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                            1 approximation of functions by solutions of second-order elliptic systems on compact sets in ℝ2

Bagapsh, A. O.; Fedorovskiy, Konstantin Yurievich

doi:10.1134/s0081543817060037

Cited by 11 publications

(5 citation statements)

References 10 publications

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“…Take a function h ∈ C(Γ ). We are going to obtain the function f of the form (5) such that Lf = 0 and f | Γ = h. To do this we substitute the expansion (5) into the equation Lf = 0 and equate to zero the factors f n for the same powers of the parameter τ . Moreover, we put f 0 | Γ = h and f n | Γ = 0 for n ≥ 1.…”

Section: Perturbation Methodsmentioning

confidence: 99%

“…Suppose Ω and h are such that h • ω ∈ C α (T) for some 1/2 < α < 1. Then for every τ ∈ [0, 1) the series (5), where f n = F n • ω −1 , n ≥ 0, and where F n are defined by (16), converges uniformly on Ω to the function f ∈ C(Ω) that satisfies the equation Lf = 0 in Ω and coincides with h on Γ .…”

Section: Perturbation Methodsmentioning

confidence: 99%

“…The paper has the following structure. In Section 2 we explain the aforementioned method of solutions constructing in the form of representations (5). Further, in Section 3 we prove the convergence of the series (5) under assumptions to h and Ω mentioned above.…”

Section: Problemmentioning

confidence: 99%

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The perturbation method for the skew-symmetric strongly elliptic systems of PDEs

Bagapsh¹

2021

Preprint

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For a Jordan domain with sufficiently smooth boundaries, the solution to the Dirichlet problem for second order skew-symmetric strongly elliptic system with constant coefficients and regular enough boundary data is constructed in the form of a power series of a small parameter describing the perturbation of the given system from the Laplace one. The coefficients of this series are the functions that are determined sequentially as solutions to special Dirichlet problems for the usual Laplace and Poisson equations. The obtained series converges uniformly in the closure of the domain under consideration.

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Section: Perturbation Methodsmentioning

confidence: 99%

Section: Perturbation Methodsmentioning

confidence: 99%

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The perturbation method for the skew-symmetric strongly elliptic systems of PDEs

Bagapsh¹

2021

Preprint

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“…for a complex-valued function 𝑔 of a complex variable 𝑧 = 𝑥 + 𝑖𝑦 with only two parameters 𝜏 ∈ [0, 1) and 𝜎 ∈ [0, 1) ∪ (1, ∞], see [4], [5].…”

Section: Introductionmentioning

confidence: 99%

Perturbation method for strongly elliptic second order systems with constant coefficients

Bagapsh

2023

Ufimsk. Mat. Zh.

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We consider a classical Dirichlet problem for a strongly elliptic second order system with constant coefficients in Jordan domains in the plane. We show that the solution of the problem can be represented as a functional series in the powers of the parameter governing the deviation of the operator of the system from the Laplacian. This series converges uniformly in the closure of the domain under the assumption that the boundary of the domain and the given boundary function satisfy sufficient regularity conditions: the composition of the boundary function with the trace of a conformal mapping of the unit circle on the domain belongs to the Hölder class with the exponent exceeding 1/2.

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“…We mention that most of the facts and statements given in this paragraph are not new. Many of them are known and can be found, for example, in [2] or [5], see also [6]. However, the complex canonical form that will be obtained in the end of this paragraph, first appeared in the works by the authors.…”

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confidence: 99%

On energy functionals for second order elliptic systems with constant coefficients

Bagapsh,

Fedorovskiy

2022

Ufimsk. Mat. Zh.

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We consider the Dirichlet problem for second-order elliptic systems with constant coefficients. We prove that non-separable strongly elliptic systems of this type admit no nonnegative definite energy functionals of the formwhere 𝐷 is the domain in which the problem is considered, 𝛷 is some quadratic form in R 4 and 𝑓 = 𝑢 + 𝑖𝑣 is a function of the complex variable. The proof is based on reducing the considered system to a special (canonical) form when the differential operator defining this system is represented as a perturbation of the Laplace operator with respect to two small real parameters, the canonical parameters of the considered system. In particular, the obtained result show that it is not possible to extend the classical Lebesgue theorem on the regularity of an arbitrary bounded simply connected domain in the complex plane with respect to the Dirichlet problem for harmonic functions to strongly elliptic second order equations with constant complex coefficients of a general form is not possible. This clarifies a number of difficulties arising in this problem, which is quite important for the theory of approximations by analytic functions.

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C 1 approximation of functions by solutions of second-order elliptic systems on compact sets in ℝ2

Cited by 11 publications

References 10 publications

The perturbation method for the skew-symmetric strongly elliptic systems of PDEs

The perturbation method for the skew-symmetric strongly elliptic systems of PDEs

Perturbation method for strongly elliptic second order systems with constant coefficients

On energy functionals for second order elliptic systems with constant coefficients

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