Integral Representations for Solutions of Some Types of the Beltrami Equations

Katz, David B.; Kats, B. A.

doi:10.3103/s1066369x18030039

Cited by 7 publications

(11 citation statements)

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“…Finally, by decomposition theorem for solutions of higher order iterated Beltrami equations, we transform the original problems to the vector-valued Riemann-Hilbert problems with shift for β-analytic functions, and further construct canonical matrix of triangular matrix functions explicitly, and thus obtain the explicit solutions and conditions of solvability. The main results extend those of [18,19,22,27,29,30,32,[34][35][36][37][38].…”

Section: Introductionsupporting

confidence: 79%

“…In this section, we recall some basic notions, notations, and facts in [30,32] which will be used in the sequel, and establish a decomposition theorem for a class of higher order iterated Beltrami equations.…”

Section: Generalized β-Analytic Functionsmentioning

confidence: 99%

“…For example, when β = 0 and L is the unit circle, Theorem 4.3 is exactly reduced to the results in [18,19,22,36,37]. Furthermore, when α(t) = t or n = 1, β = 0, α(t) = t, Theorem 4.3 is just reduced to the corresponding results in [5,7,[27][28][29][30]. Moreover, by developing a new theory of canonical matrix, Theorem 4.3 can be also applicable to the results in [38] under some suitable assumptions.…”

Section: )mentioning

confidence: 99%

“…In recent years, Blaya and his collaborators studied some jump Riemann BVPs for solutions of a special Beltrami equation with good properties, which are called β-analytic functions (see [26,27]) and were firstly proposed by Tungatarov [26]. Afterwards, some variations of these BVPs and related problems aroused a great deal of interest in some fields such as harmonic analysis, infinitesimal geometry, and fractal theory [27][28][29][30][31][32][33]. More recently, Kats made a deep study on jump Riemann BVPs for a wider class on non-rectifiable curves in [30,32].…”

Section: Introductionmentioning

confidence: 99%

“…Afterwards, some variations of these BVPs and related problems aroused a great deal of interest in some fields such as harmonic analysis, infinitesimal geometry, and fractal theory [27][28][29][30][31][32][33]. More recently, Kats made a deep study on jump Riemann BVPs for a wider class on non-rectifiable curves in [30,32]. They gave explicit solutions and conditions of solvability by introducing integration over non-rectifiable paths and establishing the corresponding Cauchy-Pompeiu formula.…”

Section: Introductionmentioning

confidence: 99%

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Riemann–Hilbert problems with shift on the Lyapunov curve for null-solutions of iterated Beltrami equations

et al. 2019

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In this article, we study some Riemann-Hilbert problems with shift on the Lyapunov curve for generalized polyanalytic functions, which are null-solutions of a class of iterated Beltrami equations. Firstly, we obtain two integral representations of these functions by using Cauchy formula associated with the Beltrami equations and explicitly constructing two weakly singular kernels. Next, we develop a theory of matrix factorization for triangular matrix functions with shift in the frame of Beltrami equations and solve the Riemann-Hilbert problems by using them and the decomposition theorem for the iterated Beltrami equations. Finally, we get explicit formulae of solutions and conditions of solvability for the Riemann-Hilbert problems.Throughout this paper, we always assume that a(s) ∈ H μ (L), where a(s) is the angle between the tangent line at t = t(s) on L and the positive real axis. Let Ω be an open set such that L ⊂ Ω, and f be a fixed analytic function in a domain satisfying inf t∈L |f (t)| > 0 and Ω ⊂ . Denote Z to be the set of all zeros of f and f . A complex-valued function

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