To the theory of semilinear equations in the plane

Gutlyanskiî, Vladimir; Nesmelova, Olga; Ryazanov, Vladimir

doi:10.1007/s10958-019-04519-z

Cited by 12 publications

(10 citation statements)

References 38 publications

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“…Note also, by the way, the connection of H with the known Pompeiu integral operator (see, e.g., relation (2.21) in [8]), that gives its representation in the explicit form…”

Section: Mathematicsmentioning

confidence: 99%

Hilbert problem with measurable data for semilinear equations of the Vekua type

Gutlyanskiî

Nesmelova

Ryazanov

et al. 2022

Dopov. Nac. akad. nauk Ukr.

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We prove the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the nonlinear equations of the Vekua’s type ∂z f (z ) = h (z )q(f (z )). The found solutions differ from the classical ones, because our approach is based on the notion of boundary values in the sense of angular limits along nontangential paths. The results obtained can be applied to the establishment of existence theorems for the Poincaré and Neumann boundary-value problems for the nonlinear Poisson equations of the form ΔU (z )=H (z )Q(U (z )) with arbitrary measurable boundary data with respect to the logarithmic capacity. They can be also applied to the study of some semilinear equations of mathematical physics modeling such processes as the diffusion with absorption, plasma states, stationary burning etc. in anisotropic and inhomogeneous media.

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“…Note also, by the way, the connection of H with the known Pompeiu integral operator (see, e.g., relation (2.21) in [8]), that gives its representation in the explicit form…”

Section: Mathematicsmentioning

confidence: 99%

Hilbert problem with measurable data for semilinear equations of the Vekua type

Gutlyanskiî

Nesmelova

Ryazanov

et al. 2022

Dopov. Nac. akad. nauk Ukr.

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“…Following this way, we proved Theorem 7 in [2] on the solvability of the Dirichlet problem for the Poisson equations U G Δ = with sources in classes ,…”

Section: To the Dirichlet Problem With Arbitrary Measurable (Over Log...mentioning

confidence: 89%

“…With a view to further developments of the theory of boundary-value problems for semilinear equations, the present paper is devoted to the Dirichlet problem for quasilinear Poisson equations in Jordan domains with arbitrary boundary data that are measurable over the logarithmic capacity, see, e.g., [2] for its definition and properties.…”

Section: To the Dirichlet Problem With Arbitrary Measurable (Over Log...mentioning

confidence: 99%

“…where M is a constant depending only on S, and, in particular, the restriction of G N to  is a completely continuous bounded linear operator (see, e.g., Lemma 2 in [2] or Theorem 1 in [3]).…”

Section: Definitions and Preliminary Remarksmentioning

confidence: 99%

“…Рязанов 1,3 , https://orcid.org/0000-0002-4503-4939 А.С. Єфімушкін 1 1 Інститут прикладної математики і механіки НАН України, Слов'янськ 2 Донбаський державний педагогічний університет, Слов'янськ 3 Черкаський національний університет ім. Богдана Хмельницького E-mail: vgutlyanskii@gmail.com, star-o@ukr.net, vl.ryazanov1@gmail.com, a.yefimushkin@gmail.com ЗАДАЧА ДІРІХЛЕ З ВИМІРНИМИ ДАНИМИ ДЛЯ НАПІВЛІНІЙНИХ РІВНЯНЬ НА ПЛОЩИНІ Вивчення задачі Діріхле з довільними вимірними даними для гармонічних функцій в одиничному колі  сходить до відомої дисертації Лузіна.…”

Section: This Work Was Partially Supported By Grants Of Ministry Of E...unclassified

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Dirichlet problem with measurable data for semilinear equations in the plane

Gutlyanskiî

Nesmelova

Ryazanov

et al. 2022

Dopov. Nac. akad. nauk Ukr.

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The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk  goes to the known dissertation of Luzin. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. With a view to further developments of the theory of boundary-value problems for semilinear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is firstly constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations ΔU =G over the sources G ∈Lp, p >1. The latter makes it possible to apply the Leray— Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form ΔU (z ) =H (z )⋅Q(U (z )) for multipliers H ∈Lp with p >1 and continuous functions Q: R→R with Q(t ) /t →0 as t →∞. These results can be applied to some specific quasilinear equations of mathematical physics, arising under a modeling of various physical processes such as the diffusion with absorption, plasma states, stationary burning, etc. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

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