate is proved provided that $E$ is a subset of a hyperplane and the (capacitary) dimension of $E$ is greater than $n-2$. The proof gives control of constants $C$ and $\alpha$.AMS 2000 Mathematics subject classification: Primary 31B35. Secondary 35B35; 35J45

On continuous solutions of a generalized Cauchy–Riemann system with more than one singularity

We study the solvability of the Riemann–Hilbert problem for a generalized Cauchy–Riemann system with several singularities and reveal several new phenomenon. For the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities; moreover, it does not depend continuously on the coefficients of the equation.

On the Variety of Solutions of Generalized Cauchy-Riemann Systems with a Singular Line

In the article it is proved that a specific generalized Cauchy-Riemann system with singularity on a circle has only the trivial solution in the class of infinitely differentiable functions.

Numerical Solutions of Cauchy-Riemann Equations for Two and Three-Dimensional Flows

For two-dimensional flows, the conservation of mass and the definition of vorticity comprise a generalized Cauchy-Riemann system for the velocity components assuming the vorticity is given. If the flow is compressible, the density is a function of the speed and the entropy, and the latter is assumed to be known. Introducing artificial time, a symmetric hyperbolic system can be easily constructed. Artificial viscosity is needed for numerical stability and is obtained from a least-squares formulation. The augmented system is solved explicitly with a standard point relaxation algorithm which is highly parallelizable. For an extension to three-dimensional flows the continuity equation is combined with the definitions of two vorticity components, and are solved for the three velocity components. Second-order accurate results are compared with exact solutions for incompressible, irrotational, and rotational flows around cylinders and spheres. Results for compressible (subsonic) flows are also included.

On annihilators of harmonic vector fields

Let Ω⊂ℝN be a smooth bounded domain. We characterize smooth vector fields g on ∂Ω which annihilate all harmonic vector fields f in Ω continuous up to ∂Ω, with respect to the pairing\(\left\langle {f,g} \right\rangle = \int\limits_{\partial \Omega } {f \cdot gd\sigma } \) (dσ denotes the hypersurface measure on ∂Ω). In addition, we extend these results to differential forms with harmonic vector fields being replaced by harmonic fields, i.e., forms f satisfying df=0, δf=0. A smooth form g on ∂Ω is an annihilator if and only if suitable extensions, u and v, into Ω of its normal and tangential components on ∂Ω, satisfy the generalized Cauchy-Riemann system du=δv, δu=0, dv=0 in Ω. Finally, we prove that the described smooth annihilators are weak* dense among all annihilators. Bibliography: 12 titles.

On the nonlinear generalized Cauchy-Riemann system on the whole plane

On the nonlinear generalized Cauchy-Riemann system on the whole plane

Coercive solvability of a generalized Cauchy-Riemann system in the SpaceL p (E)

For an inhomogeneous generalized Cauchy-Riemann system with nonsmooth coefficients separated from zero, we establish conditions for the solvability and estimation of a weighted solution and its first-order derivatives.

The variety of solutions of the singular generalized Cauchy-Riemann System

We prove that the equation $$2\bar z\partial _{\bar z} \bar w = 0_1 z \in G,$$ in whichB(z) ∈C∞(G),B0(z)=O(|z})α),α>0,z → 0, and $$b(\varphi ) = \sum\limits_{k = - m_o }^m {b_k e^{ik\varphi } } $$ does not have nontrivial solutions in the classC∞(G).

Continuous solutions of a generalized Cauchy-Riemann system with a finite number of singular points

Continuous solutions of a generalized Cauchy-Riemann system with a finite number of singular points

To the theory of a generalized Cauchy-Riemann system with a singular point

To the theory of a generalized Cauchy-Riemann system with a singular point

Reexamination of Ohm's law and a relation to generalized analytic functions

In 1876 Rayleigh found a neat theorem relating electrical conductances of a quadrilateral shaped plate having unit conductivity. He showed that the conductance between one pair of opposite edges is the reciprocal of the conductance between the other pair of opposite edges. (This is obvious for a rectangular plate.) Let u be the electrical potential function in the first situation and let v be the potential function in the second situation. Then a proof of the theorem can be given by taking u and v to be conjugate harmonic functions. The first goal of this note is to show that Rayleigh's reciprocal theorem can be generalized to hold for plates which are neither isotropic nor homogeneous. In other words the conductivity can be a tensor rather than a scalar. Moreover, the two by two conductivity tensor can vary arbitrarily from point to point of the plate. Possibly the most interesting case of this new generalization results when the conductivity tensor is required to be symmetric and have unit determinant at all points. Then it results that the complex potential functions of the form f = u + iv share many of the properties of classical analytic functions. For example: the sum, product, and quotient of complex potential functions all satisfy the same generalized Cauchy-Riemann system.

A representation of the solutions of the generalized Cauchy-Riemann equations and its applications

A method of integral representations for the generalized Cauchy-Riemann system in terms of an arbitrary analytic function, similar to the well-known Whittaker-Polozhii representation /1/, is developed. The representation includes various well-known results as special cases, and the limiting case leads to the classical representation of the theory of a generalized axisymmetric potential. The representations established are used to reduce mixed problems for the system to paired equations and then to a Fredholm equation of the second kind. At the same time, a device is described for regularizing paired equations, and a case in which a closed solution exists is presented. The results are extended to a sytem of more-general form and also to second-order equations, whose type and dimensionality are not essential. It is shown that the integral operators constructed here convert the solution of a parabolic or hyperbolic equation with variable coefficients into a solution of the classical equations of heat conduction and wave propagation, thus furnishing an explicit representation for solutions of the corresponding Cauchy problems. The effectiveness of the approach is demonstrated with reference to the problem of inflow in a fissure in an inhomogeneous layer of finite width. Simple formulae for the pressure and discharge of fissures are presented.

A certain class of generalized Cauchy-Riemann systems

One investigates a generalized Cauchy-Riemann system of the special form where the constant matrices A, B, C of order k are selected in such a manner that any solution of this equation is a harmonic function (with values in ℝk). For such harmonic fields one introduces the Hardy class HP(ℝ+3) and one discusses its connection with the Hardy class HP(ℝ2), introduced by E. M. Stein and G. Weiss. One obtains also the following analogue of W. Rudin's theorem: Any closed set E ∇(ℝ2 of zero measure is an interpolation set for the space .

Characterization of solutions to the generalized Cauchy-Riemann system

For a generalized biaxially symmetric potential U on a semi-disk D +, a harmonic conjugate V is defined by the generalized Cauchy-Riemann system. There is an associated boundary value theory for the Dirichlet problem. The converse to the Dirichlet problem is considered by determining the boundary functions to which U and V converge. The unique limits are hyperfunctions on the ∂D +. In fact, the space of hyperfunctions is isomorphic to the spaces of generalized biaxially symmetric potentials and their harmonic conjugates. A representation theorem is given for U and V terms of convolutions of certain Poisson kernels with continuous functions that satisfy a growth condition on the ∂D +.

A difference method for solving boundary value problems for generalized analytic functions

WE consider a difference method for solving a Riemann-Hilbert boundary value problem for a generalized Cauchy-Riemann system in a multiply connected region when n + 1 > m, where n is the index of the problem and m + 1 is the connectivity of the region. We know from [1] that the inhomogeneous problem is always solvable in this case, while the corresponding homogeneous problem has 2 n + 1 − m linearly independent solutions and an extra 2 n + 1 − m functionals have to be defined in order for the solution to be unique. Auxiliary conditions in the form of integral functionals are obtained in Section 1 and the initial problem with these auxiliary conditions is shown to be correctly posed in a certain metric, so that the order of the error of the difference solution can be estimated and its convergence proved. The general form of the integral functionals making the problem uniquely solvable was found in [2] with no restrictions on n ⩾ 0 and m ⩾ 0, but a modified statement of the problem was considered, namely, the right-hand side in the boundary condition was specified up to an additive constant in each of the internal contours, and accordingly 2 n + 1 extra conditions were specified. Extra conditions of an integral type were considered in [3, 4] for a simply connected region. In Section 2 we replace the initial problem with the auxiliary conditions by a difference problem. To obtain the difference set of equations the problem is reduced to the equivalent variational problem of finding the minimum of a quadratic functional by the method of least squares. The difference set of equations is obtained from the minimization condition for the functional in the class of special functions, defined on the difference mesh and satisfying the auxiliary conditions mentioned. We show that the difference problem has a unique solution, estimate its error and prove its convergence. We show that a solution of the initial problem (with no auxiliary conditions) can be found approximately and point out the analogy between the initial problem and the corresponding difference problem as regards solvability and the manifold of solutions of the corresponding homogeneous problems.