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

Abstract

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.

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

“…the function Q, and the domain D. Finally, arguing similarly to the last item in the proof of Theorem 2 in[3], we show that (18) implies (16).Remark 3. By the construction in the above proof, the source G G c =   , where c is a conformal mapping of D onto  and : G →    is a fixed point of the nonlinear opera-…”

“…Arguing similarly to the first item in the proof of Theorem 2 in[3], we see that restriction to the boundary of the homeomorphic extension of c to D onto  . clear by the hypotheses of Theorem 2 that H  has compact support in  and belongs to the class ( ) p L  .…”

“…and the Hilbert operator g  from Section 1 in [3], generating generalized analytic functions : f →   with the sources g and the angular limits…”

“…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 ПУАНКАРЕ З ВИМІРНИМИ ДАНИМИ ДЛЯ НАПІВЛІНІЙНИХ РІВНЯНЬ ПУАССОНА НА ПЛОЩИНІ Крайова задача Гільберта належить до найважливіших з огляду на її численні застосування, зокрема, до крайових задач Діріхле, Пуанкаре та Неймана в гідромеханіці.…”