Logarithmic Potential and Generalized Analytic Functions

Dopov. Nac. akad. nauk Ukr.

Nesmelova

Ryazanov

et al. 2022

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.

mentioning

confidence: 60%

“…In paper [1], we considered generalized analytic functions f with sources , >2, p g L p  that have generalized first derivatives by Sobolev and satisfy the equation…”

Section: Mathematicsmentioning

confidence: 99%

“…For this purpose, let us analyze the construction of solutions of Eq. ( 1) with the Hilbert boundary condition (2) from the proof of Theorem 1 in [1].…”

Section: Mathematicsmentioning

confidence: 99%

Section: On the Hilbert Problem For Semilinear Equations In A Unit Diskmentioning

confidence: 99%

See 2 more Smart Citations

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

Dopov. Nac. akad. nauk Ukr.

Nesmelova

Ryazanov

et al. 2022

“…On completely c ontinuous Poincaré operators. In Section 7 of [1], we considered the Poincaré boundary-value problem in terms of directional derivatives for the Poisson equations…”

mentioning

confidence: 99%

Poincaré problem with measurable data for semilinear Poisson equation in the plane

Dopov. Nac. akad. nauk Ukr.

Nesmelova

Ryazanov

et al. 2022

We study the Poincaré boundary-value problem with measurable in terms of the logarithmic capacity boundary data for semilinear Poisson equations defined either in the unit disk or in Jordan domains with quasihyperbolic boundary condition. The solvability theorems as well as their applications to some semilinear equations, modelling diffusion with absorption, plasma states and stationary burning, are given.

On the Hilbert problem for semi-linear Beltrami equations

Ryazanov²,

Nesmelova³

et al. 2023

J Math Sci

The presented paper is devoted to the study of the well-known Hilbert boundary-value problem for semi-linear Beltrami equations with arbitrary boundary data that are measurable with respect to logarithmic capacity. Namely, we prove here the corresponding results on the existence, regularity, and representation of its nonclassical solutions with a geometric interpretation of boundary values as the angular (along the nontangential paths) limits in comparison with the classical approach in PDE. For this purpose, we apply completely continuous operators by Ahlfors-Bers, first of all to obtain solutions of semi-linear Beltrami equations, generally speaking with no boundary conditions, and then to derive their representation through the solutions of the Vekua-type equations and the so-called generalized analytic functions with sources. Besides, we obtain similar results for nonclassical solutions of the Poincaré boundary-value problem on directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to semi-linear equations of the Poisson type. The obtained results are applied to some problems of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states, and stationary burning in anisotropic and inhomogeneous media.